2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet

Transcription

1 2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet

2 CORE Voe du Roman Pays 34 B-1348 Louvan-la-Neuve, Belgum. Tel (32 10) Fax (32 10) E-mal:

3 CORE DISCUSSION PAPER 2008/8 An ntegrated model for warehouse and nventory plannng Géraldne STRACK 1 and Yves POCHET 2 February 2008 Abstract We propose a tactcal model whch ntegrates the replenshment decson n nventory management, the allocaton of products to warehousng systems and the assgnment of products to storage locatons n warehousng management. The purpose of ths artcle s to analyse the value of ntegratng warehouse and nventory decsons. Ths s acheved by proposng two methods for solvng ths tactcal ntegrated model whch dffer n the level of ntegraton of the nventory and warehousng decsons. A computatonal analyss s performed on a real world database and usng multple scenaros dfferng by the warehouse capacty lmts. Our observaton s that the total cost of the nventory and warehousng systems can be reduced drastcally by takng nto account the warehouse capacty restrctons n the nventory plannng decsons, n an aggregate way. Moreover addtonal nventory and warehouse savngs can be acheved by usng more sophstcated ntegraton methods for nventory and warehousng decsons. Keywords: mult-tem nventory model, tactcal warehouse model, ntegrated model, Lagrangan relaxaton. 1 CORE and LSM, Unversté catholque de Louvan, Belgum. E-mal: Geraldne.strack@uclouvan.be 2 CORE, Unversté catholque de Louvan, Belgum. Ths paper presents research results of the Belgan Program on Interunversty Poles of Attracton ntated by the Belgan State, Prme Mnster's Offce, Scence Polcy Programmng. The scentfc responsblty s assumed by the authors.

4 1 Introducton Nowadays, managers are faced wth the need to delver a hgh level of servce wth mnmal warehouse and nventory cost. As t has been shown n surveys (WERC 1986 and [1]), the order pckng actvty represents 65% of the total cost and 50% of the workforce of a warehouse. Ths proporton s even more mportant f we consder dstrbuton warehouses where the man actvty (the only added value) s to receve pallets of tems from vendors, stock them and delver customer orders contanng dfferent tems. In addton, wth the mprovement n nformaton technology, t becomes possble to develop tools whch can help managers to handle warehouse and nventory ssues more effcently. At all classcal levels of decson (strategc, tactcal and operatonal) [13],[14],[9], warehouse managers have to tackle problems whch can be dvded nto two broad classes: warehouse management and nventory management problems. Regardng warehouse management ssues, managers have to decde where to assgn the products nsde the warehouse. Strategc decsons concern ssues such as the sze of the warehouse and the techncal specfcatons of the warehouse. Tactcal decsons concern ssues such as the layout of the warehouse and the szng of the varous areas nsde the warehouse [6],[16]. Fnally, operatonal decsons deal wth control polces and routng problems. Concernng nventory management, managers must decde whch product, and how much of each product need to be stored n the warehouse. In ths class, strategc decsons concern the sze and the desgn of the warehouse (a common decson wth warehouse management) and more specfc decsons such as the confguraton of the nventory management decson systems. The tactcal decsons concern ssues such as the operatng hours, the replenshment polces and work force sze. On the last level (operatonal), problems such as what to produce or delver, when and on whch machne or by whom are consdered. (see also [11] for more detals) All those decsons are nterrelated but are dealt wth ndependently [9]. Up to now, those ssues (strategc, tactcal and operatonal decsons) are handled n a pyramdal top-down approach where the flexblty of decsons decreases from top to bottom. Strategc decsons are frst taken and then create lmts to decsons taken at the tactcal and operatonal levels. For example, once the sze and the desgn of the warehouse are fxed, these decsons wll have to be respected when replenshment polces have to be desgned as well as when the sze of the dfferent warehousng areas has to be optmzed (see [9], [5] for more examples). On top of ths, decsons taken at each level of the pyramd are also handled ndependently and sequentally [13]. For example, concernng warehousng decsons taken at the tactcal level, Jeroem P. van den Berg [13] has ntroduced a classfcaton of the dfferent problems faced by managers. He has proposed four dfferent classes of decsons. The frst class tackles ssues related to the assgnment of products across warehousng systems. Warehousng systems dffer by the level of automaton used. The author [13] gves three levels of automaton: manual order pckers where pckers rde along the pckng poston (pcker to product systems), AS/RS order pcker 1

5 or carousel where products are pcked by machnes and sorted by people (product to pcker systems) and totally automated order pckers (robot technology). In addton, warehouses are often dvded n areas accordng to the unt load retreval. Usually, a forward area s used for order pckng of unts of tems frequently ordered and a reserve area s used ether for replenshment of the forward area or for order pckng of cartons or pallets of tems or for products not ordered frequently enough. The second class of problems concern the clusterng of correlated products n such a way that products that are frequently ordered together are assgned to locatons close to each other. The thrd class of problems concerns the workload balancng problem across the warehouse and the last class concern the assgnment of products to storage locatons n order to mnmze the dstance traveled for order replenshment and pckng. We propose a tactcal model whch ntegrates more phases of the decson process: the replenshment decson n the nventory management, the allocaton of products to warehousng systems and the assgnment of products to storage locatons n the warehousng management. We consder a pcker to product dstrbuton warehouse whch s dvded n a forward and a reserve area. Our objectve s to mnmze all relevant warehousng and nventory costs by optmzng the quantty of each product allocated to the forward area (by reducng the work load related to order pckng), the locaton of the product nsde the forward area and the nventory replenshment polces. Our tactcal model takes the sze of the warehousng systems as gven (strategc decson level). Our am s to test whether or not an ntegrated approach to take these nventory and warehousng decsons has some addtonal value, compared to the classcal sequental approach. In the second secton, we wll make a bref revew of the lterature avalable on the subject. The thrd secton wll ntroduce the model formulaton and the varous assumptons made. A descrpton of the methodology used to solve the ntegrated model wll follow n Secton 4 and lastly the varous computatonal results wll be presented for an ndustral test case n Secton 5. 2 Lterature revew We gve references to the dfferent models n the feld of warehouse and nventory management avalable n the lterature. As wrtten n Secton 1, most tactcal ssues n warehouse and nventory management are tackled ndependently and sequentally. In consequence, the models developed n those two felds are presented separately. 2.1 Forward-reserve models The Forward-reserve problem (FRP) s the problem of assgnng products to the forward and reserve areas n order to reduce the overall work content n order pckng [2]. Nowadays, most warehouses are dvded n two areas: forward and reserve. The 2

6 forward area s used for broken-case and full-case pckng and the reserve area s used for pallet pckng and reserve storage. Once a product s stored n the forward area (respectvely the reserve area), all pcks must be performed from the forward area (respectvely the reserve area). When the nventory of an tem stored n the forward area reaches ts reorder pont an nternal replenshment s performed (from the reserve area to the forward area).the forward area s usually a smaller area than the reserve area where order pckng takes less tme and s then less costly. The crtcal decson concerns the choce of the products whch wll be stored n the forward area. Indeed, f all products are located n the forward area, the sze of ths area ncreases and the advantage of lower order pckng cost vanshes. The other decson s to allocate space n the forward area for the dfferent products. Hackman and Rosenblatt [10] were the frst to address the ssues of decdng whch product to store n the forward area (assgnment ssue) and how much to store (allocaton ssue). They consdered a warehouse composed of a small area (forward area) where pckng of products s based on an effcent (less tme consumng) AS/RS automated storage and retreval system. The reserve area s a large area (nfnte capacty) handled through an neffcent manual/ sem-automated materal handlng system. Recepton of products s made through the manual/ sem automated area and can be used to satsfy customers orders or to make nternal replenshment of the AS/RS area. The queston tackled n ths artcle s to decde whch and how much product must be stored n the forward area takng nto account pckng costs and nternal replenshment costs and the capacty constrant of the forward area. They solve the problem through the greedy heurstc where the products are assgned to the forward area accordng to some pror rankng of the products untl the space s full. Ths rankng s based on the comparson of the savngs due to effcent pckng n the AS/RS area and the cost of nternal replenshment. They prove a suffcent condtons for optmalty. Frazelle et al. [3] propose a model that tackles three warehouse decsons: the sze of the forward area and the allocaton/ assgnment of products to the forward area. They propose a model whch mnmzes the total warehousng costs, whch depends on the sze of the forward area (replenshment cost of the forward area, Reserve/forward pckng cost and the cost of captal (shelvngs)), under forward capacty and congeston constrants. Frstly, they show that the congeston constrant s redundant. Consequently, the optmal quantty assgnment/allocaton soluton can be found based on Hackman and Rosenblatt (1990)[10] work. Secondly, they show that the optmal assgnment for the products (forward or reserve area), consderng the lnear relaxaton of ther model, s the rankng gven by Hackman and Rosenblatt 1990 [10] whch s ndependent of the sze of the forward area. They proposed an algorthm whch gves a near optmal soluton to ther model based on the lnear relaxaton of ther model. J.P. van den Berg et al.[2] propose a model to solve the forward-reserve problem n the case of unt load replenshment. Those replenshments can occur durng busy or dle pckng perods. The objectve s to mnmze the number of urgent or concurrent replenshments of the forward area arrvng durng the busy perods. Such replenshment are needed n case of shortage durng the busy perod but 3

7 should be avod because congeston can result. Instead, replenshment actvtes are encouraged to take place durng the prevous dle perod. The resultng forwardreserve model s a bnary programmng problem whch s solved usng a greedy knapsack heurstc procedure based on a lnear relaxaton of the ntal model. In a second part of the artcle, they modfy the model to ncorporate a lmt on the amount of concurrent replenshment. 2.2 Inventory models The am of nventory management s to mnmze total operatng costs whle satsfyng customer servce requrements [4]. In order to accomplsh ths objectve, an optmal orderng polcy wll be determned by answerng to questons such as when to order and how much to order. The operatng costs taken nto account are the procurement costs, the holdng costs and the shortage costs whch are ncurred when the demand of a clent can not be satsfed (ether lost sales costs or backorder costs)[4] [7]. There exst dfferent nventory polces [7] : perodc revew polcy and the contnuous revew polcy. The frst polcy mples that the stock level wll be checked after a fxed perod of tme and an orderng decson wll be made n order to complete the stock to an upper lmt (order up to pont), f necessary. In the second polcy, the stock level wll be montored contnuously. A fxed quantty wll be ordered when the stock level reaches a reorder pont. The order quantty wll only be delvered after a fxed lead tme and shortage can exst f the nventory s exhausted before the recept of the order quantty. Those basc polces can be adapted to take nto account specal stuaton such as sngle or mult tem models, sngle or mult perod models, determnstc or stochastc demands, lost sales or backorder...(see [7],[11],[4] for more detals and examples) 3 Model formulaton 3.1 Problem Descrpton We consder a warehouse composed of a reserve area and a forward area. The forward area s dvded nto locatons and each product n the forward area s allocated to a number of locatons. Before the pckng perod, the forward area s replenshed through advance replenshments from the reserve area. The level of the advance replenshment for each product corresponds to the fllng of the allocated space n the forward area. Nevertheless, f the stock level n the forward area reaches some reorder pont, to avod shortages, concurrent replenshments wll take place. Meanwhle, the enterprse receves external supply for all products. The ssues that we address smultaneously are the decson of the routes taken by the dfferent products n the warehouse (external supples to the reserve area or drectly to the forward area) and the quantty of the products allocated to the forward and/or the reserve area (warehouse management ssues). In addton, the optmal frequency of the external supples wll be optmzed as well as the safety stocks requred to offer 4

8 an adequate customer servce level (nventory management ssues). These ssues are nterrelated because the external supply cost wll depend on the routes taken by the product on one hand and the locaton of the product n the warehouse wll depend on the quantty ordered on the other hand. 3.2 Assumptons Frst of all, we assume that the layout of the warehouse s gven. By ths, we mean that the sze of the warehouse and of the dfferent warehousng systems are gven (forward and reserve areas). Nevertheless, we suppose that t s possble to rent external storage space f the space avalable n the warehouse s not suffcent to store all the products. Ths addtonal capacty s usually rented at a hgher cost than the cost of the nternal warehouse capacty. We suppose also that ths addtonal capacty mples the same costs (recepton cost, storage cost..) than the ones lnked to the reserve area. Dfferent storage polces may exst: random storage polcy and dedcated storage polcy[4]. In a random storage polcy, products are randomly assgned to a locaton n the warehouse. In a dedcated storage polcy, each product s assgned to a specfc locaton. In the latter case, f the product s not avalable n the warehouse then the locaton of ths product wll be empty and there wll be unused space. The forward area, due to ts purpose, wll be handled through a dedcated storage polcy. Concernng the reserve area, we wll consder the two storage polces, the dedcated or the random. Durng the pckng perod, dfferent actvtes can occur. We have consdered sx man actvtes : concurrent and advance replenshments of the forward area from the reserve area, pckng from the forward area and the reserve area, external supply of the forward area and the reserve area. We formulate some assumptons for each of those actvtes. Concernng the advance replenshment of the forward area, we suppose that ths actvty occurs durng some dle perod just before the pckng perod and does not mply any congeston cost. Whereas, concurrent replenshments occur durng the pckng perod when there are not enough tems of a product n the forward area to satsfy the demand of that product and therefore nduce congeston. Concurrent replenshment wll be performed mmedately when the reorder pont s reached. The level of the reorder pont corresponds to the average demand durng the concurrent replenshment lead tme, plus some safety stock. We suppose that the nternal safety stock s fxed and known for all products. Concernng the pckng actvty, we suppose that the tme t takes to pck a product from the forward area (respectvely the reserve area) does not depend on the locaton of the product nsde the forward area (respectvely the reserve area) because products are typcally pcked durng standard pckng tours through the whole areas. Therefore, we won t make a dstncton between the varous locatons nsde the dfferent areas. Nevertheless, the number of products that we can put n a locaton of the forward area wll depend on the sze or volume of that product. Each product can be pcked ether from the forward area or from the reserve area. If 5

9 the product has been allocated to the forward area (respectvely the reserve area) then all the pcks have to be performed from the forward area (respectvely the reserve area). Several unts of a product can be pcked n a sngle pck. The cost of pckng s proportonal to the number of pcks. Fnally, concernng the external supply of the products, we assume that the warehouse manager want to use an nventory control polcy based on contnuous revew polcy (reorder pont system). Therefore the order quantty s constant, the reorder pont s constant, the delvery tme s fxed and the demand of the varous products durng the supply lead tme s probablstc. 3.3 Model The ndexes used are : 1,..., I to denote products and j : 1,..., J to denote a number of locatons n the forward area. Next, we descrbe the data and varables used n the model. For each element, we gve the unts of measure between brackets. Data: α : number of unts of product that can be stored n a sngle locaton of the forward area.[unts] CostRepA : cost of advance replenshment. [euros/replenshment] CostRepC : cost of concurrent replenshment. [euros/replenshment] P ckcostf : pckng cost n the forward area.[euros/pck] P ckcostr : pckng cost n the reserve area.[euros/pck] SSI : nternal safety stock for products n the forward area whch are replenshed through the reserve area. [unts] CostR : recepton cost for the reserve area. [euros/recepton] CostF : recepton cost for the forward area. [euros/recepton] CostCar : nventory Carryng cost [euros/unts/pckng perod] CostAcqu : acquston cost of product [euros/unts] CostShort : shortage cost [euros/unts] CostCapasupp : addtonal capacty cost [euros/unts] L : supply lead tme [pckng perods] CapaF : capacty of the forward area.[locatons] CapaR : capacty of the reserve area.[unts] U : random varable representng the demand of product durng one pckng perod wth expected value E[U ][unts] d : random varable representng the demand of product durng the supply lead tme.[unts] σ L : standard devaton of demand of product durng the supply lead tme[unts] µ L : average demand of product durng the supply lead tme[unts] : random varable representng the number of pcks of product per pckng p δ j perod[pcks] : expected number of concurrents replenshments of product per pckng perod, f j locatons are allocated to product n the forward area. Then δ j can be computed as δ j = s=1 P (U s (jα SSI) because there s one concurrent replenshment each tme that (jα SSI) unts of products have been pcked. 6

10 u j : Usng the defnton of varable δ j, we defne u j = δ j δ j 1 Varables: x,j = y = z = 1 f the product s suppled to the reserve area, pcked from the forward area and f j locatons at least are allocated to product n the forward area 0 otherwse 1 f the product s suppled drectly to the forward area from the supplers and pcked from the forward area only 0 otherwse 1 f the product s assgned to the reserve area and pcked from the reserve area only 0 otherwse Capasupp = number of external storage locaton rented [unts]. (These are dentcal to the locaton n the reserve area) Q = replenshment quantty of product [unts] = reorder pont of product [unts] r Before statng the objectve functon, note that x j have been chosen to be bnary varables nstead of nteger varables because we beleve that we get a stronger formulaton. 7

11 The objectve functon s the expected warehousng and nventory costs per pckng perod and s defned as follows: mn CostRepA x 1 (1) J CostRepC x j uj (2) j CostR z E(U) Q CostR x 1 E(U) Q CostF y E(U) Q (3) P ckcostf E(p ) (x 1 y ) (4) P ckcostr E(p ) z (5) CostCapasupp Capasupp (6) CostCar ( Q 2 r µl ) (7) ( ) E(U) CostAcqu Q (8) Q ( ) E(U) CostShort (d r ) f(d )dd (9) Q r Concernng the warehouse costs, followng our assumptons, we have taken nto account the cost of advance replenshment of the forward area (1) and the cost of concurrent replenshment of the forward area (2). The cost of advance replenshments of product occurs once per pckng perod f product s assgned to the forward area (.e, f x 1 =1). The cost of concurrent replenshment depends on the number of concurrent replenshments whch occur durng the pckng perod. We have used the defnton of u j to obtan the concurrent replenshment cost as expressed n (2). The warehouse cost contans also pckng cost n the forward area (4) (respectvely the reserve area (5)). The rental cost of the addtonal storage capacty s expressed n (6). The tradtonal nventory costs are composed of nventory carryng cost (7), acquston cost (8) and shortage cost (9). We have also recepton costs as defned n (3). 8

12 Constrants : x j x j 1, j : j 2 (10) ( J ) ( ) Q r µ L x j y CapaF (11) j=1 [ ( Q 2 r µl α ) ( ) Q z 2 r µl x 1 [ ( ) ( ) Q r µ L z Q r µ L x 1 ] J α x j CapaR Capasupp j=1 (12) ] J α x j CapaR Capasupp (13) (x 1 z y ) = 1 (14) Capasupp 0 (15) x j, y, z, Q, r 0, j (16) j=1 There are sequencng constrants (10) specfyng that a j th locaton can be assgned to product only f j 1 locatons have already been assgned. The number of products stored n the dfferent areas (forward and reserve area) must not exceed the total amount of space avalable and depends on the storage polcy: (11) for the dedcated storage n the forward area, (12) n case of random storage polcy n the reserve area and (13) n case of dedcated storage polcy n the reserve area. So dependng on the polcy only one constrant n (12) or (13) should be actve. In addton, followng the assumptons, the product can only follow one route n the warehouse (14). Fnally, all the varables must be non negatve(15)(16). 4 Methodology The global model composed of the warehouse and the nventory decsons and constrants presented n Secton 3.3 s a mxed nteger non lnear model. Gven the complexty of solvng ths model to optmalty, our am s to fnd a procedure to solve heurstcally ths model n order to ntegrate decsons concernng the nventory and warehouse felds. We propose two heurstcs methods to solve ths problem offerng dfferent levels of decsons ntegraton. The frst method s a heurstc sequental soluton procedure. The second method gves a hgher level of ntegraton and s smlar to the method used n the teratve procedure proposed by C.J. Vdal and M. Goetschalchx [15] for solvng blnear models. 4.1 Heurstc Sequental Soluton In order to solve the model, we solve sequentally the nventory model then the warehouse model. Frstly, we solve a relaxaton of the nventory model then the 9

13 soluton obtaned for the nventory varables wll be fxed and used to solve the warehouse model. The nventory sub model (.e., the orgnal model wthout costs and constrants related to the warehouse problem) s the mult-tem nventory control model wth two capacty constrants 1 defned by the mnmzaton of nventory costs (recepton nventory, carryng, shortage costs) under nventory and storage capacty constrants. The formulaton of the nventory sub model s therefore: mn ( ) Q CostCar 2 r µ L ( ) E(U ) CostAcqu Q Q ( ) E(U ) CostShort Q CostR z E(U ) Q CostCapasupp Capasupp under the constrants J j=1 x j ( Q 2 r µ L Capasupp 0 x j, y, z, Q, r 0 ( Q r µ L α ) r (d r ) f(d )dd CostR x 1 E(U ) Q y CapaF ) ( ) Q z 2 r µ L x 1, j CostF y E(U ) Q J α x j CapaR Capasupp Ths s a non lnear mxed nteger model where the warehouse varables stll appear n order to model the orderng/recepton costs and capacty constrants. To render ths model ndependent of the warehouse decsons varables, we wll perform a relaxaton and an approxmaton. Frst of all we wll approxmate the objectve functon. The recepton cost, whch depends on the routes taken by the products, wll be approxmated by the followng 1 the constrant concernng the storage capacty of the reserve area depends on the storage polcy of the reserve. We wll develop n ths secton the methodology concernng the random storage polcy. The methodology s easly adaptable n case of dedcated storage polcy. j=1 10

14 CostRecp E(U ) Q where CostRecp s the cost of recepton whch s ndependent on the route taken by the product and s defned as an average of the hstorcal reserve and forward recepton cost. Secondly, we wll approxmate the capacty constrants (reserve and forward constrants). By ths, we mean that nstead of havng two capacty constrants for the dfferent areas n the warehouse, we wll consder only one global capacty constrant for the entre warehouse. Ths global capacty constrant s the aggregaton of the forward and the reserve capacty constrant, and s defned as follows : ( Q r µ L ) Capa Capasupp The new value Capa s the global aggregated warehouse capacty and s defned as the sum of αcapaf and CapaR, where α s the average hstorcal number of products n one locaton of the forward area. The objectve functon so obtaned s ndependent on the routes taken by the varous products. Consequently, the objectve functon s ndependent of the warehouse decsons taken. The global capacty constrant s also ndependent of the routes taken by the dfferent products n the warehouse. Nevertheless, the orderng quantty and reorder pont of each product wll be dependent on the amount of space globally avalable n the warehouse but not on the sze of the dfferent areas n the warehouse, the number of locatons allocated to each product n the forward area and on the routes of the varous products. Ths nventory model s therefore ntegratng a decson from nventory and warehouse felds (through the global capacty constrant). By dualzng the global capacty constrant wth multpler λ and the addtonal capacty non-negatvty constrant wth multpler µ, we obtan the lagrangan of ths mult product nventory model wth three unknown elements, Q, r for all..i and CapaSupp and no constrant: 11

15 L(λ, µ) = mn ( ) Q CostCar 2 r µ L ( ) E(U ) CostAcqu Q Q ( ) E(U ) CostShort Q CostRecp E(U ) Q r (d r ) f(d )dd CostCapasupp Capasupp ( ( ( λ Capa Capasupp Q r µ L ) )) µ Capasupp The frst order necessary condtons are used to derve the optmal value for the orderng quantty (Q ), the reorder pont (r ) for all = 1..I and the addtonal capacty (Capasupp) for fx λ and µ[8]. The standard necessary condtons for optmalty gve the followng results 2 : Q = 2 E(U ) (CostRecp CostShort r (d r ) f(d )dd ) P rob(d r ) = Q (CostCar λ) CostShort E(U ) (CostCar 2 λ) (17) (18) CostCapasupp λ µ = 0 (19) We omt the non negatvty constrant on Q and r because they are satsfed by the frst order condton. The complementary slackness and feasblty condtons are used to determne the optmal value of the addtonal capacty(capasupp) and the Lagrangan multplers (λ, µ): 2 In the rest of the paper, for notatonal convenence, the optmal value of the varables wll be ndcated by an upper bar 12

16 ( ( ( λ Capa Capasupp Q r µ L ) )) = 0 (20) µ Capasupp = 0 (21) ( Q r µ L ) Capa Capasupp (22) Capasupp 0 (23) µ 0 (24) λ 0 (25) By combnng the necessary optmalty condton (19), the complementary slackness condton (21) and the feasblty condton (24) we obtan : (CostCapasupp λ) Capasupp = 0 (26) λ CostCapasupp (27) whch replaces (19), (21) and (24). The resultng frst order necessary condtons are defned as : Q = 2 E(U ) (CostRecp CostShort r (d r ) f(d )dd ) (CostCar 2 λ) (28) P rob(d r ) = Q (CostCar λ) (29) CostShort E(U ) ( ( ( λ Capa Capasupp Q r µ L ) )) = 0 (30) ( Q r µ L ) Capa Capasupp (31) 0 λ CostCapasupp (32) Capasupp 0 (33) (CostCapasupp λ) Capasupp = 0 (34) To fnd all possble soluton to (28) - (34), we must dstngush three possble cases : 13

17 1. λ = 0 Followng (34), the value of the addtonal capacty (Capasupp) must be equal to zero. The optmal order quantty and reorder pont can be calculated by equatons (28) and (29). The value of Q and r for all = 1..I can then be used to see f (31) s satsfed (.e. f there s a soluton wth Capasupp = 0 and the constrant satsfed). If the constrant s not satsfed then there s no soluton wth λ=0. If the constrant s satsfed, we have a soluton to the lagrangan. 2. λ > 0 Followng (30), the capacty constrant s saturated. Followng (32) and (34), we must dstngush two dfferent cases. (a) λ = CostCapasupp The optmal order quantty and reorder pont can be calculated by equatons (28) and (29). From (34), capasupp 0 and can be calculated wth (31)(takng nto account the fact that the constrant s saturated). If the value of Capasupp obtaned s greater or equal to zero then we have a feasble soluton otherwse we have no soluton. (b) 0 < λ < CostCapasupp From equaton (34), the value of the addtonal capacty Capasupp s equal to zero. The optmal value of the lagrangan multpler can be calculated by solvng a system composed of three equatons wth three unknown elements. Indeed, we obtan a lagrangan composed of three unknown elements ( Q, r and λ). In addton, we know that the global capacty constrant s bndng. In ths case, we can derve the three necessary condtons for optmalty ([8],[11]) and calculate the value of the three unknown elements. The frst order necessary condtons are as follows: Q = 2 E(U ) (CostRecp CostShort r (d r ) f(d )dd ) (CostCar 2 λ) P rob(d r ) = Q (CostCar λ) CostShort E(U ) ( Q r µ L ) = Capa (35) Those three possble soluton cases wll be analyzed, a possble soluton wll be calculated and the best one wll be selected(.e. the one whch mnmze the most the objectve functon of the frst subproblem). 14

18 In each of the possble soluton cases presented above, the necessary condtons for optmalty (28,29) must be calculated for fxed values of λ. But as the order quantty depends on the reorder pont and conversely, an teratve procedure s used to fnd the optmal value of the two unknown elements. The teratve procedure wll stop when the value for the varables from one teraton to the other s relatvely stable. In case of 2.(b), we must n addton fnd the value λ such that (35) s satsfed. Therefore, we need to update λ, resolve (28)-(29) untl (35) s satsfed. When the optmal soluton of the nventory model s obtaned, the optmal order and reorder quantty are used to solve the warehouse model. The resultng warehouse model (where the value of the nventory varables (Q, r for all = 1..I) are fxed based on the soluton of the nventory model wth one capacty constrant) s a mxed nteger problem whch s solved usng a Branch&Bound procedure. In the warehouse model, the two capacty constrants are taken nto account n order to obtan a feasble soluton to the warehouse model and the optmal value of the addtonal capacty (CapaSupp) s reoptmzed. The warehouse model s defned as follows: mn CostRepA x 1 J CostRepC x j uj j CostR z E(U ) Q P ckcostf E(p ) (x 1 y ) P ckcostr E(p ) z CostCapasupp Capasupp CostR x 1 E(U ) Q CostF y E(U ) Q under the followng constrants: 15

19 x j x j 1, j : j 2 J ( x j Q r µ L ) α j=1 ( Q 2 r µ L (x 1 z y ) = 1 CapaSupp 0 x j, y, z 0, j y CapaF ) ( ) Q z 2 r µ L x Integrated Heurstc method J α x j CapaR Capasupp Accordng to C.J. Vdal and M. Goetschalchx [15], global optmzaton for b-lnear problems s only possble for small nstances. Medum and large scale supply chan problems such as warehouse plannng and nventory management problems need to be solved through a heurstc approach. They propose a heurstc based on an teratve soluton of the two lnear subproblems. We use the same heurstc approach to solve our non lnear MIP model. We decompose the global model accordng to the dfferent varables and we solate two groups of varables: the nventory and the warehouse varables. Each of these groups of varables wll defne a subproblem. The frst subproblem wll be composed of the nventory varables and constrants and the values of the warehouse varables wll be fxed (by the value obtaned at the prevous teraton). Consequently, the orderng cost and the locaton of the products nsde the warehouse s fxed by the value of the warehouse varables. We obtan a classcal mult product nventory model wth two capacty constrants for each area n the warehouse (forward and reserve area). The objectve functon of the frst nventory subproblem s : j=1 16

20 ( ) Q mn CostCar 2 r µ L ( ) E(U ) CostAcqu Q Q ( ) E(U ) CostShort (d r ) f(d )dd Q CostR z E(U ) Q CostCapasupp Capasupp Under the constrants : J j=1 x j ( Q 2 r µ L Capasupp 0 Q, r 0, j r ( Q r µ L α CostR x 1 E(U ) Q ) y CapaF ) ( ) Q z 2 r µ L x 1 CostF y E(U ) Q J α x j CapaR Capasupp By dualzng the two capacty constrants wth multplers λ F (forward capacty constrant) and λ R (reserve capacty constrant) and the non-negatvty addtonal capacty constrant wth multpler µ, the lagrangan can be wrtten as followed: j=1 17

21 L(λ F, λ R, µ) = mn CostCar ( ) Q 2 r µl ( ) E(U) CostAcqu Q Q ( ) E(U) CostShort (d r ) f(d )dd Q CostR z E(U) Q r j=1 CostR x 1 E(U) Q CostCapasupp Capasupp [ [( J ) ( λ F Q r µ L (CapaF x j λ R (CapaR Capasupp µ Capasupp [ α ) y ]] ) CostF y E(U) Q [ ( ) ( ) Q 2 r Q µl z 2 r µl x 1 J α x j ]]) Frst of all, as we have done n the sequental heurstc soluton procedure, the necessary optmalty condton on the varable Capasupp and the complementary slackness and feasblty condtons on the addtonal capacty non-negatvty constrant can be used to derve the dfferent possble values of the varable Capasupp and the lagrangan multplers µ and λ R. j=1 CostCapasupp λ R µ = 0 (36) µ Capasupp = 0 (37) µ 0 (38) Capasupp 0 (39) From the above equatons, we can derve the followng: Capasupp 0 (40) (CostCapasupp λ R ) Capasupp = 0 (41) 0 λ R CostCapasupp (42) Ths system (41) and (42) gves the dfferent possble value for λ R (and consequently µ) and Capasupp whch solve the lagrangan. We can then wrte the frst order necessary optmalty condtons defnng the optmal order quantty and reorder pont for each product for a fxed value of λ R and λ F : 18

22 2 E(U ) (CostR z CostR x 1 CostF y CostShort E(U ) Q = r (d r ) f(d )dd 2 λf y α 2 λ R z 2 λ R x 1 CostCar (43) prob(d r ) = Q λf (CostCar y λ CostShort E(U ) α R z λ R x 1) (44) In order to solve (40) - (44), we must dstngush three possble stuatons: 1. λ R = 0 We obtan a lagrangan where one capacty constrant (forward capacty constrant) has been dualzed. Ths lagrangan can be solved by the same methodology used n the sequental heurstc procedure. 2. λ R > 0 (a) λ R = CostCapasupp As the value of λ R s fxed, we obtan agan a lagrangan wth one capacty constrant (forward capacty constrant) whch can be solved based on the sequental heurstc procedure. (b) 0 < λ R < CostCapasupp The optmal value of the lagrangan multplers (λ R, λ F ) wll be determned by lagrangan relaxaton where the two capacty constrants (forward and reserve constrants) wll be dualzed. The optmal value of the two lagrangan multplers wll be found by applyng the subgradent optmzaton algorthm on the lagrangan dual. The resultng lagrangan dual s as follows: W LD = Max { L(λ F, λ R ) : λ F, λ R 0 } Then, the optmal order quantty and reorder pont wll be calculated wth the system (43) and (44) for fxed values of λ R and λ F We have decded that, due to the non monotoncty of the objectve functon, we wll stop the subgradent optmzaton phase when we encounter ten teratons wthout an mprovement n the soluton obtaned. In the second subproblem, the warehouse varables wll be optmzed takng nto account the warehouse capacty, allocaton and the routng constrants whle the values of the nventory varables wll be fxed. The model obtaned s a mxed nteger problem where the reorder pont and the orderng quantty are fxed (at 19

23 the value obtaned at the prevous teraton). Ths problem s solved by Branch & Bound. Ths mxed nteger model s the same as the one used to solve the model wth the sequental heurstc procedure. Those two subproblems are solved teratvely (Fgure 1), where the output of one of the subproblem wll become the nput of the other subproblem at the next teraton. Fgure 1: The teratve procedure n the ntegrated model We decde to stop the teratve process after a lmted number of teratons where the best nventory and warehouse cost has been recorded at each teraton. Indeed, ths stoppng crteron s based on two facts. Frst of all, we want to keep a reasonable computng tme. Secondly, we have observed, n prelmnary test, that the mprovement n warehouse and nventory cost was occurrng n the frst steps of the teratve process. Ths procedure offers a hgher level of ntegraton of warehouse and nventory decsons because we do not only take nto account the sze of the warehouse n 20

24 the nventory model but also the sze of the dfferent areas n the warehouse, the routes taken by the products and the sze of the locaton nsde each areas of the warehouse. 5 Computatonal results In order to present the value of ntegratng warehouse and nventory decsons, the dfferent soluton methods are tested on a real world database. The dfferent methods tested dffer accordng to ther level of ntegraton. 5.1 Database The evaluaton of the heurstcs proposed n Secton 4 wll be realzed through tests whch wll be performed on 400 products comng from real world data of the retal sector 3. In order to be able to mplement the heurstcs, some nformaton are needed concernng the products. Concernng the demands of the products, we have assumed that pckng perods demand of product n the warehouse follows a normal dstrbuton N(µ, σ ). Ths assumpton was made because, n order to derve the demand probablty dstrbuton, we needed the hstogram of the expedtons per pckng perod and ths nformaton was not avalable. In addton, those products are delvered regularly whch supports the assumpton. Secondly, the nventory model that we have consdered s a contnuous revew, mult-tem reorder pont wth lost sales. Ths lost sales assumpton s made because, n case of shortage, an urgent order s expedted from another warehouse (so the order s lost for the warehouse under study) n order to obtan a 100% servce level. The dfferent costs composng the objectve functon were not avalable n the company. Those costs have been fxed accordng to the assumptons made n the model descrpton Benchmark methods The dfferent methods used and compared as benchmark are presented below n ncreasng level of ntegraton (from left to rght). The frst benchmark method s the one that most companes use today. In the frst step, an uncapacted nventory model s solved for each tem, where the orderng quantty and the reorder pont s calculated wthout takng nto account the sze of the warehouse, the sze of the dfferent areas n the warehouse and the routes of the products. The nventory decsons are taken ndependently of the warehouse decsons, based on orderng and nventory costs trade-offs. 3 for more nformaton on the dataset see the appendx 4 a senstvty analyss s performed n Secton 5.4 n order to analyse the mpact of changes n the objectve functon costs 21

25 Uncapacted Heurstc sequental Integrated method method method nventory Mult product Mult product Mult product models nventory control nventory control nventory control model wthout model wth model wth capacty constrant one capacty constrant two capacty constrants warehouse MILP MILP MILP models Table 1: Benchmark models Once the uncapacted mult tems nventory control model s solved, the warehouse model s solved based on the value of the nventory varables found. Ths warehouse model s the same as the one use for the sequental heurstc procedure and the ntegrated method. The other two benchmark methods have been presented n the Secton Descrpton of the computatonal results For the presentaton of the results obtaned wth the varous methods descrbed n Secton 5.2, we gve the dfference n cost (expressed n percentage) when we use a more ntegrated method. For ease of presentaton, we are gong to ntroduce two new defntons: - Frst ntegraton savngs The mprovement obtaned n the nventory and warehouse costs when we use, on the same dataset, wth the same capacty and storage polcy, the heurstc sequental procedure nstead of the uncapacted procedure. Ths mprovement wll be expressed n percentage. A postve value for the mprovement means an mprovement n cost (decrease n cost) when usng the sequental procedure nstead of the uncapacted procedure. - Second ntegraton savngs The mprovement obtaned n the nventory and warehouse costs when we use, on the same dataset, wth the same capacty and storage polcy, the ntegrated procedure nstead of the uncapacted procedure. Ths mprovement wll be expressed n percentage. A postve value for the mprovement means an mprovement n cost (decrease n cost) when usng the ntegrated procedure nstead of the uncapacted procedure. 22

26 5.3.1 Unlmted reserve capacty When there s unlmted capacty n the warehouse (unlmted reserve area, but lmted sze of the forward area), the sequental heurstc procedure and the uncapacted method gve, as expected, the same result for the nventory and the warehouse decsons. Indeed, the only dfference between the two models s that the sequental heurstc procedure takes nto account a global capacty constrant whch s redundant when there s enough capacty n the warehouse. We can conclude that the frst ntegraton gves 0% mprovement n cost. The only mprovement n warehouse and nventory cost s observed when the ntegrated model s used. In the ntegrated model, t s stll possble to optmze the amount of products whch can be stored n a locaton of the warehouse area and optmze the routes taken by the varous products. Consequently, the avalable space n the warehouse s used more approprately. Therefore, tables 2 and 3 show the results for the second ntegraton. Warehouse Cost Savngs (%) Replenshment Reserve Forward Total pckng pckng warehouse cost &recepton &recepton cost cost cost Second Integraton Table 2: Unlmted capacty warehouse results (expressed n percentage). Inventory Cost Savngs (%) Recepton Carryng Shortage Total nventory cost cost cost cost Second Integraton Table 3: Unlmted capacty nventory results(expressed n percentage). Wth the ntegrated model, the optmal order quanttes have ncreased compared wth the result obtaned wth the heurstc sequental procedure. Ths mples a decrease n the recepton cost and an ncrease n the storage cost. Concernng the shortage cost, an ncrease n the order quantty mples a decrease n the number of replenshment cycles. Nevertheless, ths decrease s compensated by a decrease n the safety factor whch ncreases the expected shortage per replenshment cycle. The decrease of the recepton cost can also be explaned through the reoptmza- 23

27 ton of the routes n the ntegrated procedure. The ncrease n the order quanttes mples that some products, whch were, wth the heurstc sequental procedure, suppled drectly to the forward area, must be suppled through the reserve area because there s not enough space anymore. Consequently, more product are pcked from the reserve area whch decreases the replenshment cost and decreases the forward recepton and pckng cost. The amount of space consumed n the reserve area has slghtly ncreased. Ths ncrease s manly due to the shft n routes. The forward capacty s used totally and better wth the ntegrated procedure because each locaton n ths area s flled totally Lmted warehouse capacty In the case of lmted warehouse capacty (n the forward and n the reserve area), savngs can be obtaned by usng the frst ntegraton and the second ntegraton. Warehouse Cost Savngs (%) Replenshment Reserve Forward addtonal Total pckng pckng capacty warehouse cost &recepton &recepton cost cost cost cost Frst ntegraton Second ntegraton Table 4: Lmted capacty warehouse results (expressed n percentage). Inventory Cost Savngs (%) Recepton Carryng Shortage Total nventory Cost Cost capacty Cost Cost Frst ntegraton Second ntegraton Table 5: Lmted capacty nventory results (expressed n percentage). 24

28 The mprovement obtaned by the frst ntegraton As there s lmted space n the reserve area, we can observe a reducton n the order quantty when the sequental heurstc procedure s used. We have shown that the global capacty constrant used n ths soluton method s assocated to a lagrangan multpler (see Secton 4.1) whch can be nterpreted as an mplct storage cost. Indeed, when there s not enough capacty n the warehouse, the carryng cost ncreases due to ths lagrange multpler (see equaton 28 n Secton 4.1) whch results n a decrease n the order quantty. Frstly, ths order quantty reducton mples hgher recepton and shortage cost. Nevertheless, ths ncrease n recepton and shortage cost s compensated by the decrease n carryng cost and mostly the decrease of addtonal capacty cost. Also observed that the carryng becomes so expensve (the lagrange multpler ncrease because of lmted capacty) that t s more nterestng to have hgher shortage cost (reducng servce level). Secondly, ths reducton n the order quantty mples that some product could be suppled drectly n the forward area (when the reducton n the order quantty allows t) nstead of beng suppled through the reserve area. Consequently, we can observe a change n the routes, whch combned wth the reducton n the order quantty, allows to decrease drastcally the cost of rentng addtonal capacty and replenshment cost. Concernng the pckng cost, there s an ncrease n the reserve pckng cost and a decrease n the forward pckng cost whch s due to the reoptmzaton of the routes. The reducton n the order quantty, when usng the heurstc sequental procedure, s drastc. Ths can be explaned by the type of relaxaton used for the nventory model (see 4.1). Indeed, the global capacty constrant used n ths method was based on a dedcated storage polcy whch s more restrctve than the real capacty constrants. Ths mples that the storage capacty s unadequatly used. The mprovement obtaned by the second ntegraton. Wth the ntegrated procedure, the quantty allocated to each locaton n the forward area s optmzed. Ths results n a decrease n the order quantty compared to the uncapacted method whch results n an ncrease n the recepton cost and shortage cost and a decrease n the carryng cost. Nevertheless, ths decrease n the order quantty s less mportant than the one obtaned wth the frst ntegraton because the capacty constrants are more adequately represented n the ntegrated method. Consequently, the avalable space n the warehouse s more adequately used. Concernng the routes taken by the varous products, ths change n the order quantty allows to fll n better the locatons of the forward area for certan products. For other product, ths ncrease n the ordered quantty does not justfy a locaton n the forward area anymore whch frees some place for other product whch fll n better the space. Ths optmzaton of space nvolves a change of routes whch mples that there s more drect recepton through the forward area and less recepton through the reserve area. Consequently, 25

29 the replenshment cost decreases compared to the uncapacted method (ntroduced n Secton 5.2) Value of ntegraton Lmted Reserve Capacty Unlmted Reserve Capacty Frst ntegraton Second ntegraton Table 6: Total savngs (n percentage) wth lmted and unlmted capacty. Table 6 summarzes the percentage of total savngs whch can be obtaned by ntegratng more decsons of the warehouse and nventory felds. The amount of savngs whch can be acheved depends on the capacty avalable n the warehouse. When there s enough capacty, only the ntegrated model allows one to mprove the nventory and warehouse costs due to a better management of the dfferent locatons n the forward area. Ths nvolves a reoptmzaton of the routes. Nevertheless, the amount of savngs realzed s relatvely small compared to the savngs obtaned n the case of lmted capacty. When the capacty avalable n the warehouse s lmted, the major part of the savngs s ncurred by usng the heurstc sequental procedure. Ths s manly due to the decrease n addtonal storage capacty rented. Wth the ntegrated method, space s utlzed better than when the heurstc sequental method s used due to the fact that the capacty constrant are better represented and taken nto account. 5.4 Senstvty analyss The computatonal tests have been performed wthout havng the real value of the dfferent cost coeffcents composng the objectve functon. In addton, n busness, changes n the objectve functon cost coeffcents can occur. For example, changes n the productvty of the pckers whch wll nfluence the pckng cost or changes n the carryng cost due to changes n the value of the product [12]. Therefore, entreprses may be nterested n knowng f those changes wll have an mpact on the optmal nventory and warehouse confguraton (.e optmal order quantty, reorder pont and optmal routes for each product). In order to answer to those questons, a senstvty analyss s acheved. Ths senstvty analyss wll be performed by testng varous scenaros under lmted 26

30 and unlmted reserve capacty 5. The scenaros wll dffer from one another by the varaton appled to the warehouse and nventory cost coeffcents. In order to have results whch can be nterpreted, the varaton appled to the dfferent cost coeffcents wll be no more than 20% Warehouse senstvty test The frst senstvty test concerns the mpact of changes n the warehouse cost coeffcents on the total savngs and on the warehouse and nventory confguraton. There exsts a dependency between the warehouse cost coeffcents. Therefore, the value of the warehouse objectve functon cost coeffcents have been fxed accordng to ther defnton and sgnfcance. The warehouse objectve functon s composed of forward and reserve recepton costs, forward and reserve pckng costs and advance and concurrent replenshment costs. We know that the forward pckng cost s lower than the reserve pckng cost. Indeed the forward area s a smaller area than the reserve area where the pckng actvty can be preformed faster (cheaper) than n the reserve area. On the other hand, the reserve recepton cost s lower than the forward recepton cost because the replenshment of the reserve area can be techncally acheved more easly than n the forward area (by larger szes or full pallets). Concernng the replenshment actvty, advance replenshment s less costly than concurrent replenshment because ths actvty does not result n creatng congeston. Those relatons are mportant otherwse t would not be nterestng to have a forward and a reserve area. In addton, the assumptons made on our model would not be vald.(see Secton 3.2 for more detals) Therefore, n the senstvty analyss, we are nterested n the mpact on the warehouse and nventory cost and on the nventory and warehouse confguraton of the relatve cost dfference between the forward and reserve recepton cost, the forward and reserve pckng cost and the advance and concurrent replenshment costs. Therefore, we have three relatve cost dfferences n our frst senstvty scenaro, each takng three possble values. For each of these twenty seven senstvty scenaros, we have decded to consder two extreme values. Table 7 gves the mnmum, average and maxmum total savngs, n percentage, obtaned wth the dfferent senstvty scenaros (.e by varyng the value of the warehouse objectve functon cost coeffcents) and wth dfferent capacty lmts. Most of the dfference n the total savngs (except n the case of the second ntegraton wth lmted capacty) are relatvely stable wth no more than 1% dfference between the maxmum and mnmum total savngs for a certan capacty lmt and a certan type of ntegraton. By analyzng n more detals the results obtaned wth the dfferent senstvty scenaros, we observe that the optmal order quantty and reorder pont do not change from one scenaro to the other. Only n the case of lmted capacty, we can observe that the routes taken by the varous products 5 The appendx gves a complete descrpton of the dfferent scenaros 27