European Journal of Operatonal Research 181 (2007) 239 251 Producton, Manufacturng and Logstcs A jont optmsaton model for nventory replenshment, product assortment, shelf space and dsplay area allocaton decsons Moncer A. Harga a, *, Abdulrahman Al-Ahmar b, Abdel-Rahman A. Mohamed b a Amercan Unversty of Sharjah, Engneerng Systems Management, P.O. Box 26666, Sharjah, Unted Arab Emrates b Industral Engneerng Department, College of Engneerng, Kng Saud Unversty, P.O. Box 800, Ryadh 11421, Saud Araba Receved 8 June 2005; accepted 20 June 2006 Avalable onlne 26 September 2006 www.elsever.com/locate/ejor Abstract In ths paper, we propose an optmsaton model to determne the product assortment, nventory replenshment, dsplay area and shelf space allocaton decsons that jontly maxmze the retaler s proft under shelf space and backroom storage constrants. The varety of products to be dsplayed n the retal store, ther dsplay locatons wthn the store, ther orderng quanttes, and the allocated shelf space n each dsplay area are consdered as decson varables to be determned by the proposed ntegrated model. In the model formulaton, we nclude the nventory nvestment costs, whch are proportonal to the average nventory, and storage and dsplay costs as components of the nventory costs and make a clear dstncton between showroom and backroom nventores. We also consder the effect of the dsplay area locaton on the tem demand. The developed model s a mxed nteger non-lnear program that we solved usng LINGO software. Numercal examples are used to llustrate the developed model. Ó 2006 Elsever B.V. All rghts reserved. Keywords: Inventory replenshment; Assortment; Space allocaton; Optmzaton 1. Introducton Due to the recent competton n the retalng ndustry, retalers are strvng to mprove ther operatons n order to run ther stores more effcently. Product assortment, product dsplay area selecton, shelf space allocaton, and nventory control are crtcal retalng operatons havng major mpact on the fnancal performance of retal stores. Managng these three operatons ndvdually wll obvously result n sub-optmal overall retal store s proft. Therefore, the decson-makng process regardng these three operatons should be ntegrated to ncrease the retaler proftablty. * Correspondng author. E-mal addresses: hargam@yahoo.com, mharga@aus.edu (M.A. Harga). 0377-2217/$ - see front matter Ó 2006 Elsever B.V. All rghts reserved. do:10.1016/j.ejor.2006.06.025
240 M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 The product assortment problem s concerned wth the optmum use of shelves to provde a complete assortment of products meetng the shopper s preferences. In other words, the problem s to determne the varety of products to be dsplayed on the shelves that have the greatest value to customers so as to maxmze the store s total proft. After decdng whch products to be ncluded n the product assortment, the retal manager has to determne the locatons wthn the store to dsplay each of the selected products. Ths s a vtal decson havng great mpact on the sales performance of the retal store especally durng the promoton perod when some products may be dsplayed n more than one locaton. As kndly ponted to us by one of the referees, the dsplay locaton decson s very ted to the category management ssue. In fact, some retalers adoptng the category management perspectve have the problem to change the category tree (the assortment categorzaton s crtera) and the dsplay. For ths reason they are compelled to have the same tem dsplayed n two dfferent categores. The followng example kndly gven by the same referee clarfes ths pont. Tesco decded to ntroduce the new Snack and Pop Factory category (everythng you need for organzng a party) n the Tesco Extra Format. After dong that they have to manage the problem about the soft drnk category. Should the coca cola can tem be dsplayed only n the soft drnk category, or also n the Snack and pop factory and n other promotonal area? Our proposed mathematcal model could be nterestng for retalers facng ths knd of problem emergng from category management mplementaton. Moreover, because of the shelf space lmtaton, retalers cannot dsplay all products avalable from supplers. Shelf space s, therefore, the scarcest and one of the most mportant resources for attractng more customers n retal stores. An effcent management of shelf space cannot only decrease nventory and dsplay costs but also provde hgher customer satsfacton as well. The shelf space allocaton problem refers to the problem of allocatng the avalable shelf space among the selected tems to be ncluded n the product assortment based on each tem s proftablty. The product assortment, dsplay area selecton and shelf space allocaton problems are therefore crtcal ssues n retal operatons management. The control of nventory s another crtcal problem that plays an mportant role n the effcency and proftablty of retal stores. It s a recurrng problem faced by retalers when decdng about the tmng and the sze of the orders to be placed for each of the dsplayed products. In the nventory lterature, ths problem s often referred to the replenshment problem. Several marketng research works provded emprcal evdence that store s sales depend on the amount of nventory dsplayed on the shelves. Therefore, many researchers have estmated the demand usng space elastc functons. These functons have power-form mathematcal expressons where the coeffcents are called space elastctes. The elastcty coeffcent s defned as the senstvty of the customer to the nventory dsplayed n terms of the quantty bought. Gven that the assortment, allocaton, and replenshment problems are obvously nterrelated problems, the man objectve of ths paper s to develop an ntegrated model that wll result n global optmal proftablty nstead of the local proftablty obtaned by optmsng these problems ndvdually. The remander of the paper s organzed as follows. Secton 2 revews the research works related to the problem addressed n ths paper. In Secton 3, we present the necessary notaton and assumptons and formulate the ntegrated model as a mxed nteger non-lnear program. Secton 4 llustrates the developed model through some numercal examples and the last secton concludes the paper. 2. Lterature revew Inventory control s one of the most actve areas of research n Industral Engneerng and Operatons Research. The large number of nventory control publcatons that appeared n the lterature snce the development of the frst nventory model n 1934 can be attrbuted to the followng two facts. Gven ther practcal aspects, nventory problems are mathematcally attractve and most of them are consdered as open problems to ths date. Moreover, the nteracton of nventory control wth other areas such as producton plannng, schedulng, mantenance, faclty locaton, qualty, transportaton, marketng, and retalng have attracted many researchers to study the relatonshp between nventory control and these areas. Inventory control has wtnessed many streams of studes among whch the followng streams can be cted as the ones related to the topc n ths paper:
M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 241 1. Pure nventory systems wth the orderng decsons (tmng and sze) as the only decson varables. 2. Extenson of some of the pure nventory models by modellng the demand as functon of the nventory level. 3. Integraton of pure nventory problems wth assortment and shelf space allocaton problems. For the frst stream of nventory studes, a collecton of nterestng pure nventory models can be found n the excellent book by Slver et al. (1998). These are related to sngle and multple tems wth determnstc or stochastc demands under several assumptons regardng, for examples, the cost structure (constant or varable), plannng horzon (nfnte or fnte), and demand (constant or tme-varyng). In the second stream of studes, several authors tred to extend some of the pure nventory models by assumng that the demand s a functon of the nventory level. In fact, t has been shown emprcally that sales n a retal store tend to be proportonal to the quantty dsplayed on the shelves. Baker and Urban (1988) were the frst to ntroduce a class of nventory models n whch the demand rate s a functon of the nstantaneous nventory level of an tem. They developed an economc order quantty (EOQ) model for a power-form nventory level-dependent demand. Several other works extended Baker and Urban s model under other nventory stuatons such as deteroratng tems, dfferent classes of customers, presence of defectve tems, effects of nflaton and tme value of money, and stochastc demand. Datta and Paul (2001) assumed that the demand rate s functon of the nstantaneous nventory level untl a gven nventory level s acheved, after whch the demand rate becomes constant. Ths s qute the reverse stuaton that usually occurs n retal stores. The nventory level dsplayed on the shelves s usually kept constant for a certan perod untl the quantty n the backroom faclty s completely depleted. Durng ths perod of tme the demand s constant snce the same quantty s dsplayed on the shelf. Startng from the moment when the backroom nventory s exhausted, the demand wll be affected by the varyng dsplayed quantty on the shelf. All these models mplctly assumed that the entre orderng quantty s avalable for sales. Ths s a rather smplfyng assumpton snce, practcally speakng, n most retal stores, the orderng lot sze s frst receved n a backroom faclty and then transferred n smaller szes to the dsplay area to replace the quantty wthdrawn from the shelves. Therefore, only the dsplayed quantty s n the shoppers vew and the quantty stocked n the backroom has no mpact on sales. In the thrd stream of studes, some authors attempted to nclude the assortment and shelf space allocaton problems n the nventory decson-makng process. Hayya (1991) stated that some nventory problems, such as the nteracton wth marketng, as n the allocaton of shelf space n supermarkets, remans a murky area. Bregman (1995) dscussed the need to ntegrate the marketng, operatons, and purchasng functons to compete n today s marketplace. Curhan (1972) assumed that space elastcty (the rato of relatve changes n unt sales to relatve change n shelf space) s a functon of several product-specfc varables, ncludng physcal propertes, merchandzng characterstcs, and use characterstcs. Usng nearly 500 grocery products, he developed a regresson model to test the mpact of shelf space changes on unt sales. Although he was unable to explan observed varatons n space elastcty, hs model showed a postve relatonshp between shelf space and unt sales. Corstjens and Doyle (1981) developed a nonlnear programmng model for the shelf space allocaton model n whch the demand rate s a functon of shelf space allocated to the product. The model maxmzes the proft subject to constrants on avalably supply for each product and lower and upper bounds on the space assgned to each tem. He found that allocaton of shelf space between products depend on each tem s proftablty, ts space-elastcty of demand, and cross-elastctes between products. Zufryden (1986) proposed the use of dynamc programmng to solve the shelf space allocaton problem snce t allows for the consderaton of general objectve functon and provdes nteger solutons. Hansen and Hensbroek (1978) consdered jontly the space allocaton problem and the product selecton problem. They proposed a model for the smultaneous optmal selecton among a gven set of products and the allocaton of shelf space to these products. Ther model s formulated under three constrants: (1) the space allocated to a product must correspond to an nteger number of facng, (2) the mnmum quantty of the shelf space that must be allocated to a product ncluded n the product assortment, and (3) the total avalable shelf space s lmted.
242 M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 To be more realstc they also put n consderaton the lead-tme t needs to fll the shelf wth a product. A zero lead tme mples the avalablty of a backroom stock and the shelf stock s replenshed the nght after the shelf check. On the other hand, f the lead tme s postve, then the product must be ordered from the central warehouse snce a backroom stock s not avalable. Urban (1998) proposed comprehensve models that ntegrate assortment, allocaton, and replenshment decsons. Hs models present a more realstc formulaton of the demand by dstngushng between the backroom nventory and the dsplayed nventory as well as ncorporatng the effect of shelves that may not be kept fully stocked. He frst generalzed the sngle-tem nventory-level-dependent demand nventory model to model the demand rate as a functon of the dsplayed nventory level only. He then ncluded a product assortment decson by extendng hs frst model nto the mult-tem, constraned stuaton. Due to the complexty and nherent non-lnearty of the ntegrated mult-tem model, he proposed two heurstcs; namely a greedy heurstc and a genetc algorthm. All of the above shelf space allocaton and assortment models were formulated under the assumpton that the demand s a functon of the allocated shelf space. Moreover, these optmsaton models neglected the effect of dsplay locaton wthn the retal store. However, based on a seres of feld experments, Dreze et al. (1994) found that dsplay locaton had a large mpact on sales, whereas changes n the allocated shelf space had much less mpact as long as a threshold was mantaned. Our works dffers from the above revewed papers n a number of respects. We wll make a clear dstncton between showroom and backroom nventores. We wll nclude n the cost functon the nventory nvestment costs, whch are proportonal to the average nventory, for both showroom and backroom facltes. The storage and dsplay costs are also used as other components for the nventory costs for the showroom and backroom, respectvely. Fnally, we wll consder the effect of the dsplay locaton on the tem demand. We wll also consder the shelves where each tem ncluded n product assortment wll be dsplayed as a decson varable to be determned by the ntegrated model. 3. Model formulaton As mentoned earler, we consder a retal store sellng multple tems and havng lmted spaces n the backroom and dsplays areas. The demand functon for each tem ncorporates the man and cross effects of shelf space as well as the locaton effects. The objectve of the developed model s to determne the subset of tems to be ncluded n the product assortment, the locatons of the shelves wthn the store where they should be dsplayed, the quantty to be dsplayed n each locaton, and the orderng quantty for each product n order to maxmze the total net proft. Note that, durng the perods wth no sales promoton, the model developed n ths paper can be easly extended to the case when the prce s also a decson varable by ncludng n the demand functon the man and cross effects of the prce. The model s formulated under a set of assumptons smlar to the one n Urban (1998). The nventory system nvolves a set of tems, N = {1,2,...,n}, m locatons wthn the store each wth a dsplay capacty F k (k =1,2,...,m), and a backroom faclty wth lmted storage capacty F b. Replenshments to the system are ndependent for each tem (no jont replenshments). They are sent drectly to the backroom nventory, and are nstantaneous wth a known and constant lead tme. The dsplay areas are parttoned nto dedcated areas for each product hence, no product uses the area dedcated to another product, even f the nventory s nsuffcent to fll t completely and s contnuously replenshed from the backroom nventory. The demand rate of the tem s determnstc and s a known functon of ts dsplayed nventory level, the dsplayed nventory level of other products to account for substtutons or complementartes. Shortages are not allowed. Note that when all tems are to be ncluded n the product assortment then the objectve functon of the model wll be smply the sum of the ndvdual proft functons. However, gven that t s not necessarly prof-
table to nclude all tems n the assortment, we must ntroduce bnary varables to determne whether a partcular tem should be part of the product assortment. Ths bnary varable s defned as follows: 1 f tem s ncluded n the product assortment; y ¼ 0 otherwse: Moreover, f tem s selected to be dsplayed n the store then we assume that ts orderng quantty q wll be dvded nto m sub-lots q k (k =1,2,...,m where m s the number of locatons where the th tem s dsplayed wth m 6 m) to be used for the kth dsplay locaton. Note that gven that a selected tem wll not necessarly be dsplayed n all locatons, then we need to ntroduce another bnary varable to determne ts dsplay locaton. We defne ths second type of bnary varable as follows: 1 f tem s dsplayed n locakton k; y k ¼ 0 otherwse: These two types of bnary varables should satsfy the followng constrants: y k 6 y for ¼ 1; 2;...; n; and k ¼ 1; 2;...; m; ð1þ y 6 Xm y k 6 NL for ¼ 1; 2;...; n: ð2þ Constrants (1) ensure that f the th tem s not selected to be part of the product assortment (y = 0), then t wll not be dsplayed n any locaton (y k = 0). On the other hand, constrants (2) ensure that f tem s ncluded n the product assortment, then t wll be dsplayed at least n one locaton and at most n NL locatons. Gven that more than two dsplay locatons for the same product are really not frequent n retalng, NL can be set to two for most realstc cases. In other stuatons, such that retal stores wth two exts (we observed ths case n some local retal stores) or durng promoton perods, some products may be dsplayed n more than two locatons. In these cases, NL can be larger than two. The demand for any product dsplayed on shelf k can be estmated by D k ðtþ ¼a I b k sk ðtþ Y M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 243 j2n þ j6¼ I dj sj ðtþ; where, a space scale parameter for the th tem, b k man space elastcty of product on shelf k, d j cross-elastcty between products and j (percent change of unts of product due to 1% change n the number of unts dsplayed of product j. d j need not equal d j and d j can be zero, postve, or negatve dependng on products and j beng ndependent, complementary, or substtute, I sk (t) quantty of tem dsplayed on shelf k at tme t, I sj (t) total quantty of tem j dsplayed n the store at tme t, N + set of tems to be ncluded n the assortment. In order to follow the varaton of the dsplayed quantty of tem on shelf k over tme, we assume that the retal store operates as follows. After recevng quantty q n the backroom area and reservng a quantty q k for shelf k, s k (maxmum dsplay space allocated to tem on shelf k) unts are moved to the kth dsplay area. From that tme on, the allocated shelf space s contnuously replenshed from the backroom to mantan t fully stocked untl the tme at whch the backroom s nventory s completely depleted. Durng ths tme perod the dsplayed quantty s kept constant at level s k. Thereafter, the allocated shelf s no longer reflled from the backroom untl all dsplayed quantty s sold. At that tme, a new nventory cycle wll start agan. Fg. 1 shows the varaton of the dsplayed quantty for three tems for a sngle locaton case. It can be observed from ths fgure that snce D k (t) s a functon of the dsplayed nventory levels of all tems ncluded n the assortment, the demand rate of an ndvdual tem wll change for every nstance when the dsplayed quantty n any locaton of any other tem falls below ts shelf space allocaton, s jk. For example, note that the demand at tme zero
244 M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 Dsplayed quantty, I sk (t) Item 1 s 11 T11 Item 2 s 21 Item 3 s 31 T21 tme 0 t1 t2 T31 Fg. 1. Varaton of the dsplayed quantty over tme for 3-tem, one locaton case. s D 11 ð0þ ¼a 1 s b 11 11 sd 21 21 sd 31 31, the demand at tme t 1 s D 11 ðt 1 Þ¼a 1 s b 11 11 I d 21 s21 ðt 1Þs d 31 31 and at tme t 2 t s gven by a 1 s b 11 11 I d 21 s21 ðt 2ÞI d 31 s31 ðt 2Þ. As n Urban (1998), n order to smplfy the evaluaton of D k (t), we wll approxmate t such that t depends only on the total shelf space allocaton, not the nstantaneous nventory level n a gven dsplay area, of all other products n the assortment. Ths can be done by replacng n the above demand expresson I sj (t) bys j, where s j ¼ Xmj s j ¼ Xm s jk ; y jk s jk : m j 6 m; or; Ths mples that the demand of product s a functon of ts dsplayed nventory and the space allocated for all other products n the assortment. Therefore, the demand functon for tem n the kth dsplay locaton can be rewrtten as D k ðtþ ¼a I b k sk ðtþ Y j2n þ j6¼ s dj j : A more explct expresson for D k (t) where we make use of the bnary varable y j s gven by D k ðtþ ¼a I b Yn skðtþ ð1 y j þ s j y j Þ dj : j¼1 j6¼ Note that at tme zero, the demand s gven by D k ð0þ ¼a s b k k Y n j¼1 j6¼ ð1 y j þ s j y j Þ dj : Therefore, the demand at any tme t can be rewrtten as D k ðtþ ¼D k ð0þ I bk sk : ð6þ s k ð3þ ð4þ ð5þ
After dervng an approxmate expresson of the demand functon over tme, we next develop an expresson for the total proft per unt of tme. The dfferent components of the net proft realzed by product n locaton k are as follows: (1) Gross proft per cycle. The gross proft generated over an nventory cycle s equal to (P C )q k, where P and C are the unt sellng prce and purchasng cost for the th tem. Note we do not consder here other revenue sources for the retaler such as przes or entrance fees. (2) Orderng cost = A. (3) Inventory nvestment cost per cycle for the th product n the kth locaton. Suppose that ntally (tme 0 of the nventory cycle) an order of sze q k s receved n the backroom area and at the same tme s k unts are moved to the kth dsplay area. Thereafter, the allocated shelf space s contnuously replenshed from the backroom as soon as unts are sold to mantan t fully stocked untl tme s k,at whch the stock n the backroom area s completely depleted. Durng ths tme nterval [0, s k ], the demand rate s constant and equal to D k (0) snce the amount dsplayed on the shelf s kept equal to the shelf space allocated, s k. The varaton of nventory n the backroom storage area over tme s depcted n Fg. 2. Note that T k s the nventory cycle tme and represents the tme to consume all the quantty q k. The average nventory at the backroom storage area from tme 0 to tme s k s smply the trangle n Fg. 2, whch s gven by I bk ¼ ðq k s k Þs k : 2 Gven that s k ¼ q k s k D k ð0þ ; M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 245 snce s k s the tme to consume the q k s k unts at a rate of D k (0) per unt of tme, then I bk ¼ ðq k s k Þ 2 : ð7þ 2D k ð0þ Durng the tme nterval [s k,t k ], the dsplay area can no longer be replenshed snce the stock n the backroom area s completely depleted and consequently the demand rate wll start to decrease as the dsplayed quantty on the shelf s decreasng. The stock varaton n the shelf over tme s shown n Fg. 3. The nventory varaton over the perod [s k,t k ] s governed by the followng dfferental equaton, whch states that the dsplay quantty s decreasng due only to the demand effect: di sk ðtþ ¼ D k ðtþ; s k 6 t 6 T k wth a fnal condton I ks ðt k Þ¼0 dt Inventory Level q k s k (0) D k T k τk Fg. 2. Varaton of the nventory level over tme n the backroom faclty.
246 M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 Inventory Level D k (t) T k τk Fg. 3. Varaton of the nventory level over tme n the dsplay faclty. or, usng (6): di sk ðtþ dt ¼ D kð0þ I b k s b k sk ðtþ; k The soluton to the dfferental equaton s " # 1 I sk ðtþ ¼ D 1 b k kð0þ ð1 b k ÞðT tþ : s b k k s k 6 t 6 T k wth a fnal condton I ks ðt k Þ¼0: Note that, from Fg. 3, when t = s k, we have I sk (s k )=s k. Therefore, usng Eq. (8), we get ð8þ s k T k ¼ s k þ D k ð0þð1 b k Þ : Substtutng s k ¼ q k s k D k n the last equaton yelds ð0þ T k ¼ q k s k þ s k = ð1 b k Þ : D k ð0þ When tem s not dsplayed n locaton k, the expresson of the cycle tme T k wll result n an ndetermnate value (0/0) snce q k, s k and D k (0) are all zeros. To remedy ths stuaton, we change the last expresson of the cycle tme to T k ¼ q k s k þ s k =ð1 b k Þ ð1 y k þ y k D k ð0þþ : ð9þ The average nventory per cycle n the kth dsplayed area s I sk ¼ s k s k þ Z T k s k I sk ðtþdt; whch, after substtutng I sk (t) of Eq. (8) becomes I sk ¼ s k D k ð0þ q k 1 b k s k : ð10þ 2 b k Fnally, the total nventory nvestment cost for the th product n the kth locaton s IIC k ¼ h ði bk þ I sk Þ; where h s the unt nventory nvestment cost per unt of tme for the th product. After substtutng Eqs. (7) and (10), h 1 h ð q 2 k s k Þ 2 þ s k q k 1 b k 2 b k s k IIC k ¼ : ð11þ D k ð0þ Note that when tem s not dsplayed n locaton k, then IIC k should be zero. However, usng the last expresson for the nventory nvestment cost, we end up wth an ndetermnate rato of the form 0/0. Therefore, to avod ths numercal problem, we use nstead the followng expresson for IIC k :
IIC k ¼ h 1 h ð q 2 k s k Þ 2 þ s k q k 1 b k ð1 y k þ D k ð0þy k Þ 2 b k s k (4) Backroom storage cost per cycle n the kth locaton = h b (q k s k )T k, where h b s the storage cost n the backroom faclty proportonal to the maxmum nventory. (5) Dsplay cost per cycle n the kth locaton = h sk s k T k, where h sk s the dsplay cost for tem n the kth locaton proportonal to the allocated shelf space. Next, lettng T = Max{T k : k =1,2,...,m}, the proft per unt of tme for product can be wrtten as P m ½ðP C Þq p ðs k ; q k Þ¼ k IIC k h b ðq k s k ÞT k h sk s k T k Š A y : ð13þ ð1 y þ T y Þ Note that when tem s not selected as a member of the product assortment, then expresson (13) wll result n zero value. The total proft per unt of tme over all products s smply pðs k ; q k Þ¼ Xn p ðs k ; q k Þ: ð14þ ¼1 The mxed nteger non-lnear programmng formulaton for the multple product assortment problem wth shelf space allocaton and dsplay locaton effects can be stated as follows: Max pðs k ; q k Þ; Subject to: M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 247 X n ¼1 X n ¼1 q ¼ Xm f s k 6 F k ; k ¼ 1; 2;...; m; ð15þ f q 6 F b ; s ¼ Xm ð12þ ð16þ y k q k ; ¼ 1; 2;...; n; ð17þ y k s k ; ¼ 1; 2;...; n; ð18þ q 6 q max ; ¼ 1; 2;...; n; s 6 s max ; ¼ 1; 2;...; n; q k 6 y k q max ; ¼ 1; 2;...; n; and k ¼ 1; 2;...; m; ð19þ y k s mn 6 s k 6 y k s max ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; ð20þ s k 6 q k ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; D k ð0þ ¼a s b k k Y n j¼1 j6¼ ð1 y j þ s j y j Þ dj ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; T k ¼ q k s k þ s k =ð1 b k Þ ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; ð1 y k þ y k D k ð0þþ h 1 h 2 k s k Þ 2 þ s k ðq k 1 b k 2 b k s k Þ IIC k ¼ ð1 y k þ D k ð0þy k Þ ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; 0 6 T k 6 y k T ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; ð21þ T 6 y T max ; ¼ 1; 2;...; n; ð22þ y k 6 y ; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; ð23þ
248 M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 Table 1 Sze of the optmsaton problem Varable Number of bnary varables Varable Number of contnuous varables y n q n y k nm s n T n q k nm s k nm T k nm IIC k nm Total n + nm 3n +4nm y 6 Xm y k ; ¼ 1; 2;...; n; y k ¼f0; 1g; ¼ 1; 2;...; n and k ¼ 1; 2;...; m; y 2f0; 1g; ¼ 1; 2;...; n: Constrants (16) ensure that the dsplay space capacty (F k ) n each locaton s not volated, where f s the amount of shelf space requred per unt of tem. Constrant (17) s related to the backroom storage space capacty where F b s the avalable total storage capacty s the backroom faclty. Constrants (19) and (20) make sure that when tem s not dsplayed n locaton k then q k and s k are set to zero, respectvely. Note that f an tem s not ncluded n the assortment (y = 0), then by constrants (23) all the y k s are zeros and, consequently, all q k s and s k s are zeros whch make q and s equal to zero by constrants (17) and (18), respectvely. In constrants (21) T s determned as the maxmum value of the T k s. Constrants (22) set T equal to zero when tem s not ncluded n the product assortment. The above optmzaton program contans n + nm bnary varables and 3n + 4nm contnuous varables. Moreover, ths formulaton resulted n 6n + m + 8nm + 1 constrants (see Table 1). For an n-tem, m-locaton problem, suppose we select j tems n the assortment, then there are (2 m 1)j n possble assgnments to the m locatons. Gven that there are ways of selectng j tems out of n tems, j then the total number of possble product locaton combnatons s ð2 m 1Þ P n n j¼1 j. j 4 For example, for n=4 and m=2, the number of combnatons s 3 þ 4 2 þ 4 3 þ 4 4 ¼ 96: 1 2 3 4 However for n = 6 and m = 4, ths number becomes 3780 combnatons. These two examples show that the exhaustve method search can be very tme consumng snce we have to solve the problem usng only contnuous varables for that large number of combnatons. We propose to use any avalable optmsaton software package such as LINGO package. In fact, models smlar n complexty to one developed n ths paper can be handled easly wth the advent of hgh speed computers and very effcent optmsaton solvers. 4. Numercal examples In ths secton, we llustrate the above optmzaton model by solvng two 4-tem problems wth 2 and 4 dsplay areas, respectvely. The data for cross elastcty parameter s shown n Table 2. The data for product dependent parameters (h,h b,...,f ) are shown n Table 3, and the data for product locaton parameters (h sk, b k ) are shown n Table 4. We also assume that the backroom capacty s 200 unts of space, and the shelf space capacty s 12 unts of space for each locaton. Table 5 summarzes the results obtaned by solvng the problem usng the LINGO Software. The output results show that only tems 1 and 3 are selected and are dsplayed on both shelves, resultng n a total proft per unt of tme 1807. Note that more unts of the frst tem are reserved for the frst shelf locaton,
M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 249 Table 2 Cross elastcty parameters Item j Cross elastcty between and j 1 2 3 4 1 0.065 0.071 0.025 2 0.048 0.041 0.031 3 0.098 0.037 0.058 4 0.037 0.023 0.042 Table 3 Product dependent parameters Item j h h b P C A f j a s max s mn q max q mn 1 2.5 1.2 18 10 50 1 28 12 1 200 0 2 3.75 1.8 27 15 50 1 15 12 1 200 0 3 5 2.4 36 20 50 1 25 12 1 200 0 4 2 0.96 14.4 8 50 1 10 12 1 200 0 Table 4 Product locaton dependent parameter Item j Dsplay cost h sk Shape elastcty b k 1 2 1 2 1 1.4 1.8 0.4 0.18 2 2.1 2.7 0.2 0.25 3 2.8 3.6 0.3 0.34 4 1.12 1.44 0.24 0.29 Table 5 Results of the optmzaton problem wth 4 tems and 2 locatons Item y y k q q k s s k 1 2 1 2 1 2 1 1 1 1 132.8 83.4 49.4 12 7.8 4.2 2 0 0 0 0 0 0 0 0 0 3 1 1 1 67.2 29.8 37.4 12 4.2 7.8 4 0 0 0 0 0 0 0 0 0 Table 6 Product locaton dependent parameters for a 4-tem, 4-locaton problem Item j Dsplay cost h sk Shape elastcty b k 1 2 3 4 1 2 3 4 1 1.4 1.8 1.5 1.6 0.4 0.18 0.16 0.13 2 2.1 2.7 2.25 2.4 0.2 0.25 0.23 0.18 3 2.8 3.6 3 3.2 0.3 0.34 0.32 0.27 4 1.12 1.44 1.2 1.28 0.24 0.29 0.26 0.20 q 11, than for the second locaton. Smlarly, more unts of tem 1 are dsplayed n the frst shelf locaton, s 11, than n the second dsplay area. However, the converse can be observed for the thrd tem. We also solved the problem for a 4-tem, 4-locaton problem. The nputs for ths problem are shown n Table 6.
250 M.A. Harga et al. / European Journal of Operatonal Research 181 (2007) 239 251 Table 7 Results of the optmzaton problem wth 4-tem and 4-locaton Item y y k q q k s s k 1 2 3 4 1 2 3 4 1 2 3 4 1 1 1 1 1 1 96.4 36.2 20.7 20.1 19.3 12 7 1.9 1.7 1.4 2 1 1 1 1 1 32.2 8.1 8.0 8.0 8.1 4 1 1 1 1 3 1 1 1 1 1 71.4 17.6 19.0 18.3 16.5 12 2.9 3.4 3.2 2.5 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The optmal soluton s to select tems 1, 2, and 3 and to dsplay them on all shelves wth a total proft of 3381, see Table 7) It can be notced from Table 7 that, except for tem 1 for whch more unts are dsplayed on the frst shelf locaton, almost equal unts (after roundng to the nearest nteger values) are dsplayed on all shelves for the remanng two tems. 5. Concluson Ths paper s concerned wth the development and analyss of an ntegrated model for the nventory lot szng, dsplay area and shelf space allocaton, and product assortment problem. The developed model s a complex mxed nteger nonlnear program that was solved usng LINGO software. In the model development, we dd not consder some other mportant factors, such as the strategc mportance of an tem, that can nfluence decsons on product assortment, shelf space and dsplay area allocatons. In fact, some tems cannot be mssng on shelves because of ther mportance n product portfolo whle other products can have substtute or can be out of stock for a whle wthout any harmful effects. As recommended by on of the referees, such an ssue can be consdered as a potental future development of the developed model. Moreover, our research work can be extended by developng effcent heurstcs to solve the optmzaton problem for large szed problems. Another possble extenson s to consder the stochastc case for the demand. A frst avenue of ths extenson s for pershable products that can be dsplayed for short perod of tme on the shelves. In ths case, t can be formulated as a sngle-perod nventory stochastc problem wth the mean of the demand dependng on the shelf space. For non-pershable products, the problem can be addressed as a contnuous revew problem, whch wll consttute the second avenue for the proposed extenson. Acknowledgement We are very grateful to the referees for ther comments and suggestons whch dstnctly mproved the content and the presentaton of the paper. References Baker, R.C., Urban, T.L., 1988. A determnstc nventory system wth and nventory-level-dependent demand rate. Journal of the Operatonal Research Socety. 39, 823 831. Bregman, R.L., 1995. Integratng marketng, operatons, and purchasng to create value. Omega 23, 159 172. Corstjens, M., Doyle, P., 1981. A model for optmzng retal space allocatons. Management Scence 27, 822 833. Curhan, R.C., 1972. The relatonshp between shelf space and unt sales n supermarkets. Journal of Marketng Research IX, 406 412. Datta, T., Paul, K., 2001. An nventory system wth stock-dependent prce-senstve demand rate. Producton Plannng and Control 12, 13 20. Dreze, X., Hoch, S., Purk, M., 1994. Shelf management and space elastcty. Journal of Retalng 70, 301 326. Hansen, P., Hensbroek, H., 1978. Product selecton and space allocaton n supermarkets. European Journal of Operatonal Research 3, 474 484. Hayya, J.C., 1991. Research ssues: On nventory. Decson Lne 22, 14 16.
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