Opinion imedpub Jounals www.iedpub.co DOI: 10.21767/2394-9988.100060 Intenational Jounal of Applied Science - Reseach and Review Abstact Optial Policies fo Peishable Ites when Deand Depends on Feshness of Displayed Stock and Selling Pice Display of stock plays an effective ole to boost up the deand as it encouages custoe to buy oe. Fo peishable good feshness is also an ipotant facto affecting its deand, as consues look fo fesh ites. Selling pice is also a ajo facto affecting the deand. This pape poposes an inventoy odel whee deand depends on selling pice, feshness of ite and displayed stock. The taditional assuption of zeo ending inventoies is elaxed to a non-zeo ending inventoy. As it ay be pofitable to have a closeout sale at a akdown pice, and always keep on-hand fesh displayed stocks if the deand is feshnessand-stock dependent. The objective is to axiize the total pofit with espect to thee decision vaiables (i.e., unit pice, cycle tie and ending-inventoy level). Nueical exaples ae pesented to validate the odel and sensitivity analysis of inventoy paaetes is done to undestand thei effect in deteining optial policies. Keywods: Non-zeo ending inventoy; Peishable goods; Selling pice; Stockdependent deand Poona Misha 1* and Azhauddin Shaikh 2 1 Depatent of Matheatics, School of Technology, Pandit Deendayal Petoleu Univesity, Raisan Gandhinaga-382007, India 2 Institute of Infastuctue Technology Reseach and Manageent, Ahedabad, Gujaat-380026, India *Coesponding autho: Misha P poona.isha@sot.pdpu.ac.in Depatent of Matheatics, School of Technology, Pandit Deendayal Petoleu Univesity, Raisan Gandhinaga-382007, India. Tel: +91-9228850060 Received: Septebe 21, ; Accepted: Octobe 23, ; Published: Octobe 30, Intoduction Deteinistic inventoy odels usually conside the deand ate to be eithe constant o tie-vaying, but independent of level of inventoy i.e. level of stock. Howeve, it has been noted, especially in the etail aket, that these inventoy odels ae not suitable fo epesentations of the eality of inventoy contol situation in the etail field. Levin et al. quoted that lage piles of goods attact oe custoes this is teed as stock-dependent deand [1]. To incopoate this scenaio into inventoy odels, a vaiety of stock-dependent deand odels have been poposed. Bake and Uban established an Econoic Ode Quantity (EOQ) inventoy odel by specifying the stock-dependent deand as a powe function with diinishing etun of displayed stock level [2]. Uban extended the classical EOQ inventoy odel fo zeo ending inventoies to non-zeo ending inventoy when the deand is dependent on the aount of displayed stocks [3]. Uban coputed optial ode quantity when deand is stock-dependent [4]. Shah and Pandey pesented an EOQ odel whee deand was dependent on stock displayed and advetiseent fequency [5]. An EOQ odel with delay in payents and stock dependent deand was given by Saka [6]. Citation: Misha P, Shaikh A () Optial Policies fo Peishable Ites when Deand Depends on Feshness of Displayed Stock and Selling Pice. Appl Sci Res Rev Vol. 4 No.2:10 Saka and Saka developed bette odel fo tie-dependent deteioating inventoy when deand is stock dependent [7]. Shah et al. foulated an integated inventoy odel with stock dependent deand. Shah and Shah developed an EOQ odel fo deteioating ites with ipefect quality and stock dependent deand unde the effect of inflation [8,9]. Saka et al. pesented a odel fo pesevation of deteioating seasonal poducts with stock-dependent deand [10]. Misha and Shaikh foulated a two waehouse integated inventoy odel fo stock dependent deand ate with ode size dependent tade cedit [11]. Misha et al. developed an EOQ odel with pice sensitive stock dependent deand with shotages. This iplies that holding highe inventoy level will pobably ake the etaile ean oe pofit by selling oe ites. Unde this situation, the deand ate should depend on the level of inventoy [12]. Recently, it is noted that custoes ae becoing oe health conscious than befoe because of thei continuously ipoving lifestyle. Hence, the deand fo fesh ites (e.g., vegetables, fuits, baked goods, bead, ilk and seafood) has inceased Unde License of Ceative Coons Attibution 3.0 License This aticle is available in: http://www.iedpub.co/applied-science-eseach-and-eview/ 1
in ecent yeas. In today s aket, deteining pice and ode quantity jointly fo etailes to bette anage peishable poducts is ecognized as an ipotant way to incease pofitability and aintain copetitiveness in a supply chain. The taditional EOQ odels assue that poducts can be stoed indefinitely to eet futue deand. Howeve, any peishable poducts such as fuits, vegetables, edicines, and volatile liquids degade o deteioate continuously due to evapoation, spoilage, obsolescence o soe othe easons. Most eseaches odeled the effect of peishability in the past only fo the etaile s pespective as the on-hand stocks shink due to daage, spoilage, dyness, vapoization, etc. Reseaches aely developed the effect of peishability into thei odels fo the costue s pespective. In fact, feshness is one of the ost citical citeia affecting custoes puchasing decisions. Fujiwaa and Peea took the effect of poduct feshness on consue deand into consideation [13]. Subsequently, Sake et al. studied the inventoy policies fo peishable poducts when the deand is negatively ipacted by the age of the on-hand stocks [14]. Bai and Kendall then poposed an EOQ inventoy odel fo peishable goods by extending the deand to be feshness and shelf-space dependent [15]. Many peishable goods (e.g., packed food, fuit and vegetable salads, donuts, ilk, etc.) have only a few days of shelf life, and thei deand ates gadually decline to zeo as the expiation dates appoach. By taking the expiation date into consideation, Wu et al. established the etaile s optial eplenishent cycle tie and ending stock level when the deand ate is a ultivaiate function of the poduct feshness and displayed stock level. Chen et al. futhe expanded the inventoy odel by Wu et al. to obtain the etaile s optial lot-size, shelf-space, and ending-stock by consideing the shelfspace size as anothe decision vaiable [16,17]. Othe ecent eseach publications elated to axiu lifetie span ae studied in Sake, Wang et al., and Wu et al. To bette anage food nea its expiy date at the etail stage, Aiello et al. studied the optial tie to withdaw the food fo the shelf [18-20]. It has been noted that the deand of an ite is affected by its pice, consue ay opt fo an altenative whee he gets bette deal. So in today s copetitive akets, deteining pice and ode quantity jointly fo etailes is ecognized as an ipotant way to incease pofitability and aintain copetitiveness in a supply chain. The deand ate is usually influenced by pice of an ite. Uban and Bake futhe genealized the deand ate as a ultivaiate function of pice and displayed stocks level [21]. Teng and Chang expanded the EOQ inventoy odel with pice and stock-dependent deand to an Econoic Poduction Quantity (EPQ) inventoy odel fo deteioating ites [22]. Feng et al. pesented an EOQ odel with deand as a ultivaiate function of pice, stock and feshness of the poduct [23]. So the eseach in aea of pice and stock dependent deand is extensive. This pape develops an EOQ odel in which the deand of poduct is dependent on the feshness of ite its selling pice and displayed stock. Selling pice cucially affects not only the deand ate but also the total pofit. As a esult, the selling pice is consideed as a decision vaiable in this pape. The objective of this study is to deteine jointly the optial unit pice, cycle tie and nonzeo ending inventoy that axiize the total pofit of the etaile. The eainde of this pape includes notations and assuptions used in this pape, then the atheatical odelling is done by incopoating the deand as a ultivaiate function of unit selling pice, displayed stocks and feshness of the ite. Theeafte, nueical exaple is pesented to validate the odel. Sensitivity analysis fo key paaetes is done to obseve thei effect on decision vaiables. Lastly, we have descibed the conclusion of this pape with futue eseach diections. Notation and Assuptions The poposed odel uses the following notation and assuptions. Notation C Retaile s unit puchase cost h Holding cost pe annu A Retaile s odeing cost pe ode δ Pecentage discount in selling pice fo salvage value Maxiu life tie of ite It () Inventoy level with etaile at tie t. Decision vaiables E Ending inventoy level in units E 0 S Retaile unit selling pice, S > C T Replenishent cycle tie Functions t at () = ; Age dependent feshness at tie t, which is deceasing function of tie lying between [0,1] DSt (,) Deand ate depends on the feshness of the stock displayed and unit & selling pice;, whee α > 0 scale deand, stock dependent paaete and is fixed life tie of ite. π ( EST,, ) Retaile total pofit The objective of the odel is expessed as below: Max π ( EST,, ) Subject to, S> C and EST,, 0 Assuptions 1. Inventoy syste consists of single deteioating ite possessing fixed life tie. At the beginning of the cycle inventoy ites ae fully fesh (i.e. of feshness value 1) and thei feshness deceases with tie (i.e. depletes to zeo by the end of thei life span). 2. Deand of the poduct is dependent on the feshness of the stock displayed and its unit selling pice. Hee the standad assuption of zeo ending inventoy is given up as It ight be beneficial to close out the sale at a nonzeo ending inventoy and have fesh stock fo sale, because feshe ites would attact oe. 3. The nonzeo ending inventoy is sold out with attactive discount ate. 4. Replenishent is infinite, lead tie is zeo and shotages ae not allowed. 2 This aticle is available in: http://www.iedpub.co/applied-science-eseach-and-eview/
Matheatical Model In the poposed odel, we conside a etaile dealing with a single ite. The deand is dependent on feshness of the displayed stock and its selling pice. It ight be beneficial to close out sale at a lowe ate, and keep on-hand fesh displayed stocks, because the deand is feshness and stock dependent. At the end of cycle tie inventoy level depletes to non-negative ending inventoy i.e. E 0, which is sold fo salvage value. The following diffeential equation epesents the status of inventoy duing inteval [0, T ] : di() t t λ = ( α + βi( t) ) e S, 0 t T dt with I(0) = Qand IT ( ) = E. The solution of using IT ( ) λs λs λs λs e ( 2 ) e ( 2 ) e ( 2 ) e ( 2 ) 1 β T + T β T + T β t t β t t 1/2 1/2 1/2 1/2 It () = Ee β + e α e α e β Using the othe condition I(0) = Q and, we have (1) = Eis, (2) Hee the axiu pofit is π = $407.26 fo cycle tie is S = $10.37 selling pice S = $10.37. The odeing quantity is E = 3.64 units with E = 3.64 as nonzeo ending inventoy. The concavity of the given pofit function is also shown in Figues 1-3. Sensitivity analysis With the values of inventoy paaetes taken in exaple 1, sensitivity analysis of the optial solution with espect to inventoy paaetes is done by changing one paaete at a tie. The esults obtained ae shown in Table 1. The obsevations ade fo these calculations ae: Incease in scale deand inceases the ode quantity, total pofit and ending inventoy; wheeas the selling pice and cycle tie deceases (which iplies highe deand shotens cycle tie and inceases pofit). Fo incease in stock dependent paaete λs λs e T( 2 T) e T( 2 T) 1 β + β + 1/2 1/2 Q = Ee β + e α α β The inventoy holding cost is given by, T HC = h I() t dt (4) 0 Theefo the total pofit pe unit tie is given by, 1 Sales evenue + Salvage value - Puchase π ( ust,, ) = T cost - Odeing cost - Holding cost (5) In this odel total pofit of the fi is to be axiized with espect to nonzeo ending inventoy ( E ), selling pice ( T ) and cycle tie ( T ). Theefoe, the optiization poble is, Maxiize, Subject to, S> C and EST,, 0 Next, we follow the following steps listed below to have an optial solution. Algoith Step 1: Assign nueical values to all the inventoy paaetes. π π π Step 2: Set = 0, = 0, = 0 and solve siultaneously fo E S T u,s and T. Fo the obtained solution check the concavity though gaphs. Nueical Exaples Exaple 1: Conside α = 1500, β=15%, λ= 0.22, =0.2 yea, C = $5 pe unit, δ = 50%, A = $20 pe ode and h = $2 inventoy. The concavity of the given pofit function is also shown in Figues 1-3. Unde License of Ceative Coons Attibution 3.0 License (3) Figue 1 Figue 2 Concavity of pofit function fo S=$10.37. Concavity of pofit function fo T=0.0956 yea. 3
Figue 3 Concavity of pofit function fo E=3.64 units. Paaete Value E* S* T* Q* π 1450 3.588 10.377 0.09734 14.489 386.776 α 1500 3.647 10.371 0.09566 14.806 407.26 1550 3.704 10.365 0.09406 15.118 427.863 0.12 3.653 10.373 0.09565 14.803 407.145 0.15 3.647 10.371 0.09566 14.806 407.26 β 0.18 3.641 10.369 0.09567 14.809 407.374 0.2 1.341 10.335 0.08705 14.282 551.445 λ 0.22 3.647 10.371 0.09566 14.806 407.26 0.24 5.305 10.411 0.10532 14.881 293.395 0.18 3.365 10.354 0.09114 13.84 386.386 0.2 3.647 10.371 0.09566 14.806 407.26 0.22 3.917 10.387 0.09989 15.715 425.113 0.48 1.779 9.973 0.09592 13.98 413.042 δ 0.5 3.647 10.371 0.09566 14.806 407.26 0.52 5.469 10.804 0.09574 15.623 397.152 4.5 1.34 9.35 0.09077 14.806 486.971 A 5 3.647 10.371 0.09566 14.806 407.26 5.5 5.151 11.395 0.10152 14.422 332.914 19 3.55 10.361 0.09317 14.532 417.852 A 20 3.647 10.371 0.09566 14.806 407.26 21 3.741 10.38 0.09809 15.069 396.937 1.8 3.541 10.334 0.09594 14.814 408.982 h 2 3.647 10.371 0.09566 14.806 407.26 2.2 3.75 10.408 0.09538 14.796 405.526 cycle tie, odeing quantity and pofit inceases; wheeas the selling pice and ending inventoy deceases. With incease in pice efficiency of deand λ, ending inventoy, selling pice, cycle tie, odeing quantity incease and total pofit deceases. Table 1 Nueical esults. Incease in axiu lifetie of ite elevates all the decision vaiables, leading to oe pofit. Incease in pecentage discount inceases ending inventoy, selling pice, cycle tie and ode quantity esulting decease in pofit. Incease in puchase cost 4 This aticle is available in: http://www.iedpub.co/applied-science-eseach-and-eview/
pe unit inceases the ending inventoy, selling pice and cycle tie; which in esult deceases ode quantity and total pofit. Incease in odeing cost hikes up all decision vaiables loweing the total pofit. Lastly, incease in holding cost inceases the ending inventoy, selling pice; wheeas cycle tie, ode quantity and total pofit deceases. Nueical esults in Table 1 deonstates that the total pofit is oe sensitive to scale deand λ, pice efficiency of deand λ, axiu lifetie of ite, discount ate and the puchase cost of ite. Hence, the etaile needs to pay oe attention to these paaetes than othes (Table 1). Conclusion In this pape, the objective is to deteine optial policies fo etaile which axiizes his pofit. The deand of an ite is consideed to be dependent on stock displayed, its selling pice and the feshness of the ites. The standad assuption of inventoy level depleting to zeo is elaxed hee, instead it ight be beneficial fo the etaile to close out sale and stock up fesh ites which boosts his deand. The etaile pofit is axiized by consideing cycle tie, selling pice and nonzeo ending inventoy decision vaiables Poposed odel can be extended by allowing shotages, tade cedit and investent in pesevation technology. Model can be eviewed with stochastic deand also. Refeences 1 Levin RI, Mclaughlin CP, Laone RP, Kottas JF (1972) Poductions/ Opeations Manageent Contepoay policy fo anaging opeating systes. Macgaw-Hill, New Yok, USA. 2 Bake RC, Uban TL (1988) A deteinistic inventoy syste with an inventoy level dependent deand ate. J Ope Res Soc 39: 823-831. 3 Uban TL (1992) An inventoy odel with an inventoy-leveldependent deand ate and elaxed teinal conditions. J Ope Res Soc 43: 721-724. 4 Uban TL (2005) Inventoy odels with inventoy level dependent deand: A Copehensive eview and unifying theoy. Eu J Ope Res 164: 792-304. 5 Shah NH, Pandey P (2009) Deteioating inventoy odel when deand depends on advetiseent and stock display. 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