EXTENSION OF EOQ MODEL

Transcription

1 EXTENSION OF EO MOEL WITH EMERGENCY ORERS AN EXPLICIT ENERGY COST CONSIERATIONS A Mater Tei reented to te Faculty of te Graduate Scool at te Univerity of Miouri In Partial Fulfillment Of te Requirement for te egree Mater of Science By KARA BONO r. Wooeung Jang, Tei Suervior MAY 011

2 Te underigned, aointed by te dean of te Graduate Scool, ave examined te tei entitled EXTENSION OF EO MOEL WITH EMERGENCY ORERS AN EXPLICIT ENERGY COST CONSIERATIONS Preented by Kara Bono, a candidate for te degree of Mater of Science, And ereby certify tat, in teir oinion, it i worty of accetance. Profeor Wooeung Jang Profeor Jame Noble Profeor Antonie Stam

3 ACKNOWLEGEMENTS To te entire faculty a well a my friend in te Indutrial Engineering deartment of te Univerity of Miouri, I tank you. If it were not for a few great at and current faculty member including Sally Scwartz and r. Cerry Klein, I would not ave decided to tart or tay in te Indutrial Engineering deartment, let alone continue graduate education in te deartment. Alo, if it were not for a few friendly, uortive, and knowledgeable rofeor like r. Wooeung Jang and r. Jame Noble, I would not ave tayed an extra tree emeter at te Univerity of Miouri to tart and fini ti reearc and to attain a dual Mater of Science in Indutrial Engineering in addition to a Mater of Buine Adminitration. Tank you, r. Wooeung Jang, for encouraging me and uing me in all my endeavor. Your oeration reearc clae required me to not only learn te material, but alo undertand te material and aly it to te real world. Becaue of ti, I coe oeration reearc a my emai in my graduate degree and became fortunate to ave you a my advior. Tank you alo, for allowing me to reearc two different roject troug te Center for Engineering Logitic and itribution under te uerviion of you and r. Jame Noble. I would not ave ad ti reearc toic if it were not for te CELi roject wit Boeing Cororation. I can never tank you enoug. Your guidance trougout my reearc, my degree, and my life a been immeaurable. Tank you, r. Jame Noble, for allowing me to reearc two different roject troug te Center for Engineering Logitic and itribution under te uerviion of you and r. Wooeung Jang. A I ave aid many time, I would not ave ad ti ii P a g e

4 reearc toic if it were not for te CELi roject wit Boeing Cororation. Tank you for your uerviion trougout ti roject, eecially during r. Jang abbatical. I am alo very tankful for te team tat wa organized to ait in ti roject endeavor. Te team included Keley Kotur and Adam Rubemeyer, bot of wic ave now become great friend. Again, tank you for your guidance trougout te roject wit Boeing Cororation and tank you for articiating on my diertation committee. I would alo like to tank a few eole over in te buine college at te Univerity of Miouri. Firt of all, I want to tank rofeor emeritu r. avid Wet my grandfater. Tank you for your encouragement and uort trougout all my life. I definitely would not ave decided to continue my education beyond an undergraduate degree if it were not for you. Maybe omeday, I will be like you and attain a PH; but for now, I will be atified wit a dual Mater degree. Tank you alo for uggeting tat my committee member outide my Indutrial Engineering deartment could be te very knowledgeable and friendly, r. Antonie Stam. Lat, but not leat, I want to tank r. Antonie Stam for gladly acceting te oition of te outide member of my diertation committee. Toug I ave taken but only one cla from r. Antonie Stam, I know tat e i a great rofeor a rofeor tat will go above and beyond for i tudent. Tank you, r. Antonie Stam, for being a member of my diertation committee. I oe I can eak your interet in inventory olicie a muc a you eak mine interet in analytic. iii P a g e

5 TABLE OF CONTENTS ACKNOWLEGEMENTS.. ii LIST OF FIGURES AN TABLES... vi ABSTRACT...x Cater 1. INTROUCTION Background & Motivation Problem ecrition Problem Motivation: Boeing Cororation Effect of Energy on Logitic Value & Contribution 13. LITERATURE REVIEW INVENTORY MOEL Model ecrition 1 3. Model efinition Model Formulation Total Cot for eac Releniment Scenario Objective Function Solution Procedure Firt-Order erivative Condition Second-Order erivative Condition Summary of Solution Condition 55 iv P a g e

6 4. NUMERICAL ANALYSIS & RESULTS Numerical Analyi Parameter Suly Procurement Parameter Production Parameter Tranortation Parameter Energy Parameter Micellaneou Parameter Summary of Model Parameter Numerical Solution Aroac given Uniform itribution Objective Function Otimal Solution Condition Numerical Solution Procedure Comarative Solution for Traditional EO Model Numerical Solution Aroac given Exonential itribution Objective Function Otimal Solution Condition Numerical Solution Procedure Comarative Solution for Traditional EO Model Numerical Analyi & Reult Effect of Model Parameter wit reect to Energy Key Parameter wit reect to Energy Effect of Product Weigt wit reect to Energy Effect of Regular emand wit reect to Energy Effect of Emergency emand wit reect to Energy Summary of Analyi & Reult CONCLUSIONS Concluion Extenion 119 BIBLIOGRAPHY. 11 v P a g e

7 LIST OF FIGURES AN TABLES Figure or Table Page Figure 1.1 Interrelationi of Tranortation eciion in Suly Cain. 7 Figure 1. Interrelationi of Eential eciion in Suly Cain.. 9 Figure 1.3 etailed Interrelationi of Eential eciion in Suly Cain 11 Figure 1.4 etailed Interrelationi of Eential eciion in Suly Cain wit Emai on Inventory Policy eciion 1 Figure 3.1 Releniment Scenario for te Prooed Inventory Policy 5 Table 4.1 Suly Procurement Parameter 59 Table 4. Production Parameter 60 Table 4.3 Tranortation Parameter.. 61 Table 4.4 Energy Parameter. 63 Figure 4.1 Fuel Surcarge Rate wit reect to Fuel Cot 64 Table 4.5 Micellaneou Parameter.. 65 Table 4.6 Model Parameter for Numerical Analyi 66 Table 4.7 Effect of Model Parameter on Cange to Inventory eciion wit reect to Energy Cot for te Prooed Inventory Policy 84 Table 4.8 Effect of Model Parameter on Cange to Total Cot of Eac Logitic Activity wit reect to Energy Cot for te Prooed Policy Table 4.9 Effect of Model Parameter on Cange to Total Tranortation Cot wit reect to Energy Cot for eac Inventory Policy. 88 vi P a g e

8 Table 4.10 Effect of Model Parameter on Cange to Total Cot of all Logitic Activitie wit reect to Energy Cot for eac Inventory Policy. 88 Table 4.11 Effect of Model Parameter on Cange to te Proortion of Total Cot Allocated to eac Logitic Activitie wit reect to Energy Cot.. 90 Table 4.1 Effect of Model Parameter on Cange to te Cot Effectivene of te Prooed Policy in lace of Traditional Policy wit reect to Energy 93 Table 4.13 Parameter Level to Validate Significant Effect of Key Parameter 94 Figure 4. Cange in Exected Total Tranortation Cot er ay wit reect to Energy Cot and Product Weigt for te Exonential itribution.. 96 Figure 4.3 Cange in te Proortion of Total Cot Allocated to Tranortation wit reect to Energy Cot and Product Weigt for te Exonential itribution 97 Figure 4.4 Cange in Safety Stock wit reect to Energy Cot and Product Weigt for te Exonential itribution. 99 Figure 4.5 Cange in te Inventory Cycle Lengt and Cange in te Sceduled Order uantity wit reect to Energy Cot and Product Weigt for te Exonential itribution Figure 4.6 Cange in te Probability of an Emergency emand and Cange in te Probability of an Emergency Order wit reect to Energy Cot and Product Weigt for te Exonential itribution. 101 vii P a g e

9 Figure 4.7 Cange in te Cot Effectivene of te Prooed Inventory Policy at Reducing te Total Tranortation Cot er ay and te Total Cot of all Logitic Activitie er ay wit reect to Energy Cot and Product Weigt for te Exonential itribution. 10 Figure 4.8 Cange in Exected Total Tranortation Cot er ay wit reect to Energy Cot and Regular emand for te Uniform itribution 104 Figure 4.9 Cange in te Sceduled Order uantity wit reect to Energy Cot and Regular emand for te Exonential itribution Figure 4.10 Cange in Safety Stock and Cange in te Probability of an Emergency Order wit reect to Energy Cot and Regular emand for te Exonential itribution Figure 4.11 Cange in te Cot Effectivene of te Prooed Inventory Policy at Reducing te Total Tranortation Cot er ay and te Total Cot of all Logitic Activitie er ay wit reect to Energy Cot and Regular emand for te Exonential itribution 109 Figure 4.1 Cange in Safety Stock wit reect to Energy Cot and Average Emergency emand for te Exonential itribution. 111 Figure 4.13 Cange in te Inventory Cycle and Cange in te Sceduled Order uantity wit reect to Energy Cot and Emergency emand for te Uniform itribution.. 11 Figure 4.14 Cange in te Probability of an Emergency emand and Cange in te Probability of an Emergency Order wit reect to Energy Cot and Average Emergency emand for te Exonential itribution viii P a g e

10 Figure 4.15 Cange in te Cot Effectivene of te Prooed Inventory Policy at Reducing te Total Tranortation Cot er ay and te Total Cot of all Logitic Activitie er ay wit reect to Energy Cot and Average Emergency emand for te Exonential itribution. 115 ix P a g e

11 ABSTRACT A cutomer exectation continue to rie, o too do te cot of roducing and ditributing globally cometitive roduct and ervice tat are in line wit uc demanding exectation. Ti trend include not only te cot of material and labor, but alo te cot of energy to rocure, roduce, and deliver uc roduct and ervice acro te global market. In fact, te rice of ga a nearly quadruled in te lat two decade. Even o, te demand for uc non-renewable energy a well a te fear of it limited availability continue to rie and tu treaten it rice more. Given te trend in energy cot, ti reearc invetigate te effect of energy on logitic deciion by analyzing te effect of energy on an inventory ordering olicy. Te inventory model develoed and analyzed in ti aer i baed on te actual environment at a leading aircraft manufacturer. In articular, te rooed model i alicable for roduction ytem wit contant roduction rate but mall, underlying oibilitie for undeirable circumtance to treaten te intricately lanned roduction cedule. Rater tan ignoring te oibility for undeirable circumtance and ubequently fulfilling any emergency demand wit a more exenive and energy cot enitive emergency order, te rooed model rovide multile cenario to fulfill te emergency demand more cot effectively comared to te traditional EO model. Tee otion include fulfilling te emergency demand from afety tock alone, a combination of afety tock and an emergency order, and latly an emergency order alone if te regularly ceduled order i already in route to te roduction facility. x P a g e

12 Tu, te objective of te inventory model develoed in ti aer i to tructure an inventory olicy under exlicit energy cot conideration wit otimal ize for a ceduled order quantity, afety tock, and inventory cycle lengt tat minimize te total exected cot er unit time for a ytem wit a contant roduction rate but a mall, underlying oibility for undeirable circumtance to treaten te roduction cedule. Wilt varying mot of te model arameter, ti model i comaratively analyzed to te traditional EO model wic atifie te regular demand generated by te roduction ytem wit a regularly ceduled order and ignore te oibility of an undeirable circumtance treatening te intricately lanned roduction cedule. By varying mot of model arameter, te analyi reveal key roduction environment in wic inventory olicie are mot ignificantly affected by cange to energy cot a well a te environment in wic te rooed inventory model i mot cot effective comared to te traditional EO model. Tee environment, a illutrated and dicued in analyi, conit of ig level of at leat one of te following key arameter: te weigt of te roduct, te regular demand of te roduct, or te emergency demand of te roduct. A any one of te tree key factor increae, te cange in many of te inventory deciion or related logitic cot become more ignificant a energy cot cange. Moreover, te cot effectivene of imlementing te rooed inventory model in lace of te traditional EO model become more ignificant a any one of te tree key factor increae a energy cot rie. Terefore, roduction environment wit relatively ig level of at leat one of te tree key factor are articularly recetive to te rooed inventory model and it cot aving. xi P a g e

13 CHAPTER 1 INTROUCTION 1.1 Background & Motivation Indutry cange over te at everal decade ave been driven by everal factor including tecnology advancement and globalization. A a reult, market exectation around te world ave increaed ignificantly. Cutomer exect not only an even greater amount of quality and cutomization, but alo a fater delivery of many if not all roduct and ervice regardle of te lace of origin. Furtermore, tere i a eritent exectation for tee roduct and ervice to be available at a lower rice. Nevertele, te cot of roducing and ditributing globally cometitive roduct and ervice tat are in line wit uc exectation are alo increaing. Ti trend include not only te cot of material and labor, but alo te cot of energy to rocure, roduce, and deliver uc roduct and ervice. In fact, te rice of gaoline a nearly quadruled in te lat two decade. A a reult of tee increaing trend in cot and market exectation, buinee are comelled to manage teir reource and facilitie more efficiently and effectively in order to minimize cot and maintain cometitivene acro te global market. Even o, increaed global awarene of te environmental cot aociated wit conuming non-renewable energy uc a gaoline i forcing buinee to tink again. Rater tan minimizing monetary exene alone, buinee are reearcing for 1 P a g e

14 alternative otion in order to increae energy efficiency, reduce energy conumtion, and imrove teir environmental image. One of te mot romiing area to emloy alternative energy otion for an actual monetary return i tranortation. According to Leinbac and Caineri (007), energy conumtion witin te tranortation ector a increaed nearly 47 ercent in te at two decade comared to 4 ercent in te oter indutrial ector. Given ti increaed conumtion witin tranortation activitie and te increaed cot of energy ource like gaoline, buinee are comelled to conider alternative otion wit regard to energy and tranortation in order to manage teir reource more efficiently and effectively and tu comete in ti environmentally conciou market. For tee reaon, buinee ould exlicitly conider energy cot a it relate to tranortation deciion a well a many oter deciion contingent uon tranortation. Tat i, buinee ould exlicitly conider te cot of energy a it relate to all deciion eential for moving roduct from te ulier to te cutomer. Tee deciion wic are integrated for effective uly cain management include but are not limited to toe aociated wit location, roduction, inventory, and tranortation. Toug ucceful coordination of all te trategic, tactical, and oerational deciion are imortant for effective uly cain management, te focu area for ti reearc i inventory management. More ecifically, ti reearc exlicitly conider energy cot a it relate to inventory deciion including order quantity, afety tock, inventory cycle lengt, and tranortation. P a g e

15 1. Problem ecrition Te ituation under crutiny in ti tudy conit of a ingle art in a ingle tage of a buine uly cain. Ti tage conit of a roduction ytem wic a a number of caracteritic; mot imortantly, te roduction rate of te ytem i motly contant from year to year regardle of any fluctuation to te actual market demand of te roduct. Ti contant roduction rate may be due to any number of reaon, but one genuine oibility i tat te roduction caacity i limited. So, cutomer order wic are roceed very early and delivered to cutomer at a muc later date can be ceduled for roduction far in advance. Given uc a ytem wit a contant roduction rate and a trict roduction cedule, te demand for a art required by te ytem may be effortlely aumed. Tat i, ince te roduction rate i contant it may be aumed tat te demand for a art required by te ytem i alo contant. Hence, te inventory deciion for uc a ytem can follow a imle economic order olicy wit a regularly delivered order quantity and zero afety tock. However, witin almot any indutry tere are undeirable circumtance tat arie and treaten to affect te roduction cedule. Tee circumtance include failure to meet quality tandard, requet for reair, requet for maintenance, and urgent cange to te roduction cedule. Regardle of te exact ituation, if tere i zero afety tock, a notably exenive emergency order from a ulier or ditributor to te roduction ytem i required to atify te unexected emergency demand and to maintain te igly intricate roduction cedule. 3 P a g e

16 In te ituation under invetigation, tere i a mall robability of an undeirable circumtance to arie and generate unexected emergency demand. Nevertele, tere are multile otion available to atify te emergency demand witout backlogging or toing te roduction cedule. Tee otion include carrying afety tock, increaing te ize of te regularly ceduled order from te ulier, and lacing a more exenive emergency order from te ulier or oter comarable ditributor. Te ig cot of uc an emergency order i rimarily aociated wit iger andling cot, iger fuel cot, and greater energy conumtion for fater delivery via fater, le energy efficient tranortation mode uc a airlane. Given te aforementioned rie in energy cot, te cot of tee emergency order are exected to increae ignificantly. Terefore, it i neceary to exlicitly conider te cot of energy a it affect tranortation cot a well a oter cot including toe aociated wit rocurement and roduction wen etabliing inventory releniment olicie. Given te detailed roblem ituation above, te objective of ti reearc i to determine te otimal ize for te ceduled order quantity and te afety tock tat minimize te total cot of te ytem wit reect to rocurement, tranortation, inventory, and articularly energy cot of a ingle art in ti roduction ytem. In oter word, ti tudy exlicitly conider energy cot to determine te otimal inventory olicy tat minimize all aociated cot for a ingle art in a roduction ytem wit a contant roduction rate but mall robability for an undeirable circumtance to arie and generate unexected emergency demand. 4 P a g e

17 1.3 Problem Motivation: Boeing Cororation Te roblem ituation, inventory model and numerical analyi in te ubequent ection are motivated by actual callenge faced by te Boeing Cororation. Wit oeration acro 49 tate and in 70 countrie a well a cutomer in more tan 90 countrie, Boeing i callenged to oerate effectively and efficiently trougout it entire uly cain in order to minimize it cot and imrove it cometitivene. Ti callenge may become even more difficult a te cot of energy increae. Given te riing cot of ga, eecially in 008 wen ga rice exceeded four dollar a gallon, manager at Boeing quetioned te effect of increaing energy cot on many of it logitic deciion. Wit uc a large cale logitic network, it wa quetioned weter or not uc deciion like inventory ordering olicie are in fact ignificantly imacted by cange to energy cot, even if uc cange were mall. A a reult of ti luring callenge, a grou of individual at Boeing Cororation aroaced te Center for Engineering Logitic and itribution at te Univerity of Miouri wit te tak of inveting te effect of energy cot on many logitic deciion, but more ecifically on te inventory ordering olicy at Boeing Cororation. Altoug te actual inventory ordering olicy currently emloyed at Boeing i not clear, te inventory model develoed trougout ti reearc i baed at leat on te roduction environment at Boeing a well a oter aircraft manufacturer. Te roduction environment, like tat in te roblem decrition, i differentiated wit fairly contant roduction rate a caracteritic tat i quite common in te aircraft indutry. In fact, te aircraft indutry i rone to a very limited manufacturing caacity. Order for aircraft laced by cutomer of comanie like Boeing are roceed very early and 5 P a g e

18 delivered at a muc later date. Conequently, te roduction cedule are lanned far in advance. So weter or not te actual cutomer demand fluctuate from year to year, te roduction rate i fairly contant. Given uc a ytem, te demand for a art required for roduction may be frequently aumed a contant. However, undeirable circumtance do arie and treaten to imact te roduction cedule. Tee circumtance include failure to meet quality tandard, requet for reair, requet for maintenance, and urgent cange to te roduction cedule. Regardle of te exact ituation, if tere i no afety tock for te art in trouble, a notably exenive emergency iment of art from te ulier or a comarable ditributor i required to atify te unexected emergency demand for tat art and to maintain te igly intricate roduction cedule. In te ituation under invetigation at Boeing, tere i a mall robability of an undeirable circumtance to arie and generate unexected emergency demand. So, for many art, tere may be virtually no afety tock. Wen uc a circumtance arie, an emergency iment of te art in trouble i required for a notably iger cot tan tat of a regularly ceduled iment of te art. Since ti iger cot i rimarily aociated wit iger andling cot, iger fuel cot, and greater energy conumtion for a fater deliver by way of a fater, le energy efficient tranortation mode uc a an airlane, tee cot are exected to increae ignificantly a energy cot rie. Ti tudy comare te exected cot of a traditional economic ordering olicy witout afety tock wit an inventory model deigned to determine otimal inventory deciion wit reect to rocurement, tranortation, inventory, and articularly energy cot of a ingle art in a imilar roduction environment a decribed rior. 6 P a g e

19 1.4 Effect of Energy on Logitic Toug tere i little reearc tat exlicitly conider or mention energy cot a it relate to inventory deciion, tere i evidence tat energy cot a well a energy conumtion i related to many tranortation deciion relevant to uly cain. Tee deciion include but are not limited to mode of tranortation, tye of carrier, degree of conolidation, veicle route, and veicle load lan. Te diagram in Figure 1.1 ow ome of interrelationi between te aforementioned tranortation deciion. Figure 1.1: Interrelationi of Tranortation eciion in Suly Cain Te cot aociated to any of tee tranortation deciion illutrated in te diagram above include te cot to oerate a veicle or fleet and te cot to andle individual ackage tat are tranorted from an origin to a detination. One of te rimary oerating cot aociated to many of tee tranortation deciion i te cot of fuel conumtion. A fuel conumtion increae, o doe te cot of tranortation. Fuel conumtion can be reduced in multile way, but one frequently reearced aroac to reducing fuel conumtion aociated wit tranortation i ifting te 7 P a g e

20 mode of tranortation. Eac mode of tranortation uc a water, rail, road, and air vary by energy efficiency wic i directly related to te cot-effectivene of a mode. A barge for intance, i one of te mot energy efficient and tu cot effective mode of tranortation. Wit a fraction of te fuel conumed by oter mode, a ingle barge can tranort te ame quantity of material a 15 railcar or 80 truck (Mury 009). Furtermore, McKinnon cite tat te energy conumtion of road tranortation i 4.3 time iger tan tat of rail and 6.8 time iger tan tat of water (1999). So, te energy efficiency and te cot effectivene of a tranortation mode are trongly related to otimizing te tranortation mode deciion. Similar to te energy efficiency of a tranortation mode, te tranortation ditance i directly related to fuel conumtion. A tranortation ditance increae, fuel conumtion and tu tranortation cot increae; and different tranortation mode can become more cot effective. So, tranortation deciion related to ditance including veicle routing or even ackage conolidation can ignificantly affect te reultant tranortation cot. Converely, cange in energy cot can affect te degree of conolidation or oter tranortation deciion tat aim to minimize cot. Toug energy cot and energy conumtion may not directly and exlicitly affect every tranortation deciion, ince te deciion are interrelated a illutrated in Figure 1.1, all te tranortation deciion in ome way are affected by energy cot and energy conumtion. Likewie, ince te eential function in te logitic ytem are interrelated wit eac oter a well a tranortation, all te deciion revalent in te logitic ytem are affected exlicitly or imlicitly by tranortation deciion. Te interrelationi between tee eential function are illutrated in Figure P a g e

21 Figure 1.: Interrelationi of Eential eciion in Suly Cain Te relationi diagram in Figure 1. illutrate te interrelationi between all te eential deciion and function in te logitic ytem. A illutrated by te diagram, tranortation deciion are exlicitly related to te deciion revalent in trategic lanning, yical network organization, rocurement and uly management, roduction, and wareouing. Given tat tee deciion area are exlicitly related to tranortation deciion, and te cot of energy affect all tranortation deciion in ome way, it can be concluded tat tee five deciion area are alo affected to ome degree by te cot of energy. Altoug te remaining two deciion area inventory management and material andling are not exlicitly related to tranortation deciion, it can be reaoned tat 9 P a g e

22 tee deciion, and tu all deciion revalent to te logitic ytem, are alo affected to ome degree by te cot of energy. Suc reaoning i oible given te comlex ytem of interrelationi revalent in te logitic ytem and illutrated in Figure 1.. So, even toug tranortation i not directly linked to inventory management or material andling in te diagram, tee deciion area are imlicitly related to tranortation and tu ubequently affected by te cot of energy to ome degree troug oter deciion and interrelationi in te ytem. Tee relationi can be indirectly connected troug multile deciion, but it i eay to illutrate and undertand te imlicit relationi given only one degree of earation between te tranortation deciion and te remaining two deciion area. For intance, tranortation deciion can be linked to inventory management deciion troug any of tree deciion area trategic lanning, rocurement and uly management, and roduction. Likewie, tranortation deciion can be linked to material andling deciion troug te deciion area of rocurement and uly management, roduction, or wareouing. Given tat te remaining two deciion area inventory management and material andling are imlicitly related to tranortation deciion, and te cot of energy affect all tranortation deciion in ome way, it can be concluded tat te deciion aociated to inventory management and material andling are alo affected to ome degree by te cot of energy. Furtermore, ince every deciion revalent in te logitic ytem i eiter exlicitly or imlicitly related to tranortation deciion, it can be concluded tat every deciion eential in te logitic ytem are affected to ome degree by te cot of energy. 10 P a g e

23 A more detailed view of te interrelationi between te deciion witin eac of te eential functional area of a logitic ytem i illutrated in Figure 1.3. Ti detailed relationi diagram illutrate more ecifically te relationi between all te deciion revalent witin eac functional area a well a te relationi between all te deciion in te wole logitic ytem. Figure 1.3: etailed Interrelationi of Eential eciion in Suly Cain Following a imilar reaoning for te effect of energy cot on te eential deciion area illutrated in Figure 1., it i reaoned tat all te deciion witin eac functional area of te logitic ytem illutrated in Figure 1.3 are affected to ome degree by te cot of energy. Rater tan exlicitly owing te effect of energy cot on 11 P a g e

24 eac ecific deciion revalent in te logitic ytem illutrated in Figure 1.3, te uroe of ti reearc i to invetigate ti effect troug an examle logitic roblem ituation. Tat i, te objective of ti reearc i to invetigate te effect of energy cot on logitic by invetigating te effect of energy cot on a common logitic roblem te inventory order olicy and te overall cot aociated to deciion eential in te olicy. Te deciion eential to te inventory order olicy a well a te direct relationi between tee deciion and oter deciion in te logitic ytem are emaized in te relationi diagram illutrated in Figure 1.4. Figure 1.4: etailed Interrelationi of Eential eciion in Suly Cain wit Emai on Inventory Policy eciion 1 P a g e

25 1.5 Value & Contribution Te uroe of ti reearc i to invetigate te effect of energy cot on logitic deciion by more ecifically invetigating te effect of cange to energy cot on te deciion and cot related to te inventory ordering olicy. Toug tere i little reearc tat exlicitly conider or mention energy cot a it relate to inventory deciion, tere i evidence tat energy cot a well a energy conumtion i directly related to many tranortation deciion relevant to uly cain. For intance, te energy cot of a tranortation mode increae a te energy efficiency of te mode decreae. Similarly, te energy conumtion and tu te energy roortion of te tranortation cot increae a te tranortation ditance increae. Since te inventory ordering olicy i related to rocurement, inventory, and tranortation cot, it i teorized tat inventory deciion are affected by cange to energy cot and conumtion. Terefore, te inventory model develoed and analyzed in ti aer exlicitly conider te cot of energy in te formulation of te inventory model. Suc a formulation of an inventory model i contrary to many if not all te current reearc on inventory ordering olicie. Toug ome inventory model conider different tranortation mode or emergency iment, tere i little to no reearc tat exlicitly conider te cot of energy in tranortation or oter logitic cot. Alo, unlike muc of te current reearc, te inventory model develoed and analyzed in ti aer i baed on a imle economic ordering olicy even toug te demand by te roduction ytem i not entirely contant. Tat i, wile te roduction ytem may ave a fairly contant roduction rate, tere i a mall, underlying oibility for undeirable circumtance to treaten te rigidly et roduction cedule. Rater tan 13 P a g e

26 ignoring te oibility for undeirable circumtance and ubequently fulfilling any emergency demand wit a more exenive and energy cot enitive emergency order from a ulier, te rooed model rovide multile cenario to fulfill te emergency demand more cot effectively. Tee otion include fulfilling te emergency demand from afety tock alone, a combination of afety tock and an emergency order, and latly an emergency order alone if te ceduled order i already in route to te lant location. Given tee multile otion to releni te emergency demand more cot effectively, te inventory model develoed later in ti aer determine an otimal inventory ordering olicy imilarly to an economic inventory ordering olicy but wit afety tock in addition to a ceduled order quantity. Beide develoing a unique inventory olicy tat exlicitly conider energy cot and otimize te inventory deciion for a ytem wit a contant roduction rate but a mall, underlying oibility for emergency demand, te uroe of ti reearc i to dicover and undertand te roduction environment in wic inventory olicie are mot ignificantly affected by cange to energy cot a well a te environment in wic te rooed inventory model i mot cot effective. Provided a et of factor caracteritic of tee roduction environment, guideline can be develoed to direct buinee to manage teir inventory a well a oter reource more efficiently and effectively a energy cot and conumtion rie. Suc guideline are exected to be eecially beneficial for buinee wit extenive energy uage or logitic ytem. A i reented later in ti aer, te reearc analyi reveal tree factor wic are ignificant to te effect of energy cot on inventory deciion and related logitic cot. A any of te tree factor increae, te cange in many of te inventory 14 P a g e

27 deciion and related logitic cot become more ignificant wit reect to cange in energy cot. Moreover, te cot effectivene of imlementing te rooed inventory olicy in lace of imle economic ordering olicy become more ignificant a any one of te tree key factor increae wit reect to energy cot. Terefore, roduction environment wit relatively ig level of at leat one of te tree key factor are articularly recetive to te rooed inventory model and it cot aving. Before te reult are reented, a review of te current and related reearc on inventory model i dicued in Cater. Ten, te rooed model i decribed, defined, and formulated in Cater 3. Provided te olution condition alo formulated in Cater 3, rocedure for numerically olving te comlex inventory model are develoed in Cater 4. Subequently, te model i analyzed and comared to te traditional economic order quantity model wit reect to cange in everal model arameter in Cater 4 in order to develo guideline for buinee at te end of Cater 4 and in Cater 5. Latly, te concluion of te reearc and te analyi are reented in Cater 5 along wit future extenion to ti reearc. 15 P a g e

28 CHAPTER LITERATURE REVIEW Many reearcer tudy inventory ytem in wic tere i more tan one uly otion. Te objective of mot if not all te at reearc in ti area i to minimize te total exected cot er unit time, er inventory cycle, er year, or for te finite orizon. Te rimary difference between te model are te releniment olicie and te aumtion. Muc of te at inventory literature i baed on te claic economic order quantity model introduced by F.W. Harri (Giani et. al, 004). Te claic EO model aume intantaneou uly and determinitic demand. Given tat demand varie and uly i not alway intantaneou, te EO a been extended to te common (, R) order olicy wic determine te economic order quantity and reorder oint. Suc an order olicy i common in inventory ytem wit continuou review olicie. Tere ave been everal model develoed witin te cla of continuou review inventory olicie tat allow for emergency order. Wite and Wite (199) comare an extenion of te (r,) model tat allow emergency ordering and afety tock in addition to te original (r,) model wit no emergency ordering. Moinzade and Namia (1988) analyze an extenion of te (r,) olicy in wic two uly mode wit continuou lead time are available. Intead of determining only one et of ordering olicy arameter, te model develo two et of ordering olicie arameter. Tat i, te uggeted ordering olicy i of te form (r1, r, 1, ) in 16 P a g e

29 wic (r1,1) rereent te otimal order quantity and reorder oint for a regular order and (r,) rereent an otimal order quantity and reorder oint for an emergency order. Te otimal arameter are determined by minimizing te exected cot of rocurement, olding, and ortage. An order of eac tye a an aociated fixed ordering cot and unit rocurement cot. Joanen and Torttenon (1998) analyze a imilar model in wic te lead time of te emergency order i aumed to be muc maller tan te lead time of te regular order. Between releniment, te tate of te ytem i reviewed at certain time oint rater tan continuouly. Te order olicy i tu driven by te tate and te time of te ytem. In order to minimize te inventory cot rate wit tate-deendent emergency order, a tailor made olicy-iteration algoritm i deigned baed on te Markov deciion roce. Reult ow tan an emergency order otion may ave coniderable imact on te average cot for a ingle-item in te ytem. However, ti may not be te cae if ortage cot are ig comared to te emergency order cot. Many more reearc a been done witin te cla of eriodic review inventory olicie tan continuou review inventory olicie tat allow for emergency order. In eriodic review inventory olicie, te inventory level i cecked at regular eriod and order are made to raie te inventory a ecified treold. Fukuda (1964) reent a dynamic inventory roblem in wic tock i delivered by eiter a regular le exenive mode one eriod later or an emergency more exenive mode intantaneouly. In every even eriod, an otimal ordering olicy i determined baed on te current inventory tate and te regular and emergency order-u-to level. Tat i, if te inventory level i below te emergency order-u-to level, an emergency 17 P a g e

30 order i made in order to return to te emergency order-u-to level, and a regular order i made in order to return to te cumulative order-u-to level. If te inventory level, on te oter and, i between te emergency order-u-to level and te cumulative order-u-to level, tan te difference between te current inventory level and te cumulative orderu-to level i ordered. Wrigt (1969) tudie an extenion to ti model for bot a ingleroduct and a multi-roduct inventory ytem in wic tere exit a caacity on te ize of te emergency order. Te modeled develoed by Vlaco and Tagara (001) alo include a caacity on te ize of te emergency order in addition to analyzing te otion of lacing an emergency order early or late in te eriod. Unlike ome model, bot te model reented by Vlaco and Tagara (001) aume te fixed order cot for bot order tye are negligible and te variable cot of te regular order i negligible. Ti aumtion i true if te regular order i art of a larger order wit more roduct tat are delivered at every eriod regardle of a different ordering olicy. Tee aumtion can alo be made on tanding order wic are delivered wit te ame quantity in every eriod. A ecial cae of te multily-uly mode inventory ytem i one in wic a tanding order i deliver in every eriod. Roenine and Obee (1976) invetigate uc a ytem under a eriodic review inventory ytem. At eac review eriod, if te inventory fall below a known emergency tock level, an emergency order i lace and delivered intantaneouly to increae te inventory level to te otimal treold. Te dynamic inventory model determine te otimal emergency order quantity and tanding order quantity to minimize te exected total cot of te ytem. Unlike at reearc, if te total inventory exceed te maximum inventory level, te exce inventory i old. 18 P a g e

31 In general, analyi by Roenine and Obee (1976) ow tat te unit enalty cot on te urcaing rice of an emergency order decreae, o doe te lead-time at wic te tanding order ytem become more economical. Ti model wa extended by Ciang (007) wo doe not aume a minimum and maximum inventory level i known. Intead, te dynamic inventory model derived by Ciang (007) determine te dioe-down-to level and order-o-to level. If inventory in a review eriod i lower tan te order-u-to level, an emergency order i laced to raie te inventory to ti level intantaneouly; and if te inventory at a review eriod i iger tan te dioe-downto level, inventory i old down to ti level. Te model i olved for eiter te averagecot or dicounted-cot criterion a well a te backlogged or lot-ale roblem. Rater tan aving an otion for emergency order, ome reearc conider te otion of exediting an outtanding an order. Ciang (00) rooe two continuoureview ingle-item order olicie in wic exediting i allowed eiter by a certain treold time oint or witin a te lead time of an outtanding ordering. Te firt olicy extend anoter model imilar to te (r,) wic added a tird otimal ordering arameter order exediting level two te original two otimal arameter for te regular order order quantity and reorder oint. Te firt olicy extend ti model by enuring tat te exedited order doe not arrive after te regular order by auming tat te lat time oint at wic an exediting deciion can be made i te elaed time after a regular order i le tan (not equal to) te difference between te two order lead time. Te model of te firt olicy tu determine te otimal regular order quantity, reorder oint, exediting ordering oint, and treold time oint. 19 P a g e

32 Te econd olicy rooed by Ciang (00) aume te lead time of te order conit of two comonent: a manufacturing lead time and a delivery lead time. Wile te manufacturing lead time i aumed to be determinitic, te delivery eriod can be a variable interval. Te model determine te otimal order quantity, reorder oint, and exediting level by minimizing te total cot er unit time wit reect to a ervice level contraint. uran et. al (004) extend te econd olicy rooed by Ciang (00) wic aume te lead time of an order conit of two comonent. In ti model, if te inventory i below te exediting level at te end of te manufacturing lead time, te order i exedited; oterwie, te order i not exedited. 0 P a g e

33 CHAPTER 3 INVENTORY MOEL 3.1 Model ecrition Te inventory model develoed in ti cater i motivated by te actual environment at Boeing Cororation. Like many oter buinee in te aircraft indutry, Boeing a a limited manufacturing caacity. Order from cutomer are roceed very early and delivered at a muc later date. A a reult, te roduction cedule are lanned far in advance, and te roduction rate i fairly contant from year to year regardle of weter or not actual cutomer demand fluctuate. Given an environment in wic te roduction rate i fairly contant, it may be frequently aumed tat te demand for a art required for roduction i alo contant. Tu, te inventory olicy for art required by te roduction ytem can follow a traditional economic order olicy wit a regularly delivered order from te ulier or ditributor and zero afety tock. However, in ti roduction environment, tere i a oibility for undeirable circumtance to arie and treaten te intricately lanned roduction cedule. Tee circumtance include failure to meet quality tandard, requet for reair, requet for maintenance, and urgent cange to te roduction cedule. Regardle of te exact ituation, if tere i no afety tock for te art, a notably exenive emergency order of te art from a ulier or ditributor i required to atify te unexected emergency demand for tat art and to maintain te igly intricate roduction cedule. 1 P a g e

34 Te ig cot of uc an emergency order i rimarily aociated wit iger andling and fuel cot a well a greater energy conumtion for a fater delivery via a fater, le energy efficient tranortation mode uc a an airlane. Given te rie in energy cot, te cot of tee emergency order from ulier or ditributor are exected to increae ignificantly. Terefore, te inventory control olicie in environment imilar to te one decribed become muc more critical. So, it i neceary to exlicitly conider te cot of energy a it affect te rocurement cot, tranortation cot, and inventory cot to determine a cot effective inventory olicy. Baed on te decribed roduction environment, te following model exlicitly conider te cot of energy to determine an otimal inventory olicy of a ingle item wit a oibility of emergency demand in addition to a fairly contant regular demand. Given tat te regular demand i contant, a ceduled order of ize tat atifie te regular demand during te inventory cycle i delivered from a ulier after a determinitic lead time at te beginning of eac inventory cycle. It i aumed tat te lengt between conecutive ceduled order i identical and equal to T time unit, were te inventory cycle lengt T i alway greater tan te determinitic lead time ( < T). In addition to contant regular demand, tere exit a mall robability tat an undeirable circumtance uc a a failure to meet quality tandard or a requet for art reair will arie and treaten to cange te intricately lanned roduction cedule. Suc a circumtance create an unexected emergency demand for te art required by te roduction ytem. In order to atify ti emergency demand a well a to maintain te igly intricate roduction cedule wit minimum cot, tere exit ome additional inventory quantity for afety tock. P a g e

35 Witout afety tock, a more exenive emergency order from a ulier i required to maintain te roduction cedule and meet cutomer demand wit zero tock-out. Converely, too muc afety tock can lead to unneceary inventory olding cot. In order to balance ti tradeoff, it i aumed tat in addition to te afety tock, an emergency releniment otion i available for certain demand cenario. Regardle of te demand cenario, it i aumed tat eac cycle begin at an identical inventory level equal to in order to atify te exected demand witin eac inventory cycle. Tu, it i aumed tat te afety tock level i full or relenied at te beginning of eac inventory cycle. Te afety tock can be relenied by eiter adjuting te regularly ceduled order or lacing an emergency order from te ulier. Since te regularly ceduled order require a determinitic lead time to be ied from te ulier, te ize of te regularly ceduled order can only be adjuted witin te firt T time of te inventory cycle before it leave te ulier. Terefore, any emergency demand tat occur during te ceduled order lead time after te ceduled order leave te ulier require an emergency order from te ulier or a comarable ditributor to releni te afety tock and revent an imminent tock-out. An emergency order of a art may alo be laced before te ceduled order leave te ulier if te emergency demand exceed te afety tock level. In uc a cenario, an emergency order i neceary to revent a tock-out before te beginning of te next inventory cycle. In any cae, uc an emergency order require negligible lead time and tu i delivered from a ulier immediately after emergency demand occur. Te uroe of ti model i to determine otimal ize for te ceduled order quantity, te afety tock level, and te inventory cycle lengt T tat minimize te 3 P a g e

36 exected total cot er unit time of a roduct wit a robability of tocatic emergency demand between regularly ceduled order. In te model, it i aumed tat tere i roortional robability equal to t of one occurrence of tocatic emergency demand ariing witin a given time eriod of lengt t. Hence, tere i a roortional robability equal to (1 t) of no emergency demand ariing witin a given time eriod of lengt t. Te oibility of more tan one occurrence of emergency demand during any time eriod i not conidered in ti model. Toug it may be oible for multile intance of emergency demand witin a given time eriod, te robability er unit time of a ingle intance of tocatic emergency demand i aumed to be very mall o tat te robability of more tan one intance of emergency demand witin a given time eriod would be negligible. If tere i no emergency demand during an inventory cycle, a regular inventory releniment cenario occur. Witin a regular inventory releniment cenario, te regularly ceduled order quantity will be received by way of an economical ground tranortation after a determinitic lead time at te beginning of te next inventory cycle. If, on te oter and, an intance of emergency demand doe occur during an inventory cycle, one of tree irregular releniment cenario will arie deending on te time oint and te quantity of te emergency demand. Te tree irregular releniment cenario are deicted in te following figure alongide te regular releniment cenario. 4 P a g e

37 Figure 3.1: Releniment Scenario for te Prooed Inventory Policy Te firt irregular releniment cenario deicted in te figure arie wen te emergency demand x occur before te next ceduled order leave te ulier and i le tan or equal to te afety tock level. If te emergency demand i le tan te afety tock level, no inventory tock-out will occur before te next regularly ceduled order i received. Nevertele, te deleted afety tock mut be relenied by te beginning of te next inventory cycle in order to ave an adequate amount of inventory to atify te exected demand in te next inventory cycle. Since te emergency demand occur before te regularly ceduled order leave te ulier in ti cenario, te ize of ti order from te ulier i increaed by an amount equal to te emergency demand x. Tat i, at te beginning of te next inventory cycle, a ceduled order of ize ( x) will be delivered from te ulier via ground tranortation. In te econd irregular releniment cenario, a large intance of emergency demand occur before te next ceduled order i out from te ulier. In uc a cenario, te inventory level will decreae to zero before te beginning of te next 5 P a g e

38 inventory cycle becaue te emergency demand x i greater tan te afety tock level. So, an emergency order from te ulier or a comarable ditributor i required. Yet ince it i aumed tat no oter intance of emergency demand will occur in te current inventory cycle, tere i no reing need to releni te comletely deleted afety tock wit te more exenive emergency order. Intead, te comletely deleted afety tock i relenied by increaing te ize of te next ceduled order to ( ); and te emergency demand tat exceed te afety tock level i relenied by an emergency order of ize (x ) wic i triggered and delivered immediately via air tranortation. Te tird and final irregular releniment cenario deicted in te figure arie wen te emergency demand occur after te next ceduled order leave te ulier. Since it i aumed tat eac inventory cycle tart wit te inventory level ( ) in order to atify te exected demand witin any inventory cycle, an emergency order from a ulier i neceary to releni te deleted afety tock before te beginning of te next inventory cycle. Tu, an emergency order equal to te ize of te emergency demand x i triggered and received immediately after te emergency demand occur via air tranortation tat i aumed to be more exenive tan ground tranortation and ave a negligible lead time. 6 P a g e

39 3. Model efinition Te uroe of te rooed model i to develo an inventory model under exlicit energy cot conideration. Secifically, it i an inventory model wit otimal ize for te ceduled order quantity, te afety tock level, and te inventory cycle lengt T tat minimize te exected total cot er unit time wit reect to rocurement, tranortation, inventory, and energy cot of a art in te roduction tage of a uly cain. Toug te fairly contant demand of te art i atified by regularly ceduled order, tere i a mall robability of tocatic emergency demand x between order. It i aumed tat a regularly ceduled order i delivered at te beginning of eac inventory cycle and eac inventory cycle a an identical lengt of time T. Given te inventory cycle lengt T and a contant regular demand, te ceduled order quantity can be derived by te roduct of te inventory cycle lengt T and te contant regular demand. Ti order quantity, wic i one of te inventory model deciion, only atifie te contant regular demand witin eac inventory cycle. Anoter inventory model deciion, te level of afety tock, i deigned to atify te exected emergency demand at a minimal cot. Emergency demand arie wen an undeirable circumtance uc a a failure to meet quality tandard or a requet for art reair treaten to cange te intricately lanned roduction cedule. It i aumed tat tere exit a mall robability er unit time of a tocatic emergency demand uc tat te roortional robability of a ingle occurrence of tocatic emergency demand witin a given inventory cycle of time eriod of lengt T i T. Hence, te roortional robability of no emergency demand 7 P a g e

40 witin a given inventory cycle i (1 T). Ti i equivalent to te robability for a regular releniment cenario in wic no emergency demand occur. It i aumed tat at mot one occurrence of emergency demand can arie witin an inventory cycle ince te robability of multile occurrence of emergency demand i very mall. In te rooed model, emergency demand may occur before or after te regularly ceduled order leave te ulier. So te robability tat te emergency demand occur after te regularly ceduled order leave te ulier witin te lat time unit of te inventory cycle, a it doe in te tird irregular releniment cenario, i. Converely, te robability tat te emergency demand occur before te regularly ceduled order leave te ulier witin te firt (T ) time unit of te inventory cycle, a it doe in eiter te firt or econd irregular releniment cenario, i (T ). Since te occurrence of eiter te firt or te econd irregular releniment cenario deend on te ize of te emergency demand, te robability of one of te two cenario ariing i te roduct te emergency demand robability (T ) and te robability tat te emergency demand x i le tan or equal to te afety tock level (for te firt irregular releniment cenario) or te emergency demand x i greater tan te afety tock level (for te econd irregular releniment cenario). Te total cot of an inventory cycle deend on te aociated releniment cenario. Te total cot for eac releniment cenario conit of rocurement cot, tranortation cot, inventory cot, and exlicit energy related cot. Te rocurement and tranortation cot conit of a fixed cot comonent k m, a variable cot comonent e m aociated to energy cot, and anoter variable cot comonent c m aociated to everyting but energy cot. Te index m (o, g, a) of eac cot comonent rereent 8 P a g e

41 te origin of te cot. Any roduction or rocurement cot i rereented by te index m = o. So te fixed contracting or ordering cot to rocure te roduct from ulier i rereented by k o ; and te er unit non-energy-related roduction cot to rocure a roduct from te ulier i rereented by c o ; and er unit energy-related roduction cot to rocure a roduct from te ulier i rereented e o. Alternatively, any iing activity cot i rereented by te index m (g, a). Te tranortation and iing cot deend on te tranortation mode and tu te releniment cenario. Te tranortation mode for any regularly ceduled order i aumed to be an economical ground tranortation denoted by te index m = g tat take a determinitic time of lengt to deliver. An emergency order, on te oter and, i aumed to be a more exenive and fater air tranortation denoted by te index m = a tat take a teoretically negligible time to deliver. For eiter tranortation mode m (g, a), tere i a fixed cot comonent and two variable cot comonent tat comoe te total cot of tranortation and iing. In articular, tere i a fixed cot k m to i an order uc tat te fixed cot for ground tranortation i le tan te fixed cot for air tranortation (k g < k a ). Additionally, tere i a er unit cot c m to i one item via a ecified tranortation mode m (g, a) uc tat te unit iing cot via ground tranortation i again le tan te unit iing cot via air tranortation (c g < c a ). Latly, tere i an energy-related cot comonent e m uc a fuel cot to i one unit of roduct via a ecified tranortation mode m (g, a). Similar to te rior tranortation cot comonent, te energy-related iing cot er unit of roduct i le for ground tranortation tan for air (e g < e a ). 9 P a g e

42 Te final comonent of te total cot in eac inventory cycle i te inventory olding cot. Te total inventory olding cot i a function of te demand curve and te olding cot er unit roduct and er unit time. Toug te olding cot i identical for any inventory cycle, te total inventory cot i deendent on te releniment cenario. Indice m cot origin m (o, g, a) Parameter x k m c m e m Regular demand of roduct er cycle time Emergency demand of roduct er cycle time Probability of emergency demand in eac unit time Fixed cot of an order via cot origin m Non-energy related cot of one unit of roduct via cot origin m Energy cot of one unit of roduct via cot origin m Holding cot er unit roduct and er unit time Lead time of ceduled order Integer Variable Order quantity of te ceduled order Safety tock level of roduct Noninteger Variable T Lengt between conecutive cedule order or lengt of te inventory cycle 30 P a g e

43 3.3 Model Formulation Te uroe of te rooed model i to determine otimal ize for te ceduled order quantity, te afety tock level, and te inventory cycle lengt T tat minimize te exected total cot under exlicit energy cot conideration of a art wit a robability of tocatic emergency demand x between regularly ceduled order. In order to determine te exected total cot er unit time of te inventory model, te total cot and robability of eac releniment cenario mut be determined. Given te total cot and te robability of eac releniment cenario, te objective function to minimize te total exected cot can be formulated Total Cot for eac Releniment Scenario Te occurrence of a releniment cenario deend on weter emergency demand occur or not. If tere i no emergency demand during an inventory cycle, a regular inventory releniment cenario occur. Te robability of ti regular inventory releniment cenario in wic no emergency demand occur during te inventory cycle i (1 T). Witin a regular inventory releniment cenario, te regularly ceduled order quantity will be received by an economical ground tranortation after a determinitic lead time at te beginning of te next inventory cycle. Te cot of tee order are deicted in equation (1-). Wile equation (1) deict te cot to rocure te regularly ceduled order, equation () deict te cot to i te regularly ceduled order. Equation (3), on te oter and, deict te inventory olding cot of ti cenario in wic te inventory level decreae at a contant demand rate and te afety tock level 31 P a g e

44 remain contant trougout te inventory cycle. Given tee tree individually deicted cot comonent in equation (1-3), te total cot of te regular releniment cenario can be illutrated in equation (4). o ( e ) k c (1) g o o ( e ) k c () g g T T (3) T ( e c e ) ( ) ko k g co o g g (4) Werea te occurrence of te regular releniment cenario i contingent uon no emergency demand ariing during an inventory cycle, te occurrence of one of te oter tree irregular releniment cenario i deendent uon wen te emergency demand occur witin te inventory cycle and ow muc i demanded at tat time. If te emergency demand occur witin te firt (T ) time unit of te inventory cycle before te regularly ceduled order leave te ulier, eiter te firt or te econd irregular releniment cenario will arie wit a robability of (T ). If, on te oter and, te emergency demand occur witin te lat time unit of te inventory cycle after te regularly ceduled order leave te ulier, te tird irregular releniment cenario will arie wit a robability of. Since te releniment vary in eac of te tree irregular releniment cenario, eac irregular releniment cenario incur a different amount of cot. Te total cot of te firt irregular releniment cenario in wic te emergency demand x occur before te next ceduled order leave te ulier and i le tan or 3 P a g e

45 equal to te afety tock level i illutrated in equation (8). In ti cenario, no inventory tock-out occur before te next order i received ince te emergency demand i le tan te level of afety tock. However, a ortion of te afety tock i deleted and tu require releniment by te beginning of te next inventory cycle in order to atify te next cycle demand. Given tat ti occur before te ceduled order leave te ulier, te ceduled order quantity i increaed by an amount equal to te emergency demand x. In term of cot, te cange in te ceduled order quantity for te firt irregular releniment cenario tranlate to an increae in te cot of te ceduled order by an amount roortional to te emergency demand. Ti increae i incororated in equation (5-6) wic illutrate te cot of rocurement and tranortation for ti cenario. In articular, te cot to rocure te regularly ceduled order i increaed by te unit cot to rocure te emergency demand x a own in equation (5); and te cot to i te regularly ceduled order i increaed by te unit cot to i te emergency demand x a own in equation (6). Contrary to te increae in rocurement and tranortation cot for ti irregular releniment cenario, te inventory olding cot own in equation (7) decreae. In ti cenario a well a te econd irregular releniment cenario, it i aumed tat te emergency demand can occur at any time between te beginning of an inventory cycle and te time at wic te regularly ceduled order i ied. Yet on average, te emergency demand occur alf way between tee two time oint. So imilarly, te time oint at wic te inventory level i reduced by te emergency demand i on average, alf way between te time eriod (T ). Given tat te inventory level i 33 P a g e

46 reduced by am amount equal to te emergency demand for te remaining time eriod, te inventory olding cot i reduced by an amount roortional to te emergency demand x and te remaining time eriod wic i equal to alf te time eriod (T ). Given te rocurement, tranortation, and inventory cot individually deicted in equation (5-7), te total cot of te firt irregular releniment cenario can be illutrated in equation (8). Te robability of incurring te total cot illutrated in equation (8) i equal to te roduct of te following two robabilitie: te firt being te robability (T ) tat an emergency demand occur before te ceduled order leave te ulier and te econd being te robability f(x ) tat an emergency demand i le tan or equal to te afety tock. o ( x)( c e ) k (5) g ( x)( c e ) o g o k (6) g ( T ) T x T (7) k o T ( x)( c e c e ) x 1 kg o o g g T (8) Similarly, in te econd irregular releniment cenario, te robability of incurring te total cot caracteritic of ti cenario in equation (1) i equal to te roduct of te next two robabilitie: te robability (T ) tat an emergency demand occur before te ceduled order leave te ulier and te robability f(x > ) tat an emergency demand i greater tan te afety tock level. In uc a cenario, te inventory level will decreae to zero before te beginning of te next inventory cycle 34 P a g e

47 becaue te emergency demand x i greater tan te afety tock level. So, an emergency order i required to maintain te intricate roduction. Nevertele, ince it i aumed tat no oter intance of emergency demand will occur in te current inventory cycle, tere i no reing need to releni all te tock, ecifically te comletely deleted afety tock, wit te more exenive emergency order before te beginning of te next inventory cycle. Intead, te emergency order from te ulier ould only atify te emergency demand wic exceed te level of afety tock and tu would not be atified oterwie; and te comletely deleted afety tock ould be relenied by adjuting te next ceduled order.. So, in te econd irregular releniment cenario, two earate inventory releniment mut occur. Firt, an emergency order of ize (x ) will be triggered and delivered immediately from te ulier via air tranortation tat i aumed to be more exenive tan ground tranortation and ave a negligible lead time. Second, te next ceduled order tat i delivered from te ulier via ground tranortation will be increaed to te ize ( ) o a to releni te entirely deleted afety tock. Te rocurement and tranortation cot aociated to tee inventory releniment are rereented in equation (9-10), reectively; and te reultant inventory cot of ti cenario i own in equation (11). Te rocurement cot in equation (9) conit of two earately incurred cot. For one, it include te cot to rocure te regularly ceduled order wic i increaed to te ize ( ) o a to releni te deleted afety tock and tart te next inventory cycle wit an identical inventory level of ( ). Secondly, it include te cot to rocure te emergency order of ize (x ) o a to atify te emergency demand tat 35 P a g e

48 exceed te afety tock level. A comared to te rocurement cot in equation (1) of te regular releniment cenario, tee rocurement cot are increaed by a fixed cot to rocure te emergency order a well a te unit cot to rocure an amount of art equal to te emergency demand between te two releniment metod. Like te rocurement cot in equation (9), te tranortation cot in equation (10) conit of two earately incurred cot. Tat i, it include te cot to i te regularly cedule order via ground tranortation a well a te cot to i te emergency order via air tranortation. A comared to te tranortation cot in equation () of te regular releniment cenario, tee tranortation cot are increaed by tree factor: te fixed cot to i te emergency order via te more exenive air tranortation; te unit cot to i te larger regularly ceduled order wic relenie te comletely deleted afety tock wit ground tranortation; and latly, te unit cot to i te emergency order of ize (x ) wit air tranortation. Werea bot te rocurement and tranortation cot are iger in ti cenario a comared to te regular releniment cenario, te reultant inventory cot own in equation (11) i lower in ti cenario a comared to te regular releniment cenario. Ti i true for bot te firt and econd irregular releniment cenario. In fact, te reultant inventory cot for te two cenario are almot identical. Like in te firt irregular releniment cenario, it i aumed tat te emergency demand can occur at any time between te beginning of an inventory cycle and te time at wic te regularly ceduled order i ied from te ulier. Yet on average, te emergency demand occur alf way between tee two time oint at te oint in time equal to alf te time eriod (T ). So, like te firt irregular releniment cenario, 36 P a g e

49 te time oint at wic te inventory level i reduced by te emergency demand i, on average, alf way between te two oint in time. Given tat te inventory level i reduced by an amount equal to te deleted afety tock for te remaining time eriod, te inventory olding cot i reduced by an amount roortional to te deleted afety tock and te remaining time eriod wic i equal to alf te time eriod (T ). Te only difference in inventory cot between te two cenario i tat te reduction in inventory cot of te firt i roortional to te emergency demand x werea te reduction in inventory cot of te econd i roortional to te deleted afety tock. Given te inventory olding cot outlined in equation (11) a well a te rocurement and tranortation cot own earately in equation (9-10), te total cot of te econd releniment cenario can be illutrated in equation (1). o ( x)( c e ) k (9) g a o o ( )( c e ) ( x )( c e ) k k (10) g g ( T ) a a T T (11) T k o k g k a ( x)( co eo) ( )( c g eg) ( x )( ca ea) 1 (1) T Te final cot cenario for te tird irregular releniment cenario incur te total cot illutrated earately in equation (13-15). Te robability for tee cot to incur i equal to, wic i te robability for an emergency demand to occur after te ceduled order leave te ulier. Since te emergency demand occur during te ceduled order lead time, an emergency order equal to te ize of te emergency demand i neceary to begin te next inventory cycle at an identical inventory level. 37 P a g e

50 Tu, a comared to te cot of te regular releniment cenario own in equation (1-3), te cot of ti cenario own in equation (13-15) are only increaed by te rocurement and tranortation cot aociated wit releniing te inventory tat wa reduced by te emergency demand x. Tat i, a comared to te rocurement cot of te regular releniment cenario own in equation (1), te rocurement cot of ti cenario own in equation (13) are increaed by only te additional fixed and unit cot to rocure te emergency order of ize x from te ulier. Similarly, a comared to te tranortation cot of te regular releniment cenario own in equation (), te tranortation cot of ti cenario own in equation (14) are increaed by only te additional fixed and unit cot to i te emergency order of ize x via te more exenive air tranortation metod. Latly, te inventory olding cot own in equation (15) i identical to tat for te regular releniment cenario own in equation (3). In bot cenario, te inventory level decreae at a contant demand rate and te final inventory level i equal to te afety tock level. So te inventory olding cot i roortional to te inventory cycle lengt, te contant demand rate, and te contant afety tock level. Combined wit te rior cot, te total cot of ti tird cenario can be illutrated in equation (16). o ( x)( c e ) k (13) g a o o ( e ) x( c e ) k k c (14) g g a a T T (15) T ko kg ka ( x)( co eo) ( cg eg) x( ca ea) ( ) (16) 38 P a g e

51 39 P a g e 3.3. Objective Function Te objective of te rooed model i to determine te otimal ceduled order quantity and te afety tock level tat minimize te exected total cot er unit time wit reect to rocurement, tranortation, inventory, and energy cot of a art required by a roduction ytem tat a a robability of tocatic emergency demand x, wic follow f(x), between regularly ceduled order from ulier and ditributor. Te exected total cot er unit time of te inventory model own in equation (17) i te aggregated roduct of te total cot and correonding robability of eac releniment cenario divided by te inventory cycle lengt T. ( ) ( ) ( ) = T e c e c k k T T g g o o g o 1 1 ) TC(, (17a) ( )( ) dx x f T x T e c e c x k k T T g g o o g o ) ( 1 1 ) ( 0 (17b) ( )( ) ( )( ) ( )( ) dx x f T T e c x e c e c x k k k T T a a g g o o a g o ) ( 1 1 ) ( (17c) ( )( ) ( ) ( ) ( ) dx x f T e c x e c e c x k k k T a a g g o o a g o ) ( 1 0 (17d)

52 Baed on te robability of eac of te four releniment cenario, te objective function own in equation (17) i earated into eac of te four cenario (a-d). Te total cot er unit time of te cenario illutrated in (17a) i aociated wit te regular releniment cenario in wic no emergency demand occur during te inventory cycle. Te robability of uc a cenario wit no emergency demand witin an inventory cycle i (1 T). Converely, te robability for a ingle occurrence of emergency demand during te entire inventory cycle i T. Given an occurrence of emergency demand during an inventory cycle, one of tree irregular releniment cenario will arie deending on te time oint and te quantity of te emergency demand during te inventory cycle. If te emergency demand occur before te regularly ceduled order leave te ulier witin te firt (T ) time unit of te inventory cycle, eiter te firt or te econd irregular releniment cenario will arie wit a robability of (T ). Te total cot er unit time of te firt irregular releniment cenario i deicted in equation (17b). Te robability of ti cenario i equal to te roduct of te following two robabilitie: te firt being te robability (T ) tat an emergency demand occur before te ceduled order leave te ulier and te econd being te robability f(x ) tat an emergency demand i le tan or equal to te afety tock. If te emergency demand i greater tan te afety tock a it i for econd irregular releniment cenario, te robability of te cenario equal te roduct of te next two robabilitie: te robability (T ) tat an emergency demand occur before te ceduled order leave te ulier and te robability f(x > ) tat an emergency demand 40 P a g e

53 i greater tan te afety tock level. Te total cot er unit time aociated to ti irregular releniment cenario i illutrated in equation (17c). Te final iece of te objective function (17d) i te total cot er unit time of te tird and lat irregular releniment cenario. In ti final cenario, te emergency demand occur after te regularly ceduled order leave te ulier or ditributor witin te lat time unit of te inventory cycle; tu, te robability tat te tird irregular releniment cenario will arie i. Togeter, te equation illutrate te exected total cot er unit time of te inventory model of a roduct wit a robability of tocatic emergency demand x between regularly ceduled order. Given tat in eac cenario a regularly ceduled order quantity i delivered from te ulier at te beginning of eac inventory cycle, te objection function in (17) can be rewritten a following: T ( e c e ) ( ) 1 TC(, ) = ko kg co o g g (18a) T x ( c e c e ) ( T ) ( 1 ) x o o g g f ( x) dx T 0 ko ka x( co eo) (1 ) ( c ) ( )( ) g eg x ca ea f ( x) dx T ( T ) (18b) (18c) o a o o a T 0 [ k k x( c e ) x( c e )] a f ( x) dx (18d) In equation (18), te total cot er unit time of te regularly ceduled order quantity i diconnected from eac of te releniment cenario ince it i incurred 41 P a g e

54 regardle of te cenario. So, equation (18a) deict te rocurement, tranortation, and inventory cot er unit time of te regularly ceduled order quantity wic i incurred during te regular releniment cycle a well a eac of te irregular releniment cycle. Equation (18b-d), on te oter and, deict te additional exected cot er unit time of eac of te irregular releniment cenario. Te additional exected cot er unit time of te firt irregular releniment cenario i deicted in equation (18b). In ti cenario, te emergency demand tat occur before te ceduled order leave te ulier i le tan te afety tock level. So, even toug no inventory tock-out will occur before te next ceduled order i delivered from te ulier, a ortion of te afety tock i deleted and require releniment by te beginning of te next inventory cycle. Tu, te ceduled order quantity i increaed by an amount equal to te emergency demand x in order to releni te inventory level to te target ( ) by te beginning of te next inventory cycle. Te additional exected cot er unit time deicted in (18b) for ti cenario conit of an increae to te rocurement and tranortation cot but a decreae to te inventory cot. In articular, te rocurement and tranortation cot are increaed by te unit cot to rocure and i te emergency demand x by way of ground tranortation. Te inventory cot, on te oter and, decreae by an amount roortional to te emergency demand x and te lengt of time equal to alf te time eriod (T ) remaining after te emergency demand reduce te inventory level. Since te decreae in inventory cot will alway be le tan te increae in te rocurement and tranortation cot, te exected cot er unit time deicted in (18b) for ti cenario will alway be oitive an increae to te total cot. 4 P a g e

55 Te econd irregular releniment cenario, wic i imilar to te firt irregular releniment cenario, incur te additional exected cot er unit time tat are deicted in equation (18c). In ti cenario, te emergency demand tat occur before te ceduled order leave te ulier i greater tan te afety tock level. So, witout an emergency order, te inventory level will decreae to zero before te beginning of te next inventory cycle. Tu, an emergency order of ize (x ) wic atifie only te emergency demand not atified by te afety tock in te current inventory cycle i triggered immediately; and te next ceduled order i increaed by a quantity equal to te comleted deleted afety tock level. Overall, te additional exected cot er unit time illutrated in (18c) for ti cenario i greater tan tat of any oter cenario. For one, rocurement cot are increaed by a fixed cot to rocure te emergency order a well a te unit cot to rocure an amount of art equal to te emergency demand between te two releniment metod. Secondly, te tranortation cot are increaed by tree factor: te fixed cot to i te emergency order via te more exenive air tranortation; te unit cot to i te larger regularly ceduled order wic relenie te comletely deleted afety tock wit ground tranortation; and latly, te unit cot to i te emergency order of ize (x ) wit air tranortation. Even wit te decreae in inventory cot wic like te firt irregular releniment cenario i decreaed by an amount roortional to te deleted afety tock and te lengt of time remaining after te emergency demand delete te afety tock te additional exected cot er unit time for ti cenario will alway be oitive and greater tan tat of any oter cenario. 43 P a g e

56 Te tird and final irregular releniment cenario incur te additional exected total cot er unit time deicted in equation (18d). In ti cenario, te emergency demand occur after te ceduled order leave te ulier o tat witout an emergency order, te inventory level at te beginning of te next inventory cycle will be le tan te targeted ( ) level. Te reultant cot of ti neceary emergency order i additional fixed and variable cot to rocure and tranort an order ize equal to te ize of te emergency demand x via te more exenive but fater mode of air tranortation. Bot te reviou formulation of te objective function in equation (17) and (18) are tructured in a way tat can be logically undertood wit reect to eac releniment cenario. However, tee formulation can and ould be imlified. Yet before doing o, te function x(c g e g ) i added and ubtracted to art (18c-d) o a to not actually cange te reult of te objective function but aid in later imlification. [ k k ( c e c e )] ( ) 1 TC(, ) = o g o o g g (19a) T ( c e c e ) E[x] (19b) o o g g [ k k ( x )( c e c e )] 1 o a a a g g f ( x) dx (19c) T T [( k k ) ( c e c e ) E[x] ] o a a ( ) 1 T xf ( x) dx f ( x) dx T 0 a g g (19d) (19e) Given te revied objective function in equation (19), a few aumtion can be introduced to ait in imlifying te objective function. For one, te inventory cycle lengt T i a function of te regularly ceduled order quantity and te contant 44 P a g e

57 demand. Secifically, te inventory cycle lengt T i equivalent to te quotient of te ceduled order quantity and te contant demand. So, from ti oint on, every intance of T in te objective function i relaced wit te quotient of and. Anoter aroac to imlify te readability of te objective function i to relace arameter tat are ummed multile time wit rereentative ymbol. For intance, ince te um (c a e a c g e g ) aear multile time in te objective function, it i ereby relaced by te ymbol δ. Tu, every time te ymbol δ aear in te objective function, it will rereent te difference between te unit cot of air tranortation and te unit cot of ground tranortation (c a e a c g e g ). Since it i already aumed tat all te tranortation cot via air tranortation are greater tan toe via ground tranortation (c a e a > c g e g ), it can alo be aumed tat δ, wic rereent te oitive difference between te two cot, i greater tan zero (δ > 0). Likewie, te um (k o k g ) i ereby relaced by te ymbol γ; te um (k o k a ) i ereby relaced by te ymbol α; and latly, te um (c o e o c g e g ) i ereby relace by te ymbol β. Given tee relacement, te objective function can be rewritten a te following: TC(, ) = γ β ( ) (0a) βe[x] (0b) 1 [ α δ( x ) ] f ( x) dx (0c) [ α δe[x] ] (0d) 1 xf ( x) dx f ( x) dx 0 (0e) 45 P a g e

58 Equation (0) can ten be imlified into te final form of te objective function wic i own in equation (1): TC(, ) = [ γ ( α δe[ x] )] β( E[ x] ) ( ) (1a) 1 [ α δ( x ) ] f ( x) dx xf ( x) dx f ( x) dx 0 (1b) (1c) 3.4 Solution Procedure Te economically otimal olution to oerating an inventory ytem under contant demand and a robability of tocatic emergency demand i defined by te inventory olicy tat minimize te total cot er unit time. Since te inventory olicy i determined by two deciion variable afety tock and order quantity te otimal olution et contain all value of and uc tat te derivative of te total cot er unit time i equal to zero. Terefore, te otimal deciion and mut atify d TC(, ) = d 0 d and TC(, ) = 0. (-3) d Te otimal olicy mut alo atify te condition for wic te Heian matrix below i oitive-definite. Ti additional condition guarantee tat te olution i te otimal minimum cot and not a maximum cot or ome oter local otimal cot. d TC(, ) d TC(*,*) = d TC(, ) dd d TC(, ) dd d TC(, ) d (4) 46 P a g e

59 47 P a g e Firt-Order erivative Condition Te firt et of otimal condition are derived by etting te firt order derivative of te total cot er unit wit reect to te afety tock level and te ceduled order quantity equal to zero. Te derivative of te total cot er unit time wit reect to te afety tock level i own in equation (5): d d = ) TC(, (5a) ( ) [ ] ) ( ) ( ) ( ) ( ) ( 1 f F F f f δ δ α (5b) [ ] ) ( ) ( ) ( ) ( f F F f (5c) Te derivative in (5) can be rewritten in a imler form a own in equation (6). d d = ) TC(, (6a) ( ) [ ] ) ( 1 ) ( 1 F f δ α (6b) [ ] ) ( 1 F (6c) Tu, one condition te otimal olution mut atify i tat te firt order artial derivative of te total cot er unit value wit reect to te afety tock quantity be equal to zero a own in equation (7). ( ) 0 ) ( 1 ) ( 1 = F f δ α (7)

60 Te econd otimal condition i derived by etting te firt order derivative of te total cot er unit wit reect to te ceduled order quantity equal to zero. Ti firt order derivative wit reect to te ceduled order quantity i own in (8): TC(, ) = [ γ ( δe[ x] α) ] (8a) [ α δ( x ) ] f ( x) dx (8b) 1 xf ( x) dx f ( x) dx 0 (8c) After integrating equation (8), te derivative can be written a te following: TC(, ) = [ γ ( δe[ x] α) ] (9a) δ xf ( x) dx ) ( α δ)( F( ) F( ) 1 xf ( x) dx ) 0 ( F( ) F( ) (9b) (9c) For eaier imlification later, te derivative can be te rewritten a te following: d d TC(, ) = ( γ α α( 1 F( ) )) (30a) δ E[ x] xf ( x) dx δ ) ( 1 F( ) 1 xf ( x) dx ) 0 ( F( ) F( ) (30b) (30c) 48 P a g e

61 Te derivative can ten be imlified to te following: d d TC(, ) = ( γ αf ( ) ) (31a) δ ( ) xf ( x) dx 1 F( ) (31b) 0 1 xf ( x) dx ) 0 ( 1 F( ) (31c) Tu, te econd condition tat te otimal olution et mut atify in wic te firt order artial derivative of te total cot er unit time wit reect to te ceduled order quantity i equal to zero i own in equation (3): [ γ αf ( ) ] ( ) ( 1 ( )) = 0 δ xf x dx F (3) 0 So, te firt et of otimal condition tat are derived by etting te firt order derivative of te total cot er unit time wit reect to te afety tock level and te ceduled order quantity equal to zero are own again in equation (33) and (34): 1 αf ( ) δ = ( 1 F ( )) 0 (33) [ γ αf ( ) ] ( ) ( 1 ( )) = 0 δ xf x dx F (34) Second-Order erivative Condition Te econd et of otimal condition atifie te condition for wic te Heian matrix (35) i oitive-definite. Ti condition guarantee tat te olution i te global otimal minimum cot and not a maximum cot or ome oter local otimal cot. 49 P a g e

62 Tere are at leat two condition tat te Heian matrix mut atify for it to be defined a oitive definite. For one, te matrix mut be ymmetric. Tat i, te comonent in te diagonal of te matrix mut are te ame ign. For a matrix to be oitive-definite, te comonent of te rimary diagonal mut be oitive. Secondly, te determinant of te matrix mut be oitive. Te determinant of te Heian matrix in (35) i illutrated in (36). d TC(, ) d TC(*,*) = d TC(, ) dd d TC(, ) dd d TC(, ) d (35) det ( TC(*,*) ) d = d TC(, ) d d TC(, ) d dd TC(, ) (36) In order to determine weter te Heian matrix i oitive-definite, te econd order artial derivative of te total cot er unit time wit reect to te afety tock level and te ceduled order quantity mut be determined. Te econd order artial derivative of te total cot er unit time wit reect to te afety tock level alone i derived from te equation (33) and own in equation (37): d TC(, ) = 1 αf '( ) δ f ( ) d (37) Te econd order artial derivative of te total cot er unit time wit reect to te ceduled order quantity alone i derived from te equation (34) and own in equation (38). Equation (39) i te imlified form of te econd order artial derivative wit reect to. d TC(,) = [ γ αf ( ) ] ( ) δ xf ( x) dx 1 F( ) (38) d 0 50 P a g e

63 51 P a g e ( ) = ) ( 1 ) ( ) ( TC(,) 0 3 F dx x xf F d d δ α γ (39) Te econd order artial derivative of te total cot er unit time wit reect to firt te afety tock level and ten te ceduled order quantity i derived from te equation (33) and own in equation (40). Ti econd order artial derivative can be imlified to te form in equation (41). ( ) ) ( 1 ) ( ) TC(, F f dd d = δ α (40) ( ) [ ] ( ) ) ( 1 ) ( 1 ) ( ) TC(, F F f dd d = δ α (41) To verify te equation, te econd order artial derivative of te total cot er unit time wit reect to firt te ceduled order quantity and ten te afety tock level i derived from te equation (34) and own in equation (43). Ti ould be equivalent to te econd order artial derivative derived in equation (40) and imlified in equation (41). [ ] [ ] ) ( ) ( 1 ) ( ) ( ) TC(, f F f f dd d = δ α (4) ( ) ( ) ) ( 1 ) ( 1 ) ( ) TC(, F F f dd d = δ α (43) A exected, econd order artial derivative of te total cot er unit time wit reect to te afety tock level and to te ceduled order quantity in equation (41) and (43) are equivalent even toug eac wa derived from a different firt order artial derivative. Tu, te derivation tu far are at leat matematically accurate.

64 5 P a g e Now tat te econd order artial derivative of te total cot er unit time are derived, te Heian matrix can be determined and tu te econd et of olution condition. Te Heian matrix for te current model i own in equation (44). Te determinant of ti Heian i own in equation (45). ( ) ( ) ( ) = ) ( 1 ) ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( ' 1 TC(*,*) 0 3 F dx x xf F F f F f f f δ α γ δ α δ α δ α (44) ( ) = ) ( ) ( ' 1 *) TC(*, det f f δ α (45a) ( ) ) ( 1 ) ( ) ( 0 3 F dx x xf F δ α γ (45b) ( ) ) ( 1 ) ( F f δ α (45c) To atify te condition for wic te Heian matrix in (44) i oitive definite, te matrix mut firt be ymmetric. Tat i, te comonent in te diagonal of te matrix mut are te ame ign. Since te comonent in te bottom-left to to-rigt diagonal are te ame econd-order derivative of te total cot, it can be inferred tat te comonent in ti diagonal are te ame ign. Toug te ign of ti diagonal i inconequential, te ign of te comonent in te to-left to bottom-rigt diagonal mut be oitive in order for te matrix to be oitive definite. Tu, it i neceary to rove tat tee comonent are oitive. Te comonent in te to-left of te matrix i te econd order artial derivative of te total cot er unit time wit reect to te afety tock level alone a own in

65 equation (37). In order to logically determine te ign of ti diagonal comonent, te equation in (37) i earated into everal egment own in equation (46-48). Te firt egment of te equation in te to-left oition of te matrix own in (46) conit of te ratio of contant demand to order quantity. Ti ratio i equivalent to te invere of te inventory cycle lengt. Given te fact tat te ceduled order lead time i alway le tan te lengt of te inventory cycle, te roduct te rior ratio te invere of te inventory cycle lengt and te ceduled order lead time will alway be le tan one. So te firt egment of te to-left matrix comonent i alway oitive. (1 ) (46) αf '( ) (47) δ f ( ) (48) Te oter egment of te to-left matrix comonent own in equation (47-48) are multilied by te firt egment illutrated in equation (46) to obtain equation (37). For te entire comonent to be oitive, te um of te egment in equation (47-48) mut alo be oitive. It i imle to oberve tat te egment illutrated in equation (48) i alway oitive, becaue all te term and te ign in te equation are oitive. However, te egment illutrated in equation (47) cannot be eaily aumed a oitive or negative ince it comrie te derivative of a robability ditribution function. eending on te ditribution for te emergency demand, equation (47) may be oitive or negative. If te derivative of te robability ditribution function i le tan or equal to zero, ten equation (47) i oitive and tu te wole equation i guaranteed to be oitive. However, if te derivative of te robability ditribution 53 P a g e

66 function i greater tan zero, ten equation (47) i negative and te um of te equation (47) and (48) may eiter be oitive or negative. In order for te to-left comonent of te Heian matrix to be oitive and artially atify te condition for wic te Heian matrix i oitive definite, te otimal olution mut atify te neceary condition in wic um of equation (47) and (48) are greater tan zero and tu oitive a own in (49): α f '( ) δ f ( ) > 0 (49) In addition to te requirement for te to-left comonent of te Heian matrix, tere i a requirement for te bottom-rigt comonent of te Heian matrix in order for te matrix to artially atify te condition for wic te Heian matrix i oitive definite. Te comonent in te bottom-rigt of te matrix i te econd order artial derivative of te total cot er unit time wit reect to te ceduled order quantity alone a own in equation (39). In order to logically determine te ign of ti diagonal comonent, te equation in (39) i earated into two egment own in (50-51). [ γ αf ( ) ] (50) 3 δ xf ( x) dx ) (51) 0 ( 1 F( ) 3 Te firt egment of te equation a own in (50) conit of all oitive term and ign; werea te econd egment of te equation in (51) conit of all oitive term but a negative ign. Even wit te negative ign, te egment illutrated in equation (51) i alway oitive becaue te term being ubtracted te cumulative robability of a continuouly ditribution i alway le tan or equal to one. Tu, te 54 P a g e

67 egment in equation (51) i alway oitive; and te wole ortion of te bottom-rigt comonent of te Heian matrix i alway oitive. Conequently, tere are ten only two condition neceary for Heian Matrix to be oitive definite. For one, te Heian matrix in (44) mut atify te condition in (49) uc tat te diagonal of te matrix are te ame ign and te rimary diagonal from to-left to bottom-rigt i oitive. Secondly, te olution et mut atify te condition in (5) in wic te determinant of te Heian matrix i greater tan zero. 0 < 1 αf '( ) δ f ( ) (5a) γ αf ( ) δ ( ) xf ( x) dx 1 F( ) (5b) 3 0 ( ) δ ( 1 F ( ) ) α f (5c) Summary of Solution Condition Te economically otimal olution to oerating an inventory ytem under contant demand and a robability of tocatic emergency demand i defined by te inventory olicy tat minimize te total cot er unit time. Since te inventory olicy i determined by two deciion variable afety tock and order quantity te otimal olution et contain all value of and uc tat te firt-order artial derivative of te total cot er unit time are equal to zero and te Heian matrix of te econd-order artial derivative i oitive definite. Terefore, te otimal deciion and mut atify two et of otimal condition. 55 P a g e

68 56 P a g e Te firt et of otimal condition are derived by etting te firt order derivative of te total cot er unit wit reect to te afety tock level and te ceduled order quantity equal to zero. Tee condition wic are reiterated in (53) and (54) guarantee tat te total cot er unit time i eiter minimized or maximized. ( ) 0 ) ( 1 ) ( 1 = F f δ α (53) [ ] ( ) 0 ) ( 1 ) ( ) ( 0 = F dx x xf F δ α γ (54) Te econd et of condition guarantee tat te total cot er unit time i a global minimum and not a local minimum or even a global maximum. Tee condition wic are reiterated in (55) and (56) atify te neceary requirement for te Heian matrix of te econd-order artial derivative of te total cot er unit time to be oitive-definite. Tat i, condition for wic te Heian matrix i ymmetric wit oitive diagonal and te determinant of te Heian matrix to be greater tan zero. 0 ) ( ) ( ' > f f δ α (55) < ) ( ) '( 1 0 f f δ α (56a) ( ) ) ( 1 ) ( ) ( 0 3 F dx x xf F δ α γ (56b) ( ) ) ( 1 ) ( F f δ α (56c)

69 CHAPTER 4 NUMERICAL ANALYSIS & RESULTS Te uroe of te inventory model develoed in te reviou ection i to determine an otimal inventory olicy under exlicit energy cot conideration. Secifically, te objective of te model i to find an inventory olicy wit otimal ize for a ceduled order quantity, a afety tock level, and an inventory cycle lengt T tat minimize te exected total cot er unit time of a art wit a fairly regular demand but a mall robability of emergency demand. Suc an inventory olicy i exected to be alicable for roduction ytem wit contant roduction rate but mall, underlying oibilitie for undeirable circumtance to treaten te lanned roduction cedule. In order to illutrate te effect of energy on inventory olicie, te inventory model develoed in te reviou ection i numerically analyzed wit reect to cange in energy cot a well a numerou oter model arameter tat are reaonable to imilar roduction environment. Te reultant inventory olicy deciion and reective logitic cot for te variou model arameter are analyzed and comared to te traditional EO model in order to furter validate te inventory model and illutrate te cae in wic it i mot effective. 57 P a g e

70 4.1 Numerical Analyi Parameter In te numerical analyi to follow, mot of te model arameter are initially varied between only two level in order to identify te key arameter tat affect inventory olicy deciion and te reultant logitic cot. Tee arameter are organized into te following tree logitic function: uly rocurement, roduction, and tranortation. After key arameter are identified, te level at wic te key arameter vary need not be limited to te two initial level in te ubequent analyi. Nevertele, te uroe of varying mot of te model arameter by only two level in te analyi i to dicover and undertand te environment in wic te inventory olicie are mot ignificantly affected by cange to energy cot a well a te environment in wic te rooed inventory model i mot cot effective Suly Procurement Parameter Te firt et of model arameter are te urcaing cot aociated wit te rocurement of raw material or unfinied roduct from a ulier or ditributor required by te roduction ytem. Tee rocurement cot include a fixed urcaing cot k o to order any number of roduct from te ulier, a variable urcaing cot e o aociated to te energy conumed in order to uly a ingle unit of te roduct, and a variable urcaing cot c o aociated to everyting but te energy cot to uly a ingle unit of te roduct. A dilayed in Table 4.1, te fixed urcaing cot and te total variable urcaing cot (c o e o ) are varied between two level tat are reaonable to te rocurement activitie at imilar roduction environment. 58 P a g e

71 Table 4.1: Suly Procurement Parameter 4.1. Production Parameter Te econd et of model arameter are te factor aociated wit te demand of te roduction ytem. Again, te roduction ytem i caracterized wit a limited manufacturing caacity. A a reult, cutomer order can be roceed very early and delivered at a muc later date. So, roduction cedule are lanned far in advance, and te roduction rate i fairly contant from regardle of te actual cutomer demand. Even toug te roduction rate and tu te demand of raw material and unfinied roduct by te roduction ytem are fairly contant, tere i a oibility for undeirable circumtance to arie and treaten te intricately lanned roduction cedule. Te inventory model develoed in te reviou ection i exected to be alicable for roduction ytem wit contant roduction rate but mall, underlying oibilitie for undeirable circumtance to treaten te roduction cedule. Toug te oibility of more tan one undeirable circumtance occurring witin any inventory cycle i o mall it i reumed negligible in te rooed model, te volume of te emergency demand generated by te undeirable circumtance can be any number. Furtermore, te robability and te total cot for any of te irregular releniment cenario deend on te random ize of te emergency demand. So, te otimal inventory olicy i contingent uon te robability ditribution of te emergency demand volume generated by te undeirable circumtance. 59 P a g e

72 Te inventory model develoed in te reviou ection i formulated in uc a way tat any robability ditribution can be elected to rereent tat of te tocatic emergency demand. In te ubequent analyi of te inventory model, two different ditribution te Uniform ditribution and te Exonential ditribution are elected to ortray te beavior of te emergency demand in imilar manufacturing ytem. Wile te objective function and olution aroace are rewritten wit reect to eiter te Uniform ditribution or te Exonential ditribution in te ubequent ection, te model arameter for te two ditribution ued in te numerical analyi are own in Table 4.. For bot ditribution, te mean emergency demand varie between two level low and ig wic deend on te ize of te regular demand. Hence, tere are eentially four level at wic te mean emergency demand varie in te ubequent numerical analyi. Ti aumtion i reaonable given tat te volume of te emergency demand deend artially on te regular demand from te roduction ytem. Table 4.: Production Parameter In addition to te varying level of regular and emergency demand, Table 4. ow te robability of te emergency demand eld contant trougout all te numerical analyi to follow. Te value i aumed to be 0.01 in order to rereent te very mall robability of an undeirable circumtance occurring during an inventory 60 P a g e

73 cycle and te negligible oibility of more tan one undeirable circumtance occurring during an inventory cycle Tranortation Parameter Te tird et of model arameter are factor aociated to tranortation activitie. Tee arameter include te tranortation lead time of te regularly ceduled order a well a te tranortation cot. Like many of te aforementioned arameter, te tranortation lead time varie between two level ort and long a own in Table 4.3. Ground iment often require a tree to five day lead time, but ometime require an even longer lead time for variou reaon including longer iing ditance. Furtermore, te total lead time wic include te ulier or manufacturing lead time may be even longer. Table 4.3: Tranortation Parameter In addition to lead time, Table 4.3 ow te value at wic tranortation cot vary in te ubequent analyi. Since tranortation cot are deendent on everal factor including iing ditance, ackage weigt, and fuel cot, te tranortation cot in te ubequent analyi vary baed on cange to tee factor. Tat i, eac 61 P a g e

74 comonent of tranortation cot for eiter tranortation mode m (g, a) varie between two level for eac factor tat affect te reective cot comonent. Te firt tranortation cot factor own in Table 4.3 ditance i related to all te tranortation cot comonent and varie between two level near and far. Since te fixed tranortation cot k m to i any number of art via eiter tranortation mode m (g, a) i deendent only on te ditance of te iment, tee fixed tranortation cot vary between only two level tat are deendent uon te two level at wic te iing ditance varie. Alternatively, te variable tranortation cot c m aociated to everyting but te energy cot to i a ingle unit of te roduct via eiter tranortation mode m (g, a), i deendent on bot te iing ditance and te ackage weigt. Similarly to te iing ditance, te ackage weigt varie between two level ligt and eavy. So, te unit cot c m to i a roduct for eiter tranortation mode m (g, a) varie between four level two level of weigt for eac level of te two level of iing ditance. Te final tranortation cot e m aociated to te energy conumed in order to i a ingle unit of te roduct via eiter tranortation mode m (g, a) i deendent on iing ditance, ackage weigt, and energy cot. Toug ti cot comonent i incurred due to tranortation activitie, it i trongly related to energy cot and tu reented in te following ection wit energy arameter Energy Parameter Te et of model arameter aociated to energy cot and conumtion i directly related to te variable energy cot to rocure or i a ingle roduct. More ecifically, 6 P a g e

75 te energy arameter own in Table 4.4 affect te unit urcaing cot e o aociated to te energy conumed to rocure a ingle unit of roduct a well a te unit tranortation cot e m aociated to te energy conumed to i a ingle unit of roduct via eiter tranortation mode m (g, a). Table 4.4: Energy Parameter Te firt energy arameter own in Table 4.4 affect te unit energy urcaing cot e o to rocure a ingle roduct. Ti cot i related to te energy conumed by any activity not including tranortation before te roduct i to te roduction ytem. Trougout te ubequent analyi, te unit energy urcaing cot e o i modeled a a roortion of te total unit urcaing cot (c o e o ) to rocure a ingle unit of roduct, rater tan a function of te energy ource, energy cot, and energy conumtion. However, ince te total unit urcaing cot already varie between two level a own in Table 4.1 wit te uly rocurement arameter, te energy roortion of te total unit urcaing cot only affect te individual unit cot to rocure a ingle roduct and not te total unit cot to rocure a roduct. Te econd energy arameter own in Table 4.4 affect te unit tranortation cot e m aociated to te energy conumed to i a ingle unit of roduct via eiter tranortation mode m (g, a). A noted reviouly, te unit energy cot to i i deendent uon iing ditance, ackage weigt, and energy cot. Generally, 63 P a g e

76 tranortation and iing buinee bae te energy cot ortion of te total unit iing cot on a fuel urcarge rate, wic i a function of fuel cot and iing mode. An examle of uc a fuel urcarge rate utilized by a major logitic comany in 010 for two oible tranortation mode i own in Figure 4.1. Figure 4.1: Fuel Surcarge Rate wit reect to Fuel Cot In addition to a fuel urcarge rate, tranortation and iing buinee often bae te unit energy cot e m to i a roduct on a function of iing ditance and ackage weigt. Since te unit iing cot c m not aociated to energy i already a function of iing ditance and ackage weigt, tranortation and iing buinee frequently ue te roduct ti function and te function of fuel urcarge rate to rice te unit energy cot to i via a ecified tranortation mode. Accordingly, te unit energy cot e m to i a ingle unit of roduct via eiter tranortation mode m (g, a) i modeled in te ubequent analyi a te roduct of te unit iing cot c m and te fuel urcarge rate r m. Te model arameter for tee 64 P a g e

77 energy factor in te ubequent numerical analyi are own in Table 4.3 and 4.4 a well a Figure 4.1. A own in Table 4.3, te unit iing cot c m varie between two value for ackage weigt for eac of te two value at wic te iing ditance varie. Te fuel cot varie between te two level own in Table 4.4; and te arameter for te fuel urcarge rate r m excluding te fuel cot for eiter tranortation mode m (g, a) are eld contant and own in Figure Micellaneou Parameter Te final et of model arameter are toe tat are ecified for te ubequent numerical analyi but not included in te rior et of arameter. Tee include te time unit of te model and te inventory cot arameter. A own in Table 4.5, te time unit of te inventory model i eld contant to one day trougout all te ubequent analyi. So, te reultant logitic cot of te inventory deciion are reented in term of cot er day; and, te reultant inventory cycle lengt i reented in term of day. Table 4.5: Micellaneou Parameter In addition to te model time unit, Table 4.5 ow te inventory cot arameter tat are eld contant trougout te ubequent analyi. Similarly to mot textbook and reearc, te unit inventory olding cot er time eriod i modeled a a function of te total unit rocurement cot (c o e o ) and te inventory olding cot rate r er time 65 P a g e

78 eriod. Becaue te annual inventory olding cot rate varie between 5 and 50 ercent in mot textbook and reearc, te annual inventory olding cot rate in te ubequent analyi i eld contant at 30 ercent. Ti rate and tu te unit inventory olding cot i tranlated into day, owever, for te ubequent analyi Summary of Model Parameter Given te five et of arameter decribed above, a table ummarizing all te arameter for te following numerical analyi i own in Table 4.6. Table 4.6: Model Parameter for Numerical 66 P a g e

79 67 P a g e 4. Numerical Solution Aroac given Uniform itribution Auming tat te emergency demand i ditributed uniformly between a minimum oint a and a maximum oint b, te inventory model formulated in Cater 3 can be modified to te following objective function and olution condition. Given te modification, te model i numerically analyzed over a variety of arameter. Finally, te olution to te inventory model i comared to a traditional economic ordering olicy auming again tat te emergency demand i ditributed uniformly Objective Function Wit te aumtion tat te emergency demand x follow a uniform ditribution from minimum a to a maximum b, te total cot er unit time can be written a follow: ( ) ( ) = b a b a ) TC(, β δ α γ (57a) ( ) [ ] b dx a b x 1 1 δ α (57b) b a dx a b dx a b x (57c) After integrating te ditribution, equation (57) tranform to equation (58): ( ) ( ) = b a b a 1 ) TC(, β α δ γ (58a) ( ) ( ) ( ) b b b a b 1 1 δ α (58c) ( ) ( ) b a a b 1 (58e)

80 68 P a g e Te final objective function wit te aumtion tat te emergency demand follow a continuou uniform ditribution i own in equation (59): ( ) ( ) = b a b a 1 ) TC(, β α δ γ (59a) ( ) b b b a b δ α (59c) 1 1 b a a b (59e) 4.. Otimal Solution Condition For te olution of oerating an inventory ytem under contant demand and a robability of uniformly ditributed emergency demand to be economically otimal, it mut atify two et of condition. Te firt et of condition tate tat in order to minimize or maximize te total cot er unit time te otimal olution mut atify te condition in wic te firt-order derivative of te total cot er unit time are equal to zero. Te econd et of condition tate tat in order to minimize te total cot er unit time, te otimal olution mut atify te condition in wic te Heian matrix of te econd-order derivative of te total cot er unit time i oitive-definite. Te firt et of otimal condition are derived by etting te firt order derivative of te total cot er unit wit reect to te afety tock level and te ceduled order quantity equal to zero. Te condition in wic te derivative of te total cot er unit time wit reect to te afety tock level i equal to zero own in equation (60); and te condition in wic te derivative of te total cot er unit time wit reect to te ceduled order quantity i equal to zero i own in equation (61).

81 69 P a g e ( ) 0 1 = b a b δ α (60) ) ( = b a a b a a b δ α γ (61) Te econd et of otimal condition in wic te Heian matrix (35) i oitivedefinite require te econd-order artial derivative of te total cot er unit time wit reect to te afety tock level and te ceduled order quantity be determined firt. For te aumtion tat te emergency demand i uniformly ditributed, te econd order artial derivative of te total cot er unit time wit reect to te afety tock level alone i own in equation (6). Te econd order artial derivative of te total cot er unit time wit reect to te ceduled order quantity alone i own in equation (63). Finally, te econd order artial derivative of te total cot er unit time wit reect to te afety tock level and te ceduled order quantity i own in equation (64). = 1 ) TC(, δ a b d d (6) ( ) = ) TC(, b a a a b d d δ α γ (63) ( ) = b a b dd d ) TC(, δ α (64) Given te econd order artial derivative of te total cot er unit time, te Heian matrix for te inventory roblem wit a robability of uniformly ditributed emergency demand between regularly ceduled order can be written a te following: ( ) ( ) ( ) ( ) = *) TC(*, b a a a b b a b b a b a b δ α γ δ α δ α δ (65)

82 70 P a g e Te econd et of otimal condition in wic te olution et atifie te condition for wic te Heian matrix (65) i oitive-definite guarantee tat te olution i at te global otimal minimum. For a oitive definite matrix, two condition mut be atified. For one, te matrix mut be ymmetric. Tat i, te comonent in te diagonal of te matrix mut are te ame ign. Secondly, te determinant of te matrix mut be oitive. Given te emergency demand ditribution, te reviouly derived condition own in (55) and (56) can be redeveloed a (66) and (67). 0 1 > δ a b (66) < 1 0 δ a b (67a) ( ) b a a a b δ α γ (67b) ( ) b a b δ α (67c) 4..3 Numerical Solution Procedure Before a numerical olution can be generated for te inventory model given te uniformly ditributed volume of emergency demand, te afety tock level and te ceduled order quantity mut be derived. Te olution for afety tock level and ceduled order quantity are derived from te firt-order otimal olution condition own in (60) and (61), reectively, a follow:

83 α b a b = b a ( ) ( ) ( ( δ ) ) (68) 4 γ = [ 4α ( a) ( δ )( a b )] b a b a ( a b ) (69) Te equation for afety tock level and ceduled order quantity own in (68) and (69) are nonlinear. In fact, if te two equation in (68) and (69) are combined to form a function of te ceduled order quantity, te equation would be a olynomial to te ixt degree. Given ti comlexity, a numerical olution mut be determined troug an iterative roce. Te iterative olution roce for te inventory model wit uniformly ditributed emergency demand ize i a follow: STEP 0: Etimate an initial value of te afety tock level tarting at te minimum level of te emergency demand a. Label te value o. STEP 1: Calculate te ceduled order quantity from equation (69) uing te etimated initial value of afety tock o. Only calculate te oitive root to te quadratic function in (69) becaue te ceduled order quantity mut be greater tan 0 and tu not negative. Label te oitive value of te ceduled order quantity a 1. STEP : Calculate te afety tock level from equation (68) uing te value of te ceduled order quantity 1 derived in te rior te. Label te value of te afety tock P a g e

84 STEP 3: If 1 = o, olve te inequalitie in equation (66) and (67) given te ceduled order quantity 1 derived in STEP 1 and te afety tock level 1 derived in STEP to tet te econd-order olution condition. Oterwie, if 1 o, increment te initial value of afety tock o by 1% of te difference between te minimum and te maximum emergency demand and go to STEP 1 to reeat te roce. STEP 4: If te econd-order olution condition are TRUE, te numerical olution for te inventory olicy conit of te ceduled order quantity 1 derived in STEP 1 and te afety tock level 1 derived in STEP. Solve for oter reult including cot and cycle lengt. Oterwie, if te econd-order olution condition are FALSE, increment te initial value of afety tock o by 1% of te difference between te minimum and te maximum emergency demand and go to STEP 1 to reeat te roce. STOP: If no olution i found, te inventory olicy ould be identical to a traditional EO model wit zero afety tock Comarative Solution wit Traditional EO Model Te numerical olution derived in te reviou ection for te inventory model given a uniformly ditributed volume of emergency demand i comared to a traditional EO model. In te traditional EO model, te regular demand generated by te roduction cedule i atified by te contant order quantity; and any emergency demand generated by undeirable circumtance i atified by a more exenive 7 P a g e

85 emergency order from a ulier or ditributor via te more exenive mode of tranortation air. Tu, tere i zero afety tock. So, given a traditional EO model, te total cot er unit time i own in equation (70); and te otimal ceduled order quantity derived by taking te firt derivative of total cot function wit reect to te ceduled order quantity in equation (71) i own in equation (7). ( ) ( a b ) TC() = γ β α β δ (70) d d TC(, ) = γ (71) γ = (7) 73 P a g e

86 74 P a g e 4.3 Numerical Solution Aroac given Exonential itribution Te numerical olution to te inventory model in ti ection correond to te aumtion tat te emergency demand volume follow an Exonential ditribution wit mean µ -1. Given te aumtion on te robability ditribution of emergency demand ize, te inventory model formulated in Cater 3 can be modified to develo te following objective function and olution condition. Given tee modification, te model i numerically analyzed wit a variety of cange to te arameter Objective Function Wit te aumtion tat te emergency demand quantity x follow an exonential ditribution wit a mean µ -1, te total cot er unit time i a follow: ( ) 1 1 ) TC(, = µ β µ δ α γ (73a) ( ) [ ] x dx e x µ µ δ α 1 (73b) x x dx e dx xe µ µ µ µ 0 (73c) After integrating te ditribution, equation (73) tranform to equation (74): ( ) 1 1 ) TC(, = µ β µ δ α γ (74a) ( )( ) e e e e e e µ µ µ µ µ δ δ α (74b) ( ) e e e e e e µ µ µ µ µ (74c)

87 Te final objective function wit te aumtion tat te emergency demand follow a continuou Exonential ditribution i own in equation (75): 1 1 TC(, ) = γ α δ β ( ) (75a) µ µ µ 1 µ 1 α δ e ( 1 e ) 1 µ µ (75b) 4.3. Otimal Solution Condition For te olution of oerating an inventory ytem under contant demand and a robability of exonentially ditributed emergency demand to be economically otimal, it mut atify two et of condition. Te firt et of condition tate tat in order to minimize or maximize te total cot er unit time te otimal olution mut atify te condition in wic te firt-order derivative of te total cot er unit time are equal to zero. Te econd et of condition tate tat in order to minimize te total cot er unit time, te otimal olution mut atify te condition in wic te Heian matrix of te econd-order derivative of te total cot er unit time i oitive-definite. Te firt et of otimal condition are derived by etting te firt order derivative of te total cot er unit wit reect to te afety tock level and te ceduled order quantity equal to zero. Te condition in wic te derivative of te total cot er unit time wit reect to te afety tock level i equal to zero own in equation (76); and te condition in wic te derivative of te total cot er unit time wit reect to te ceduled order quantity i equal to zero i own in equation (77). 1 αµ δ 0 = (76) µ e 75 P a g e

88 76 P a g e ( ) ( ) 0 1 = e µ δ αµ µ γ (77) Te econd et of otimal condition in wic te Heian matrix (35) i oitivedefinite require te econd-order artial derivative of te total cot er unit time wit reect to te afety tock level and te ceduled order quantity be determined firt. For te aumtion tat te emergency demand i exonentially ditributed, te econd order artial derivative of te total cot er unit time wit reect to te afety tock level alone i own in equation (78). Te econd order artial derivative of te total cot er unit time wit reect to te ceduled order quantity alone i own in equation (79). Finally, te econd order artial derivative of te total cot er unit time wit reect to te afety tock level and te ceduled order quantity i own in equation (80). ( ) = e d d δ αµ µ µ 1 ) TC(, (78) ( ) = e d d µ δ µ α γ 1 1 ) TC(, 3 (79) = e dd d ) TC(, δ αµ µ (80) Given te econd order artial derivative of te total cot er unit time, te Heian matrix for te inventory roblem wit a robability of exonentially ditributed emergency demand between regularly ceduled order can be written a te following: ( ) ( ) = e e e e µ µ µ µ δ µ α γ δ αµ δ αµ δ αµ µ TC(*,*) 3 (81)

89 77 P a g e Te econd et of otimal condition in wic te olution et atifie te condition for wic te Heian matrix (81) i oitive-definite guarantee tat te olution i at te global otimal minimum. For a oitive definite matrix, two condition mut be atified. For one, te matrix mut be ymmetric. Tat i, te comonent in te diagonal of te matrix mut are te ame ign. Secondly, te determinant of te matrix mut be oitive. Given te emergency demand ditribution, te reviouly derived condition own in (55) and (56) can be redeveloed a (8) and (83). 0 > δ αµ µ µ e x (8) ( ) < e δ αµ µ µ 1 0 (83a) ( ) e µ δ µ α γ (83b) e δ αµ µ (83c) Numerical Solution Procedure Before a numerical olution can be generated for te inventory model given te exonentially ditributed volume of emergency demand, te afety tock level and te ceduled order quantity mut be derived. Te olution for afety tock level (a a function of te exonential value e) and ceduled order quantity are derived from te firt-order otimal olution condition own in (76) and (77), reectively, a follow: ( ) ( ) [ ] e δ αµ µ = (84)

90 γµ = µ ( αµ δ )( 1 e ) ( µ ( 1 e ) µ (85) Te equation for afety tock level and ceduled order quantity own in (84) and (85) are nonlinear. Nevertele, if te two equation in (84) and (85) are combined to form a function of te ceduled order quantity and not te afety tock level, te equation would be a olynomial to te fourt degree a own in (86). Suc a olynomial i olvable rovided a roug numerical roce. 4 3 A B C E F = 0 (86a) were A= ( µ ) (86b) [( )( K ) ] B= µ (86c) ( γ) C = µ K (86d) [( γµ K)( K ) K] 3 = (86e) E ( γµ K) = 4 K (86f) and K = ( αµ δ ) (86g) Given tat tere are four numerical olution to te ceduled order quantity derived by olving for te four root to te fourt degree olynomial equation (86a-g), a ort iterative roce i required to determine te final olution for te inventory olicy. Te ort iterative olution roce for te inventory model wit exonentially ditributed emergency demand ize i a follow: STEP 0: erive te four root to te fourt degree olynomial equation (86a-g) for te ceduled order quantity. Label te value of te ceduled order quantity 1,, 3, and P a g e

91 STEP 1: Calculate te afety tock level from equation (84) for eac value of te ceduled order quantity 1,, 3, and 4 derived in STEP 0. Label te value of te afety tock 1,, 3, and 4 STEP : Ceck weter eac reective air of value for te ceduled order quantity and te afety tock level found in STEP 0 and STEP 1, reectively, atify te inequalitie: 0 and 0. Go to STEP 3 wit any of te reective air tat atify te above inequalitie. Oterwie, if no air atify te above inequalitie, go to STEP 4. STEP 3: Tet te econd-order olution condition in equation (8-83) given te olution air from STEP. If te condition are TRUE, a olution to te inventory model given te current arameter a been found. Oterwie, if te condition are FALSE, go to STEP 4. STEP 4: No olution i found for te inventory olicy given te current model arameter. Tu, te inventory olicy ould be identical to a traditional economic ordering olicy wit zero afety tock Comarative Solution for Traditional EO Model Te numerical olution derived in te reviou ection for te inventory model given an exonentially ditributed volume of emergency demand i comared to a traditional EO model. Wit a traditional economic order olicy, te regular demand generated by te roduction cedule i atified by a regular order quantity; and any emergency demand generated by undeirable circumtance i atified by a more exenive emergency order from a ulier or ditributor via te more exenive mode of 79 P a g e

92 tranortation air. Tu, tere i zero afety tock. So, given a traditional economic ordering olicy, te total cot er unit time i own in equation (87); and te otimal ceduled order quantity derived by taking te firt derivative of total cot function wit reect to te ceduled order quantity in equation (88) i own in equation (89). TC(, ) 1 = γ β µ 1 α δ µ (87) d d TC(, ) = γ (88) γ = (89) Note, te traditional economic ordering olicy doe not cange wit reect to te ditribution of te emergency demand. In fact, te traditional model i not related to te emergency demand at all. Rater, te emergency demand only affect te total rocurement and tranortation cot and tu te total cot of te olicy. 80 P a g e

93 4.4 Numerical Analyi & Reult In order to illutrate te effect of energy on inventory olicy deciion and correonding logitic cot, te inventory model develoed in te reviou ection i numerically analyzed wit reect to cange in energy cot a well a numerou oter model arameter. Te reultant inventory olicy deciion and reective logitic cot for te variou model arameter are comared to toe of a traditional economic ordering olicy to furter validate te inventory model and illutrate te cae in wic te rooed model i mot cot effective. Tat i, te uroe of varying mot of te model arameter between two level for ti analyi i to dicover and undertand te ituation in wic te inventory olicie are mot ignificantly affected by cange to energy cot a well a te ituation in wic te rooed model i mot effective Effect of Model Parameter wit reect to Energy Cot In te analyi, te model arameter decribed in Section 4.1 are eac varied between two level in order to analyze te effect of eac model arameter on variou reult including inventory olicy deciion and logitic cot. Since te effect of energy cot i one of te rimary focue in ti reearc, te reult analyzed in te analyi are in term of cange wit reect to energy cot. Tat i, eac reult rereent te difference between te reult given ig energy cot and te reult given low energy cot. So, in te analyi, eac model arameter excluding energy cot i varied between two level in order to analyze te effect of eac model arameter on cange to variou reult wit reect to energy cot. 81 P a g e

94 Te reult collected from te comarative analyi are analyzed tatitically via te analyi of variance. For eac reone, wic rereent te cange in an inventory olicy deciion or logitic cot wit reect to energy cot, te analyi of variance determine a degree of ignificance for eac model arameter effect on te reone. A te degree of ignificance decreae (or increae), te effect become more imortant (or le imortant) to te reone. Only if te degree of ignificance i le tan or equal to i te effect articularly ignificant to te reone. In te ubequent reentation of te reult, te ignificance of eac effect on eac reone i reented a one of five level of ignificance. Since tere are two numerical analye conducted for eac of te two ditribution aumed to rereent te volume of te emergency demand, te ecific degree of ignificance for eac model arameter on eac reone may vary between te two analye. However, te general concluion of te two analye are reaonably identical. So, rater tan dilaying te degree of ignificance for eac model arameter on eac reone a an average between te two analye, te degree of ignificance i marked a one of five level. Te five level at wic te degree of ignificance i reented in te ubequent reentation of te reult for te analyi of variance include < 0.001, < 0.050, < 0.100, < 0.50, and > If te degree of ignificance i marked a eiter < or < 0.050, te effect i ignificant to te reone; oterwie, te effect i not ignificant to te reone. Nevertele, a degree of ignificance marked a < i till conidered noteworty in te analyi of variance reult of ti comarative analyi given te ligt variance between te reult of te two numerical analye conducted for eac of te two ditribution aumed to rereent te volume of te emergency demand. 8 P a g e

95 In addition to reenting te degree of ignificance of eac model arameter effect on cange in variou reult wit reect to energy cot, te ubequent reult for te analyi of variance dilay te relationi of eac model arameter on cange in variou reult wit reect to energy cot. Suc a relationi may eiter be oitive, negative, or negligible. If te relationi i oitive, te cange in te reult wit reect to energy cot increae a te model arameter increae; werea, if te relationi i negative, te cange in te reult wit reect to energy cot decreae a te model arameter increae. Te reult of te analyi of variance wic include bot te degree of ignificance and te relationi of eac model arameter on cange to inventory olicy deciion and logitic cot wit reect to energy cot are own in Table Wile Table reent te effect of eac model arameter on te cange in logitic cot wit reect to energy cot, Table 4.7 reent te effect of eac model arameter on te cange in inventory olicy deciion wit reect to energy cot. Since te traditional inventory olicy deciion do not cange a energy cot cange, only te reult concerning te cange in te rooed inventory olicy deciion wit reect to energy cot are own in Table 4.7. Te reone reented in Table 4.7 include te cange in te inventory cycle lengt, te ceduled order quantity, te afety tock, and te robability of an emergency order wit reect to energy cot for te rooed inventory olicy. Werea te cange in eac inventory deciion wit reect to energy cot oterwie known a te reone i reented in a et of two column, eac model arameter i reented in a ingle row. So, eac row of dilay te effect of te ecified model arameter on te 83 P a g e

96 reone(); and eac column of dilay te effect of all te model arameter on te reone. In articular, for eac reone, te left column dilay te degree of ignificance a model arameter affect te reone, and te rigt column dilay te relationi a model arameter a on te reone. Table 4.7: Effect of Model Parameter on Cange to Inventory eciion wit reect to Energy Cot for te Prooed Inventory Policy For eac reone own in Table 4.7, bot te roduct weigt and te average emergency demand are found to be ignificant to te cange in te inventory deciion wit reect to energy cot. In articular, tee two arameter, unlike te oter arameter, directly affect te tranortation cot required to fulfill an emergency demand. For examle, a energy cot increae, te difference between te unit energy cot to i a roduct via a fater, le energy efficient mode uc a air and a lower, more energy efficient mode uc a ground increae ignificantly. Similarly, a eiter one of tee two arameter increae, te difference between te total energy cot to i te emergency demand via an emergency order and a regularly ceduled order increae. Tu, te inventory deciion in te rooed inventory model cange ignificantly a eiter one of tee two arameter cange wit reect to energy cot in order to reduce 84 P a g e

97 te oibility of an exenive emergency order and likewie reduce te exected total cot of fulfilling an emergency demand. In te rooed inventory model, unlike te traditional economic ordering olicy, ti oibility of an emergency order may be decreaed by adding or increaing afety tock. However, afety tock affect not only te oibility of an emergency order, but alo te inventory cycle lengt wic ubequently affect te ceduled order quantity and bot te robability of an emergency demand and an emergency order. If te regular demand of te roduction ytem remain contant, a ignificant increae in afety tock caue a ignificant increae to te inventory cycle lengt and roortionally, a ignificant increae to te ceduled order quantity. Yet, if te regular demand of te roduction ytem cange wit reect to energy cot, a ignificant increae in te afety tock doe not ignificantly affect te inventory cycle lengt becaue te ceduled order quantity, wic i related to te inventory cycle lengt, i ignificantly affected by te cange in te regular demand. So, a te regular demand cange wit reect to energy cot, te level of afety tock become more ignificant to reducing te oibility of an exenive emergency order and likewie te exected total cot of fulfilling an emergency demand. In fact, te degree of ignificance for te regular demand on te cange in afety tock wit reect to energy cot i te lowet comared to all te oter arameter. Likewie, te degree of ignificance for te regular demand on te cange in eiter te ceduled order quantity or te robability of an emergency order i one of te lowet comared to all oter arameter. Te only reone regular demand doe not ignificantly affect i te cange in te inventory cycle lengt wit reect to energy cot. 85 P a g e

98 Terefore, te tree model arameter determined to be mot ignificant to cange in inventory deciion wit reect to energy cot for te rooed inventory olicy are te regular demand, te average emergency demand, and te roduct weigt. Te remaining arameter are not conidered noteworty to te rooed inventory olicy becaue neiter one i ignificant to more tan two of te reone own in Table 4.7. Tranortation ditance, for intance, i inignificant to eac and every inventory deciion, even a energy cot cange. Te rimary reaon tat tee model arameter are le ignificant tan te tree factor dicued rior to cange in inventory deciion wit reect to energy cot i tat tee arameter do not ignificantly affect te ize or te cot to releni an emergency demand a energy cot cange. Tu, eac of te remaining factor i inignificant to te cange in bot te ceduled order quantity and te afety tock; and te oter inventory deciion own in Table 4.7 are le ignificantly affected by te remaining factor a energy cot cange. Since neiter energy cot nor emergency demand i exlicitly conidered in te traditional EO model, any cange in energy cot i negligible to te inventory olicy deciion. Conequently, any cange in energy cot i alo inignificant to bot te total rocurement cot and te total inventory cot for te traditional olicy. Te only logitic activity affected by energy cot in te traditional olicy i tranortation. Converely, all te logitic activitie including rocurement, tranortation, and inventory are affected by energy cot in te rooed inventory olicy becaue bot te energy cot and te emergency demand are exlicitly conidered in te rooed model. So, accordingly, te effect of eac model arameter on te cange in te total cot of eac activity for te rooed inventory olicy i own in Table P a g e

99 Table 4.8: Effect of Model Parameter on Cange to Total Cot of Eac Logitic Activity wit reect to Energy Cot for te Prooed Policy Similarly to te reult own in Table 4.7, te regular demand, te average emergency demand, and te roduct weigt are determined to be te mot ignificant factor to te cange in te total cot of eac logitic activity wit reect to energy cot. In fact, tee tree arameter are ignificant to all of te logitic activitie own in Table 4.8. All of te oter arameter, on te oter and, are ignificant to at mot two logitic activitie in te rooed inventory olicy; and te tranortation ditance i inignificant to all of te logitic activitie a energy cot cange. Wile not all te individual activity cot are affected by cange to energy cot for bot te rooed inventory olicy and te traditional inventory olicy, te total tranortation cot i affected by energy cot for bot olicie. Likewie, te total cot of all te logitic activitie for bot olicie i affected by energy cot. So te effect of eac model arameter on te cange in total tranortation cot wit reect to energy cot for eiter inventory olicy i own in Table 4.9; and te effect of eac model arameter on te cange in te total cot of all activitie wit reect to energy cot for eiter inventory olicy i own in Table P a g e

100 Table 4.9: Effect of Model Parameter on Cange to Total Tranortation Cot wit reect to Energy Cot for eac Inventory Policy Table 4.10: Effect of Model Parameter on Cange to Total Cot of all Logitic Activitie wit reect to Energy Cot for eiter Inventory Policy According to te reult own in Table , te regular demand ize, te tranortation ditance, and te roduct weigt are ignificant to bot te cange in te total tranortation cot and te cange in te total cot of all te activitie wit reect to energy cot for eiter inventory olicy. All of te oter arameter, on te oter and, are not ignificant to eiter te total tranortation cot or te total cot of all activitie. Te rimary reaon for tee reult i tat te tree key model arameter are ignificant to te ize or te cot of te regularly ceduled order a energy cot cange werea te oter model arameter are not, even if tere i no intance of an emergency demand. For intance, te regular demand directly affect te ceduled order quantity and tu indirectly affect te total cot of any logitic activity, regardle of any cange in 88 P a g e

101 energy cot. A te regular demand increae, te ceduled order quantity increae. Tu, te total cot of any logitic activity to rocure, tranort, or tore te regularly ceduled order quantity trougout an inventory cycle increae. Yet, a te energy cot increae along wit te regular demand, te roortional increae to te ceduled order quantity more ignificantly affect te total energy cot to i te regularly ceduled order quantity via ground tranortation. Similarly, bot te tranortation ditance and te roduct weigt affect te unit cot to i a roduct via a ecified tranortation mode. A eiter te tranortation ditance or te roduct weigt increae, o too doe te unit tranortation cot aociated to a ecified mode. Ti i true for bot te unit tranortation cot related to energy and not related to energy. In fact, te unit tranortation cot related to energy i roortional to te unit tranortation cot not related to energy. So, a te energy cot increae along wit eiter te tranortation ditance or te roduct weigt, te unit energy cot to i a ingle roduct via a ecified tranortation mode uc a ground tranortation increae more ignificantly. A a reult, te cange in te total tranortation and likewie te total cot of all te logitic activitie wit reect to energy cot i ignificant to tranortation ditance, roduct weigt, and regular demand. Since at leat one logitic activity (tranortation) i affected by cange to energy cot in eiter inventory olicy, te roortion of te total cot allocated between rocurement, tranortation, and inventory activitie i alo affected by cange to energy cot. Te effect of eac model arameter on tee reone are own in Table According to te reult, te total unit urcaing cot, te tranortation ditance, and te roduct weigt are te only factor ignificant to te cange in te roortion of total 89 P a g e

102 cot allocated to eiter rocurement or tranortation activitie in bot inventory olicie a energy cot cange. Converely, te cange in te roortion of total cot allocated to te inventory activity i ignificantly affected by a majority at leat five of te model arameter in bot inventory olicie. However, te inignificant arameter to te inventory activity are not te ame for bot te inventory olicie. Table 4.11: Effect of Model Parameter on Cange to te Proortion of Total Cot Allocated to eac Logitic Activitie wit reect to Energy Cot Tee reult, at leat wit regard to te cange in te roortion of total cot allocated to eiter rocurement or tranortation activitie wit reect to energy cot, are imilar to te reult own in Table For intance, ince bot te tranortation ditance and te roduct weigt ignificantly affect te total tranortation cot a energy cot cange, te roortion of total cot allocated between rocurement and tranortation activitie wit reect to energy cot cange ignificantly but 90 P a g e

103 roortionally. Ti roortional relationi of te total cot allocation between rocurement and tranortation tranire becaue te two activitie comrie over 98 ercent of te total cot; yet, only te total tranortation cot i ignificantly affected by cange to energy cot. So, a eiter te tranortation ditance or te roduct weigt increae, for examle, te total tranortation cot increae, and tu te roortion of te total cot allocated to tranortation increae wile te roortion of te total cot allocated to rocurement decreae. Te effect of te total unit urcaing cot on te roortion of te total cot between rocurement and tranortation activitie i very imilar to tat of te tranortation ditance and te roduct weigt. However, were a te tranortation ditance and te roduct weigt ignificantly affect te tranortation cot oitively, te total unit urcaing cot ignificantly affect te tranortation cot negatively. Tat i, a te total unit urcaing cot increae, te ignificance of te total unit urcaing cot overadow te ignificance of te unit tranortation cot. So, a te total unit urcaing cot increae wit reect to energy cot, te total tranortation cot decreae, and tu te roortion of te total cot allocated to tranortation decreae wile te roortion of te total cot allocated to rocurement increae. Te cange in te roortion of te total cot allocated to inventory wit reect to energy cot, on te oter and, i affected by a majority of te model arameter. Te only arameter not ignificant to te cange in te roortion of total cot allocated to inventory wit reect to energy cot for te traditional inventory olicy include te average emergency demand ize and te tranortation lead time. Converely, te only arameter not ignificant to te cange in te roortion of total cot allocated to 91 P a g e

104 inventory wit reect to energy cot for te rooed inventory olicy i te roduct weigt. Even toug te majority of te arameter are ignificant to ti reone, te reone and likewie te reaon for te reone are not ignificant to te analyi. Tat i, ince te roortion of total cot allocated to inventory in eiter olicy rovided any cange in model arameter or energy cot i le tan ercent, te effect on ti reone are not ignificant to te analyi. Te final reult of te analyi of variance concern te effectivene of te rooed inventory olicy if imlemented in lace of te traditional inventory olicy. Table 4.1 reent te effect of eac model arameter on te cange in te difference between te total tranortation cot of te rooed inventory olicy and te traditional inventory olicy wit reect to energy cot a well a te cange in te difference between te total cot of te rooed olicy and te traditional olicy wit reect to energy cot. Tee cange rereent te cot effectivene of te rooed inventory olicy on reducing te total tranortation cot or imilarly te total cot of all activitie if imlemented in lace of te traditional olicy, eecially if energy cot cange. According to te reult own in Table 4.1, te regular demand ize, te average emergency demand, and te roduct weigt are te only factor ignificant to te cot effectivene of te rooed inventory olicy at reducing bot te total tranortation cot and te total cot of all te logitic activitie wit reect to energy cot if imlemented in lace of te traditional inventory olicy. By no coincidence, tee key arameter are identical to te key arameter determined to be ignificant to cange in inventory deciion wit reect to energy cot for te rooed inventory olicy, a own in Table 4.7. Becaue eac of tee key arameter ignificantly affect 9 P a g e

105 te robability of fulfilling an emergency demand wit an emergency order in te rooed inventory olicy, te exected total tranortation cot of fulfilling an emergency demand via any of te irregular releniment cenario i alo ignificantly affected wit reect to energy cot. Tat i, a te robability of an emergency order decreae, te exected total tranortation cot to fulfill an emergency demand via te irregular releniment cenario decreae. Table 4.1: Effect of Model Parameter on Cange to te Cot Effectivene of te Prooed Policy in lace of te Traditional Policy wit reect to Energy Cot Te effect on te robability of an emergency order and oter inventory deciion, owever, only tranire in te rooed inventory olicy a energy cot cange. Since neiter te energy cot nor te emergency demand i exlicitly conidered in traditional EO model, te traditional olicy doe not cange a energy cot cange. Terefore, te cot aving wit reect to fulfilling te emergency demand via te cenario in te rooed inventory olicy a ooed to te traditional inventory olicy wic only fulfill te emergency demand by te more exenive emergency order increae a eiter one of te tree key arameter increae. Ti reult i more ignificant a eiter one of te key arameter increae a energy cot cange. 93 P a g e

106 4.4. Key Parameter wit reect to Energy Cot Given te variou relationi and ignificance of model arameter on cange to inventory olicy deciion and logitic cot own in Table wit reect to energy, certain model arameter can be combined to illutrate ituation in wic inventory olicy deciion and teir reective cot are mot ignificantly affected by cange to energy cot. Furtermore, imilar combination of uc key factor can rereent ituation in wic te rooed inventory model i mot effective at reducing cot if imlemented in lace of te traditional EO inventory model. Tee key factor, according to te initial comarative analyi, include te roduct weigt, te ize of te regular demand er unit time, and lat but not leat, te ize of te emergency demand er unit time. All oter arameter including te fixed urcaing cot, te unit urcaing cot, te tranortation lead time, and te tranortation ditance are not a ignificant to cange in inventory olicie and cot wit reect to energy cot. Table 4.13: Parameter Level to Validate Significant Effect of Key Parameter Conequently, in te analyi conducted to illutrate te degree of ignificance for eac of te key arameter, te arameter wic are leat affected by cange to energy cot are eld contant at an arbitrary level own in Table 4.13; and te arameter wic are mot ignificantly affected by cange to energy cot are eac indeendently varied 94 P a g e