INTERNATIONAL JOURNAL OF INDUSTRIAL ENGINEERING RESEARCH AND DEVELOPMENT (IJIERD)

Transcription

1 INTERNATIONAL JOURNAL OF INDUSTRIAL ENGINEERING RESEARCH AND DEVELOPMENT (IJIERD) International Journal of Industrial Engineering Research and Development (IJIERD), ISSN 0976 ISSN (Print) ISSN (Online) Volume 3, Issue 2, July-December (2012), pp IAEME: Journal Impact Factor (2012): (Calculated by GISI) IJIERD I A E M E ECONOMIC PRODUCTION QUANTITY MODEL FOR PLANED SHORTAGES BY USING EQUIVALENT HOLDING AND SHORTAGE COST ABSTRACTS Prof. B.R. Kharde Amrutvahini Engineering College, Sangamner Pune University, Maharashtra, India khardebr@yahoo.com Dr. G.J. Vikhe Patil Principal, Amrutvahini Engineering College, Sangamner Pune University, Maharashtra, India gjvploni@yahoo.co.in Dr. K.N. Nandurkar Principal, KKW College of Engineering, Nasik Pune University, Maharashtra, India keshav1965@gmail.com In traditional ELS model with planned shortages, replenishment is at uniform rate. The model is little more complex and formulae for calculation of ELS or economic production quantity(epq), maximum inventory quantity, maximum shortage quantity, Holding cost, shortage cost and total inventory cost are more complex and difficult. In this manuscript, we used the concept of Equivalent Holding Cost (EHC) or Equivalent Carrying Cost. With this concept, Economic Production Quantity Model with back-ordering or Economic lot size model with planned shortages (ELSB) model is no different from EOQ model in terms of formulae and difficulty level. Surprisingly it is simplified to EOQ model. All formulae of EOQ model could be used for EPQB model only putting Equivalent Holding Cost (EHC) in place of holding cost! Our work really simplifies the EPQB model. Key Words: Equivalent holding cost (EHC), ELSB, and EPQB 26

2 1. INTRODUCTION The terms "planned shortages" or "back-ordering is permitted" are used interchangeably in literature. When Supply is at infinite rate with planned shortages, the model is viewed as EOQB. When supply is at finite rate for planned shortages or backordering is permitted; it is recognized as ERLB or EPQB. Some authors prefer carrying cost (Gillet, [1979]; Gupta and Hira,[2006]; Mahapatra,[2005]) while some authors use holding cost(kharde and Vikhe-Patil, [2011]; Sharma [2005]). Holding cost is more meaningful in inventory model context and hence is preferred in this manuscript. Inventory is waste and should be minimized. It is not possible to totally eliminate it. Inventory models mainly 1) EOQ, 2) ELS or EPQ 3) EOQ with planned shortages (EOQB) and 4) ELS with planned shortages (ELSB) should be used in order to minimize the total inventory cost wherever could be applicable. However last model-elsb is more complex, requires very complicated formulae; and is very rarely found in text books ( [1],[2],[5],[6],[7]). This is all do to complicated derivations and complex formulae involved in this model. We have developed the concept of Equivalent Holding Cost (EHC) for model (2) ELS or EPQ in Kharde, Vikhe-Patil, and Nandurkar [2012(a)] (3) EOQB in Kharde, Vikhe-Patil and Nandurkar [2012(b)] and (4) ELSB in this very manuscript As a result, of application of EHC; Inventory models have been very much simplified and surprisingly complex models have been now very easy-to-understand, easy-to-use and easy-toapply! It as simple as using EOQ model with substitution of EHC (of corresponding application) for holding cost. 2. Notation Notation Q The production lot size or production batch size or production quantity; units per set up(cycle); Q* Economic Lot Size (ELS) or Economic Production quantity (EPQ); units per set up(cycle) D Total annual demand; Demand/year; units per year M Maximum Inventory level; units per cycle S Maximum Planned shortages or back-ordered quantity; units per cycle O Set up cost per set up; $ per set up H Holding Cost or Carrying cost; $ per unit per year; B Shortage Cost or back-ordering cost; $ per unit per year; T(Q) Total annual inventory cost; $ per year O(Q) annual set up cost; $ per year H(Q) annual holding cost; $ per year B(Q) annual shortage cost or annual back-ordering cost; $ per year p Uniform (finite) production rate; units/time d Uniform(finite) demand or usage rate; units/time t 1 Length of production run when inventory builds up from 0 to maximum 'M' 27

3 t 2 Length of period after production run when inventory reduces from 'M' to maximum 0 t 3 Length of period when shortages or back-orders built from 0 to maximum shortages 'S' t 4 Length of period of production run shortages reduce from 'S' to 0 T cycle time; or time between two consecutive set-ups; = t 1 + t 2 + t 3 + t 4 LT Lead time R Re-order-level K p Factor for equivalent holding cost for uniform production = ( 1 /) K b Factor for equivalent holding cost for planned shortages or when back-orders are permitted = /( + ) H epb Equivalent Holding Cost (EHC) for EPQ model for permitted backorders or planned shortages; $ per unit per year 3. LITERATURE SURVEY Most of backordering models are complicated, with equations unlike those for the EPQ with full. Many works concentrate on partial backordering. Wee (1993) formulates EPQB model for deteriorating items and develop an economic production plan for deteriorating items with partial back-ordering. Teng et al. (2007) take into account not only the backlogging cost but also the cost of lost goodwill and claim better result in the literature. Pentico et al. (2009) develop a comparable model for the EPQ with partial backordering. Pentico and Drake, (2009) take a different approach to modeling the deterministic EOQ with partial backordering that results in equations that are more like the comparable equations for the basic EOQ and its fullbackordering extension. Taleizadeh et al. (2010) develop EPQB model for random defective items, service level constraints and repair failure. Pentico and Drake (2011) review deterministic models that have been developed over the past 40 years. Drake et al. (2011) consider lot-sizing planning for a two-stage system in which the final product is planned using an EPQB and the production of the components is controlled using basic EPQ. Toews et al. (2011) develop EPQB to allow backordering rate to increase linearly as the time until delivery decreases. Quality is global concern and we see many works in imperfect quality. Chung and Hou (2003) develop a EPQB model for a deteriorating production system and provide bounds for the optimal production run time. Wee et al. (2007) develop an EOPQ model for items with imperfect quality and claim better performance. Li Q. and Q. Zhang (2008) deal with EPQB model for imperfect quality items in the fuzzy environment. Jaber et al. (2008) develop EPQ for imperfect quality subject to learning effect. Cárdenas-Barrón, (2009) develops an EPQB model for imperfect quality and rework. Yoo, et al. (2009) propose EPQ model that incorporates both imperfect production quality and two-way imperfect inspection. Khan et al. (2011) do literature survey that has extended the 28

4 Salameh and Jaber, (2000) EOQ model for imperfect quality. Das et al. (2011) formulate EPQ model for imperfect quality as profit maximization problems in stochastic and fuzzy-stochastic environments and solve by Global Criteria Method (GCM). Some researchers worked in stochastic frame. Luo (1998) develop an EPQB model for perishable goods and also develop computer program. In real life manufacturing systems, generation of defective items and random breakdown of production equipment are inevitable. Chiu et al. (2007) present EPQ model with scrap, rework, and stochastic machine breakdowns. Leung (2008) use compete square method for case of uncertainty. Taleizadeh et al. (2010) determine the EPQB for multi-product and stochastic production rate with service constraints. We observe use of Genetic Algorithms. Pasandideh et al. (2010) develop model for a multi-product EPQB with limited warehouse-space and propose a genetic algorithm to solve it. Pasandideh, et al. (2011) develop non-linear integer-programming approach and use a genetic algorithm for EOQB. The integrated production inventory models using differential calculus to solve the multivariable problems are prevalent in operational research. Many researchers present a modified approach to compute economic order quantities without derivatives by cost-difference comparisons (Chung and Wee, 2007; Wee et al, 2009). Several researchers have recently derived formulae for economic-order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions (Kit-Nam and Leung, 2008). Ronald et al. (2004) claim that their method applies basic algebraic skill to derive the optimal solution for EPQB. Sphicas (2006) develop EPQB model using only algebra. They identify two cases, where backorders should and should not be allowed. Minner (2007) presents a different cost comparisons approach to obtain the EOQ and several extensions without taking derivatives. Chung and Wee, (2007) solve the multi-variable problems in tree stage supply chain, and simplify the solution procedure using a simple algebraic method. Teunter and Dekker (2008) derive EOQB using an easy and insightful marginal cost analysis. Seliaman and Ahmad (2009) consider n-stage, multi-customer, non-serial supply chain, and use algebraic methods to optimize. Chang and Ho (2010) applies the well-known renewalreward theorem to obtain a new expected net profit per unit time and derive the exact closedform solutions to determine the optimal lot size, backordering quantity and maximum expected net profit per unit time, specifically without differential calculus. Omar et al. (2010) propose an alternative approach of using the completing the squares method. Hsieh and Dye (2011) solve the same model (Pentico et al., 2011) without differential calculus and present a different decision procedure. Cárdenas-Barron (2011) develop EPQB for two different backorder costs using analytic geometric and algebra. Teng H., et al. (2011) study a two-echelon inventory system with returns and shortage backordering. Teng J., et al. (2011) correct an inappropriate mathematical error on the total cost in two previous published research papers and propose an arithmetic geometric inequality method for EPQ for vendor-buyer integration. Jaber et al. (2011) study EPQ model with learning effect. Zhang et al. (2011) 29

5 develop a heuristic algorithm by adjusting the replenishment frequencies of minor items to solve the model. The heuristic is tested by simulated numerical examples and shows satisfactory results. We observe that no work have been done in the area to simplify the basic complicated equations. We put the concept of equivalent holding cost and simplify the basic equations of EPQB to EOQ model. 4. ECONOMIC PRODUCTION QUANTITY FOR PLANNED SHORTAGES (EPQB) MODEL Certain assumptions are made in this model like Wilson's formulation (Gupta and Hira [2006], Sharma J. [2007], Sharma S. [2005]. o The demand for the item is certain, constant and continuous o Lead time is fixed o Holding cost (H) per unit per unit time is constant and does not change for different order quantity o Set up cost (O) per set up is constant and does not vary with number of set ups o Purchase price of the item is constant and does not change within the period of planning. No discount is applicable o The Production of the item start immediately as stock level reaches maximum shortage level at constant production rate (p) o The demand for the item is at uniform rate (d) o Stock outs are permitted are replenished when next run starts o Uniform production rate 'p' is greater than uniform demand rate 'd'. The model is shown diagrammatically in Figure 1. Production starts when maximum shortage level is reached. During period t 1 items are supplied at rate 'p', out of it 'd' rate is used and backorders are supplied at rate of 'p-d'. At point 'A' all back-orders are fulfilled and then, in period t 2, inventory I positive and accumulated at the rate of 'p-d' from zero to 'M' ( from point 'A' to 'B'). Then production is stopped at point 'B' and inventory is used at rate 'd' from point 'B' (M) to point 'C' (zero). From point 'C' to point 'D', shortages are built or permitted from zero to 'S' in period t 3. This is period maximum shortages occur at end of period 't 3 '. New set up is done and production starts at the end of 't 3 'and start of 't4'. In this period, back-orders are supplied and end of this period back-orders are completely satisfied and inventory is zero. 30

6 I n v e n t o r y M p - d B M Maximum Inventory d L e v e l O S A S D C D Time t 1 t 2 t 4 Maximum Shortages t 3 Figure 1 Economic Production Quantity with Planned Shortages model FACTORS FOR EQUIVALENT HOLDING COST (EHC) Factor for EHC for Uniform Production (K p ) Factor for uniform production is very useful for simplifying EPQ models (Kharde, Vikhe Patil and Nandurkar [2011(a)]). It is known as 'factor for uniform production, K p = =(1 ) (1) In this, time units used for uniform production rate and uniform demand must be same e.g. if production rate is specified in units/month, then demand rate should be in units/month. The time unit need not be per year for 'p' and 'd' but must be same. If time unit used is per year, it is more appropriate and less problematic. The K p is dimension-less constant. Factor for EPC for Back-ordering (K b ) When back ordering is permitted or shortages are planned, factor for EHC for backordering is very useful in simplifying the model. The factor is known as Factor for backordering, K b (Kharde, Vikhe Patil and Nandurkar [2011(b)]). = (2) + This factor is also independent of time units for holding cost per unit and back-ordering cost per unit. But for more practical approach, here units are taken Cost per unit per year for B and 'H' in this manuscript. Back-ordering cost is very difficult to estimate and is really not suitable to be considered per day or per week. Equation (3) is required to great extent for simplification 1 =1 + 31

7 1 = (3) + 6. Equivalent Holding Cost (EHC) During production, period 't 1 ' inventory built up rate is 'p - d' When production is on at rate of 'p' and demand in this period is at rate of 'd'; then inventory built up rate is 'p-d'. When production is off, inventory use up rate is 'd'. Maximum Inventory level, M = ( ) (4) =. (5) +=( + )( ) (6) +=( + ) (7) Similarly Maximum Shortages, S =. (8) = ( ) (9) Similarly Lot size, Q Cycles per year, N From equation (7) =( ) (10) = (11) =( + ) (12) + = = = 1 ( ) =1 (13) (14) (15) =( ) =( ) (16) Finding M: = = (4) (5) + = + = + = = () () + = + = (17) Demand rate is same as annual demand when express in same time per year. e.g. demand rate 100 units per month is same as 1200 units per year. In equations, units should always same. In 32

8 this manuscript, for derivation time is taken in 'year'. In equation (17) quantity M is units, K p is unit less constant. Time unit only is affected from d and D. Finding S: = (8) = (9) + = + =+ = + = (18) From (17) and (18) = + (19) From equation (16) and (19) = + = = (20) Average Inventory For period minimum inventory is zero; maximum is 'M'. Average inventory in period t 1 : = For period minimum inventory is zero; maximum is 'M'. Average inventory in period t 2 := Average Shortages For period minimum shortages is zero; maximum shortages is 'M'. Average shortages in period : = Similarly, for period minimum shortages is zero; maximum shortages is 'M'. Average shortages in period : = Average shortages in the period : = Annual Holding Cost Holding cost per cycle =( )( h )= 2 (( + )) We assume holding cost 'H' is cost per unit per year. Holding period here is only (t 1 + t 2 ); and is taken in years Annual Holding Cost Using equation (15) ()=(h )( ) = 2 ( + ) =( + ) 2 From equation (17) 33

9 Annual Holding Cost, H(Q) ()= = ()= ( ) 2 (21) Annual Shortage Cost Shortage cost per cycle =( )(h h = ( + ) We have assumed shortage cost 'S' is cost per unit per year. Period here is only (t 3 + t 4 ) and should be taken in years. Annual shortage Cost: ()=(h )( ) = 2 ( + ) =( + ) 2 From equation (18): Annual Set up Cost Annual set-up cost: ()= = (22) ()= = 7. Primary condition for optimum We now deduce the condition for optimum. The purchase cost of items is also to be taken in the total cost associated for inventory. But it has no effect on optimum quantity and hence not taken here (Price per unit * D or (Cu)(D)) Total annual inventory cost: from equations (21), (22) and 5.5 ()=()+()+() = ( ) = ( ) + +. In order to minimize total annual inventory cost, partial derivatives with the variables Q and S should be zero. Taking first partial derivative with respect to S and equating it to zero () = { 1 2 ( ) + +. } = ( 1) ( 1)+2.+0=0 2( +2=0 ++=0 (+)= = + 34

10 Using equation (3) = (1 ) = 1 (23) Equation (23) is very important is treated as primary condition for EPQB model. The proportion for optimal shortages and optimal lot size should be according to (23) as this is deduced because of first partial derivative. = 1 (24) 8. Equivalent Holding Cost for uniform production and planned shortages Considering the annual cost for holding and back-ordering; we use the equation (23) in finding the annual holding and annual shortage cost together. Hence taking optimum quantities; Q* and S*. 7.1 Annual Shortage and Back-ordering Cost, HB(Q) from equation (21) and (22): ()=()+() = ( ) + = 1 {( ) + } Using equation (23) = 1 { 2 (1 )) +( (1 )) = {(1 (1 2 )) +(1 ) }= 2 {(1 (1 )) +(1 ) } = 2 { +(1 ) } Using equation (2) and (3): = + = = = () + () = () {+} = ()= 2 (25) Equation (25) is for annual holding and annual shortage costs for EPQB model. If we compare it with the equation for Annual holding cost for EOQ model: ()= (26) 2 Here equation (25) and (26) gives an analogy that K p K b H is just an Equivalent Holding Cost in EPQB model. 7.2 Equivalent Holding Cost for EPQB model Defining Equivalent Holding Cost (EHC) in EPQB model = (27) The Annual Equivalent Holding Cost (EHC) for EPQB model, from equation (25) and (27) is ()+ () = (/2) ()= (/2) (28) It is combined Holding and back-ordering cost for EPQB model. 35

11 9. EPQB Model Derivation with EHC 8.1 Annual set-up cost: () = ( ) = (D/Q) O 8.2 Total annual inventory cost: ()= ()+ ()+ ()= HB (Q) + O (Q) Using equation (28), 8.1 and 8.2 ()= 0.5 Q + (D/Q) O In order to minimize the total annual Inventory cost, the derivative w.r.t. Q should be zero. () = = 0.5. = 2. = 2. This value is optimal, but to confirm for minimum, second derivative must be positive, () = = As all quantities on RHS of above equation are positive (O, D and Q); RHS is greater than zero. Hence the quantity Q found is the optimum or minimum cost point; denoting it as Q* For EPQB model = 2.. (29) Comparing this with traditional EOQ model = 2.. It is confirmed that only EHC for EPQB model should be used instead of holding cost; and formula remains unchanged. 8.3 Optimum Maximum Inventory Level (M*): = (20) = (1 ) = (1 (1 )) = (30) 8.4 Total Optimum Annual Inventory cost: ( )= ( )+ ( )= = 1 (2) (2).... =

12 T(Q*) = HB(Q*) + O(Q*) = (Q* /2)( H epb ) + (D)(O)/Q* = (1/2)[ 2(D)(O)/ H epb ] 0.5 ( H epb ) + (D)(O)/ [ 2(D)(O)/ H epb ] 0.5 = (1/2)[ 2(D)(O) H epb ] (D)(O) [ 2(D)(O)/ H epb ] -0.5 = (1/2)[ 2(D)(O) H epb ] [2(D)(O) H epb ] 0.5 = [ 2(D)(O) H epb ] 0.5 [1/2 + 1/2] T(Q*) = [2(D)(O)( H epb )] 0.5 (31) 8.5 Annual Holding Cost, H(Q): from equation (21) H(Q*) = [Q* K p - S* ] 2 H/(2K p Q*) = [Q* K p - K p [1 K b ] Q* ] 2 H/(2K p Q*) = (Q* 2 H/2K p Q*)(K p - K p [1 K b ] ) 2 = (Q*H/2K p )( K p K b ) 2 = (Q*/2)( K b 2 K p H = (Q*/2)( K b K p K b H = (Q*/2) K b H epb H(Q*) = (Q*/2) K b H epb (32) 8.6 Optimum Annual Shortage Cost, B(Q): from equation (22) B(Q*) = S* 2 B /(2 K p Q*) = {K p 2 [1 K b ] 2 Q* 2 B /(2 K p Q*) = K p [1 K b ] 2 Q* B /2 = (Q*/2) K p B [1 K b ] 2 = (Q*/2) K p B [H/(H+B) ] 2 = (Q*/2) K p BH 2 /(H+B) 2 = (Q*/2) K p [B/(H + B)][H/(H+B) }H = (Q*/2) K p K p [1 K b ]H = (Q*/2) [1 K b ] H epb B(Q*) = (Q*/2) [1 K b ] H epb (33) 8.7 Optimum Annual Holding and Shortage Cost, HB (Q*) HB(Q) = {Q /2} H epb from equation (28) 8.8 Table 1: EPQB model formulae 1 K p = ( p d)/p 2 K b = B/(H + B) 3 H epb = K p K b H 4 Q* = [ 2(D)(O)/ H epb ] S* = K p [1 K b ] Q* 6 M* = K p K b Q 7 T(Q*) = [2(D)(O)( H epb )] H(Q) = (Q*/2) K b H epb 9 B(Q) = (Q*/2) [1 K b ] H epb 10 HB(Q) = (Q/2)(H epb ) 37

13 Formulae comparison is shown in Table Table 2 Formulae Comparison Item EOQ 11 O(Q) = (D)(O)/Q 12 t 1 = M*/(p d) in time unit 13 t 2 = M*/d in time unit 14 t 3 = S*/d in time unit 15 t 4 = S*/(p d) in time unit 16 T = Q*/d in time unit = Q*/D year 17 N* = D/Q* EPQ (Gupta and Hira[2006]) EPQB with EHC (this Paper) K p - - K p K b - - K b H epb H H H epb Q* [ 2(D)(O)/ H ] 0.5 [(H+B)/B] 0.5 [P/(P D)] 0.5 [2DO/H] 0.5 [ 2(D)(O)/ H epb ] 0.5 S* - [D(P-D)/P] 0.5 [2HO/[H+B)B] 0.5 K p [1 K b ] Q* M* Q/2 [B/(H+B)] 0.5 [(P D)/P] 0.5 [2DO/H] 0.5 K p K b Q* T(Q*) [2(D)(O) H] 0.5 [B)/(H+B)] 0.5 [(P D)/P] 0.5 [2DOH] 0.5 [2(D)(O) H epb ] 0.5 H(Q) (Q*/2) H [(H+B)/B] 0.5 [P/(P D)] 0.5 [DOH/2] 0.5 (Q*/2) K b H epb B(Q) - - (Q*/2) [1 K b ] H epb HB(Q) - - (Q/2)(H epb ) O(Q) (D)(O)/Q* (D)(O)/Q* (D)(O)/Q* t 1 - M*/(p d M*/(p d t 2 - M*/d M*/d t 3 - S*/d S*/d t 4 - S*/(p d) S*/(p d) T Q*/D [(H+B)/B] 0.5 [p/(p d)] 0.5 [2O/HP] 0.5 Q*/D year N* Q*/D D/Q* D/Q* Such complex formulae are used for this model (column 3 in Table 2) but now we simplify formulae (column 4 in Table 2). 10. CONCLUSIONS Factor for equivalent holding cost for uniform production (K p ) is very simple to calculate. It is dimensionless. Factor for equivalent holding cost for back ordering (K b ) is also very simple to calculate. It is also dimensionless. The Equivalent Holding cost (EHC) for ELS model is calculated from these two factors (K p and K p ). The concept of EHC has simplified the ELSB or EPQB model to level of EOQ model. All formulae for EOQ model can be used for ELS with "EHC' in place of 'H'. Unbelievable simplification is resulted from this concept of EHC. There is No need to use the complicated formulae of old ELSB model as given in different literature. All required formulae is given in this paper- regarding different times; which are going to be very useful in production planning and control activities and scheduling. The cost formulae 38

14 would be very useful for cost estimation and accounting. Many formulae are not found in other literature. We derived in this manuscript. These formulae are very easy-to-understand, easy-touse and easy-to-apply. The notations used in this paper are simple and may be used as standard. Precaution need to be taken for using the time units for different costs, production and demand rate. Authors suggest going for year" as unit for time. Workdays need to be defined first if scheduling is the output required. Simple logic to note is "there is demand on every working day but no production is done on every working day". Secondly even it is not working day, inventory holding cost and shortage cost are still applicable. Hence, time unit for holding and shortage cost need to be taken per year! "Detail generation of calendar time schedule from calculated time" is being researched by the Authors and is further work to follow. REFERENCES [1]. Cárdenas-Barrón, L., Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Computers & Industrial Engineering, Volume 57, Issue 3, October 2009, Pages [2]. Cárdenas-Barron L., The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Applied Mathematical Modelling, Volume 35, Issue 5, May 2011, Pages [3]. Chang H. and C. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering, Omega, Volume 38, Issues 3-4, June-August 2010, Pages [4]. Chiu S., S. Wang and Y. Chiu, 2007 Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns, European Journal of Operational Research, Volume 180, Issue 2, 16 July 2007, Pages [5]. Chung, C. and H. Wee, Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives, European Journal of Operational Research, Volume 183, Issue 2, 1 December 2007, Pages [6]. Chung K. and K. Hou, An optimal production run time with imperfect production processes and allowable shortages, Computers & Operations Research, Volume 30, Issue 4, April 2003, Pages [7]. Das D., A. Roy, and S., Kar, A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment, Computers & Mathematics with Applications, Volume 61, Issue 9, May 2011, Pages [8]. Drake M., D. Pentico, and C. Toews, Using the EPQ for coordinated planning of a product with partial backordering and its components, Mathematical and Computer Modelling, Volume 53, Issues 1-2, January 2011, Pages [9]. Gary C., G. Lin, D. Kroll, and C. Lin, Determining a common production cycle time for an economic lot scheduling problem with deteriorating items, European Journal of Operational Research, Volume 173, Issue 2, 1 September 2006, Pages [10]. Gillet, B.E., Deterministic Inventory models In: Introduction to operations research. New Delhi, India; Tata McGraw-Hill, [11]. Gupta, P.K., Hira D.S., Inventory models. In: Operations research. New Delhi, India; S. Chand, [12]. Hsieh T. and C. Dye, A note on The EPQ with partial backordering and phase-dependent backordering rate, Omega, In Press, Corrected Proof, Available online 25 March

15 [13]. Jaber M., M. Ahmed and E. Saadany, An economic production and remanufacturing model with learning effects, International Journal of Production Economics, Volume 131, Issue 1, May 2011, Pages [14]. Jaber M., S. Goyal and M. Imran, Economic production quantity model for items with imperfect quality subject to learning effects, International Journal of Production Economics, Volume 115, Issue 1, September 2008, Pages [15]. Khan M., et al A review of the extensions of a modified EOQ model for imperfect quality items, International Journal of Production Economics, Volume 132, Issue 1, July 2011, Pages 1-12 [16]. Kharde, B.R.; G.J. Vikhe-Patil, and K. N. Nandurkar., 2012(a), Simplification of Economic Production Quantity Model by Equivalent Holding Cost, International Journal of Industrial Engineering Research and Development (IJIERD), Volume 3, Issue 1, January - June (2012), Pages [17]. Kharde, B.R.; G.J. Vikhe-Patil, and K. N. Nandurkar., 2012(b), EOQ Model for Planned Shortages by Using Equivalent Holding and Shortage Cost, International Journal of Industrial Engineering Research and Development (IJIERD), Volume 3, Issue 1, January - June (2012), Pages [18]. Kit-Nam and F. Leung, Technical note: A use of the complete squares method to solve and analyze a quadratic objective function with two decision variables exemplified via a deterministic inventory model with a mixture of backorders and lost sales, International Journal of Production Economics, Volume 113, Issue 1, May 2008, Pages [19]. Leung K., Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain, European Journal of Operational Research, Volume 187, Issue 1, 16 May 2008, Pages [20]. Li, Q., and Q., Zhang, Optimal Fuzzy Order Inventory Model with a Mixture of Shortages and Imperfect Items, Systems Engineering - Theory & Practice, Volume 28, Issue 11, November 2008, Pages [21]. Luo W., An integrated inventory system for perishable goods with backordering, Computers & Industrial Engineering, Volume 34, Issue 3, July 1998, Pages [22]. Mahapatra,P.B.,2001. Inventory control, In: Computer-aided production management. 1st ed. New Delhi, India; Prentice-Hall, [23]. Minner, S A note on how to compute economic order quantities without derivatives by cost comparisons, International Journal of Production Economics, Volume 105, Issue 1, January 2007, Pages [24]. Omar M., M. Zubir and N. Moin, An alternative approach to analyze economic ordering quantity and economic production quantity inventory problems using the completing the square method, Computers & Industrial Engineering, Volume 59, Issue 2, September 2010, Pages [25]. Pasandideh S., S. Niaki and A. Nia, A genetic algorithm for vendor managed inventory control system of multi-product multi-constraint economic order quantity model, Expert Systems with Applications, Volume 38, Issue 3, March 2011, Pages [26]. Pasandideh S., S. Niaki and J. Yeganeh, A parameter-tuned genetic algorithm for multiproduct economic production quantity model with space constraint, discrete delivery orders and shortages, Advances in Engineering Software, Volume 41, Issue 2, February 2010, Pages [27]. Pentico D. and M. Drake, The deterministic EOQ with partial backordering: A new approach, European Journal of Operational Research, Volume 194, Issue 1, 1 April 2009, Pages [28]. Pentico D. and M. Drake, A survey of deterministic models for the EOQ and EPQ with partial backordering, European Journal of Operational Research, In Press, Corrected Proof, Available online 1 February 2011 [29]. Pentico D., M. Drake, C. Toews, The deterministic EPQ with partial backordering: A new approach, Omega, Volume 37, Issue 3, June 2009, Pages

16 [30]. Ronald R., G. Yang and P. Chu, Technical note: The EOQ and EPQ models with shortages derived without derivatives, International Journal of Production Economics, Volume 92, Issue 2, 28 November 2004, Pages [31]. Sharma, J.K., Inventory control and deterministic models. In: Quantitative techniques for managerial decisions. 1st ed. New Delhi, India; McMillan India, [32]. Sharma, S.D., Inventory and production management. In: Operations research. 15th ed. New Delhi, India; Kedar Nath Ram Nath, [33]. Seliaman M. and A. Ahmad, A generalized algebraic model for optimizing inventory decisions in a multi-stage complex supply chain, Transportation Research Part E: Logistics and Transportation Review, Volume 45, Issue 3, May 2009, Pages [34]. Sphicas G., EOQ and EPQ with linear and fixed backorder costs: Two cases identified and models analyzed without calculus, International Journal of Production Economics, Volume 100, Issue 1, March 2006, Pages [35]. Taleizadeh A., H. Wee, and S. Sadjadi, Multi-product production quantity model with repair failure and partial backordering, Computers & Industrial Engineering, Volume 59, Issue 1, August 2010, Pages [36]. Taleizadeh A., S. Niaki, and A. Najafi, Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint, Journal of Computational and Applied Mathematics, Volume 233, Issue 8, 15 February 2010, Pages [37]. Teng H., P. Hsu, Y. Chiu, and H. Wee, Optimal ordering decisions with returns and excess inventory, Applied Mathematics and Computation, Volume 217, Issue 22, 15 July 2011, Pages [38]. Teng J., L. Cárdenas-Barrón and K. Lou, The economic lot size of the integrated vendor buyer inventory system derived without derivatives: A simple derivation, Applied Mathematics and Computation, Volume 217, Issue 12, 15 February 2011, Pages [39]. Teng J., L. Ouyang, and L., Chen, A comparison between two pricing and lot-sizing models with partial backlogging and deteriorated items, International Journal of Production Economics, Volume 105, Issue 1, January 2007, Pages [40]. Teunter R., and R. Dekker, An easy derivation of the order level optimality condition for inventory systems with backordering, International Journal of Production Economics, Volume 114, Issue 1, July 2008, Pages [41]. Toews C., D. Pentico, and M. Drake, The deterministic EOQ and EPQ with partial backordering at a rate that is linearly dependent on the time to delivery, International Journal of Production Economics, Volume 131, Issue 2, June 2011, Pages [42]. Wee, H Economic production lot size model for deteriorating items with partial backordering, Computers & Industrial Engineering, Volume 24, Issue 3, July 1993, Pages [43]. Wee H., J. Yu, and M. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega, Volume 35, Issue 1, February 2007, Pages 7-11 [44]. Wee H., W. Wang and C. Chung, A modified method to compute economic order quantities without derivatives by cost-difference comparisons, European Journal of Operational Research, Volume 194, Issue 1, 1 April 2009, Pages [45]. Yoo S., D. Kim, and M. Park, Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return, International Journal of Production Economics, Volume 121, Issue 1, September 2009, Pages [46]. Zhang R., I. Kaku and Y. Xiao, Model and heuristic algorithm of the joint replenishment problem with complete backordering and correlated demand, International Journal of Production Economics, In Press, Corrected Proof, Available online 4 February