An Efficient Model for Allocating Products and Designing a Warehouse

Sunderesh S. Heragu, Jason C.S. HuangDecision Sciences and Engineering Systems Department

Rensselaer Polytechnic Institute

and

Ronald J. Mantel and Peter C. SchuurUniversity of Twente

Enschede, The Netherlands

Abstract

The two primary functions of warehouse include (i) temporary storage and protectionof goods, and (ii) providing value added services such as fulfillment of individualcustomer orders, packaging of goods, after sales services, repairs, testing, inspection, andassembly. To perform the above functions, the warehouse is divided into severalfunctional areas such as reserve storage area, forward (order collation) area, and cross-docking. In this paper, we present a mathematical model and a heuristic algorithm thatjointly determines product allocation to the functional areas in the warehouse as well asthe size of each area using data readily available to a warehouse manager.

1 IntroductionA warehouse is generally divided into the following areas to perform the above functions– reserve storage area, forward area, and cross-docking area. The reserve area is wheregoods are held until they are required for shipment to the customer or for performingvalue added services or order collation. The latter is typically done in the forward area.The forward area could also be used to store fast movers that do not occupy much space.Cross-docking refers to the process in which items, cartons, or pallet loads are takendirectly from the receiving trucks to the shipping trucks. It provides a fast product flowand reduces or eliminates the costs associated with handling and holding inventory.Typically, conveyors or fork-lift trucks are used to transfer materials from receiving toshipping in the cross-docking area.

After determining the warehouse location, number and size, the designer may want todetermine what storage areas are to be included and the size of each, so that anappropriate material handling system can be selected and the warehouse laid out.Although determining the size of each functional area is a strategic level problem, itdepends upon another tactical level problem, namely, how the products will bedistributed among the functional areas. The latter is the product allocation problem. Thus,a joint solution of the functional area size determination and product allocation problems

is desirable. However, the general approach undertaken by practitioners is to solve thetwo problems sequentially by generating multiple alternatives for the functional area sizeproblem and then determine how the products can be allocated for each of thesealternatives. In this paper, we develop a higher level model that jointly determines thefunctional area sizes and the product allocation in a way that minimizes the total materialhandling cost.

2 Brief Literature ReviewAshayeri and Gelders [1] present a study that compares two different approaches,analytical and simulation, for warehouse design problems. They conclude that, in general,neither a pure analytical approach, nor an approach that uses only simulation, will lead toa practical design method. They suggest a combination of the two approaches is likely tolead to a good design method. Rouwenhorst et al. [9] present a reference framework anda classification of warehouse design and control problems. An extensive review of theliterature is also presented. They conclude that a majority of the papers is analysisoriented and provide some guidelines toward a more design oriented research approachfor warehousing problems. There are some publications concerning the structure of thewarehouse design methods. Gray et al. [4] propose a hierarchical design method anddescribe the application of their method by an example design. Bozer [2] considers theproblem of splitting a pallet rack into an upper reserve and lower forward area. Hackmanand Rosenblatt [5] present a model for forward/reserve problem that considers bothassignment (which product) and allocation (what amounts). They provide a heuristic thatattempts to minimize the total costs for picking and replenishment. Frazelle et al. [3]provide a framework for determining the size of forward area together with the allocationof products to that area. Hackman and Platzman [6] present a mathematical programmingprocedure that solves more general instances of the assignment and allocation problemsin a warehouse. Van den Berg et al., [10] consider a warehouse with busy and idle periodwhere reserve picking is possible.

3 Model for Product Allocation and Preliminary Warehouse DesignIn this paper, we consider warehouse configurations that include a subset of the followingfive functional areas: receiving, shipping, staging for cross-docking operation, reserve,and forward. In the receiving area, pallet loads or individual cartons of products arereceived. If necessary, they are staged for a short period time, and then either moved tothe shipping area directly (cross-docking operation) or to the storage area. In the shippingarea, picked order items are readied (e.g., shrink-wrapped, packed) and staged (ifnecessary), for shipping to the next destination. In the staging area for cross-docking,products are sorted and accumulated for further outbound operations. Reserve area is astorage area for bulky product items that typically reside in the warehouse for a relativelylonger duration. Normally the reserve area uses high-density storage equipment toachieve the goal of high space utilization. Forward area is a relatively smaller storage

area typically used for fast order picking or performing value-added operations or ordercollation. Thus, the following material flows are possible in a warehouse (see Figure 1):

1. Flow 1: Receiving Cross-docking Shipping2. Flow 2: Receiving Reserve area Shipping3. Flow 3: Receiving Reserve area Forward area Shipping4. Flow 4: Receiving Forward area Shipping

Figure 1 Typical product flows in a warehouse

Flow 1 is the cross-docking operation. Upon receipt, products items are either put intoa staging area for a short period of time and then moved to the shipping area, or directlymoved to the shipping area. The received products are typically presorted at suppliers’facilities. The operation here is simply to pass on the product to a customer or the nextfacility in the supply chain.

Flow 2 is a typical warehouse operation. Products are stored in reserve area andorder-picking operation is performed as required. It is assumed that typically, only thoseitems that remain in the warehouse for relatively extended periods of time and shipped asis (or with minimal value added operations) will be allocated to the reserve storage area.Flow 3 is also a typical warehouse operation. Products are first stored in reserve areatypically in pallet loads, broken into smaller loads (cartons or cases) and then moved tothe forward area for fast order picking, order consolidation or performing value addedoperations. Flow 4 can be thought of as another form of cross-docking operation.Products are received and then are directly put into forward area to perform the orderconsolidation. This type of operation is usually seen in the supplier warehouses or whenthere is a need to consolidate large orders.

In formulating the model, the following notation is used:Parameters i : Number of products i = 1, 2, …, n

Receiving

Reserve

Forward

Shipping

Flow 1 (Cross Docking)

Flow 4

Flow 2

Flow 3

j : Type of material flow; j=1,2,3,4

iλ : Annual demand rate of product i in unit loadspi : Average percentage of time a unit load of product i spends in reserve area if productis assigned to material flow 3Hij : Cost of handling a unit load of product i in material flow jCij : Cost of storing a unit load of product i in material flow j per year

iS : Space required for storing a unit load of product iTS : Total available storage spaceQi : Order quantity for product i (in unit loads)Ti: Dwell time (in years) per unit load of product i

CDCD ULLL , : Lower and upper storage space limit for cross-docking areaLLF ,ULF : Lower and upper storage space limit for forward areaLLR ,ULR : Lower and upper storage space limit for reserve area

Decision VariablesijX = 1 if product i is assigned to flow type j ; 0 otherwise

γβα ,, : Proportion of available space assigned to each functional area, α - crossdocking, β - reserve, γ - forward

Model

Minimize4

1 1

n

ij i iji j

H Xλ= =∑∑ + ( )

4

1 1/ 2

n

ij i iji j

C Q X= =∑∑ (1)

14

1=∑

=jijX i∀ (2)

( )11

/ 2n

i i ii

Q S X TSα=

≤∑ (3)

( ) ( )2 31 1

/ 2n n

i i i i i i ii i

Q S X p Q S X TSβ= =

+ ≤∑ ∑ (4)

( ) ( )3 41 1

(1 ) / 2 / 2n n

i i i i i i ii i

p Q S X Q S X TSγ= =

− + ≤∑ ∑ (5)

1=++ γβα (6)

CD CDLL TS ULα< < (7)

R RLL TS ULβ< < (8)

F FLL TS ULγ< < (9)α,β,γ ≥ 0 (10)Xij = 0or1 ∀i, j (11)

The objective function (1) minimizes the total cost of handling the average, annualloads of each product assigned to its respective area as well as the corresponding annualstorage costs. The reader should not confuse storage costs with inventory holding costs.While inventory holding costs depend only upon the value of the inventory, they are thesame whether the inventory is in reserve or forward or cross-docking area. Storing costs,on the other hand, depend upon the area in which the product is stored and these coststend to carry a premium for the cross-docking and forward areas (because these areconsidered prime real estate in a warehouse) and are relatively not that expensive for thereserve area. Of course, the handling costs are different (in fact, the opposite) for theseareas and thus our model trades off storage costs against handling costs. Note that Xij

tells us whether or not product i is assigned to flow j, and QiXij /2 gives us the averagenumber of the corresponding unit loads in inventory.

Constraint (2) ensures that each product is assigned to only one type of material flow.If the same product could be allocated to multiple flows due to different demand patterns,then our model requires that the manager at least knows or is able to estimate thepercentage of this product that could be assigned to two or more of the four flows.Constraints (3)-(5) ensure that the space constraints for the cross-docking, reserve, andforward areas are met. The right hand side includes three additional variables whose sumis required to be 1 (constraint 6). This is to ensure 100 percent of the space available isallocated to the three areas. Constraints (7)-(9) serve to enforce upper and lower limits onthe space that can be allocated to cross-docking, forward and reserve areas.

A more detailed version of the above model and two efficient heuristics to solve it,along with extensive computational details for very large datasets, e.g., 150,000 products,are presented in Heragu et al. [8].

4 Heuristic AlgorithmAlthough large instances of the warehouse design model presented in the previous sectioncan be solved directly using an available branch-and-bound based algorithm for mixed-integer programming problems (see tables 3 and 4 in section 6), when the problem isseverely constrained so that there are a limited number of feasible solutions, or when thenumber of products is in the hundreds of thousands, the number of binary integervariables increases considerably and solving the resulting model takes significantcomputational time. Hence, we propose below an efficient heuristic for solving themodel.Heuristic Algorithm:Step 1: For each i=1,2,…,n, find Min

j=1,2,3,42qijHijλi + qijCijQi /2( ).

Let Minj=1,2,3,4

2qijHijλi + qijCijQi /2( )occur for j=j*.

Set Xij*=1, remaining Xik = 0, k =1,2,3,4, k ≠ j *.

Step 2: If the current assignment violates one or more of the space constraints (3)-(5),examine each area j=1,2 and 4 for which the space constraint is violated, one at atime. For each i such that Xij=1 and k ≠ j, find:∆ ik = 2qijHijλi + qijCijQi /2( )− 2qikHikλi + qikCikQi /2( ).Sort the ∆ ik values in non-increasing order and consider the smallest sorted∆ ikvalue. Let it occur for i=i* and k=k*. If assigning product i* to area k* doesnot violate space constraints for k*, set Xi*k*=1 and Xi*l=0,∆ i*l = +∞, l =1,2,3,4 l ≠ k *. Otherwise, find the next smallest sorted ∆ ik value andrepeat step 2 until constraints (3)-(5) are satisfied.

Step 3: For the feasible solution obtained in step 2, apply simulated annealing to improvethe solution.

Three aspects of the above algorithm are worth a discussion. First, it is easy to showthat the solution obtained in step 1 is optimal for the model in section 4 excluding thespace constraints (3)-(10). Thus, it provides a very tight lower bound on the objectivefunction value for the model specified by constraints (1)-(11). It is thus no surprise thatthe general purpose branch and bound algorithm in LINDO is able to solve problems with3000 products. Second, no feasible solution may be available at the end of step 2. Thiscould be because the problem itself has no feasible solution, or because the changing ofone assignment at a time, may not examine all feasible solutions. Hence, in ourimplementation, we actually modify step 2 to consider changing the assignment of pairsof products at a time, followed by three products at a time, then four products at a time,etc., until a feasible solution is obtained. Third, the simulated annealing algorithm used instep 3 uses the procedure outlined in Heragu [7]. The feasible solution at the end of step 3can be represented as a binary matrix [Xij] where a 1 (0) indicates product i is (not)assigned to flow j. The simulated annealing algorithm systematically considers swappinga product currently assigned to flow j to another flow as well as swapping pairs ofassignments provided such an exchange yields a feasible solution. For example, ifproduct i is assigned to flow j and p is assigned to flow q, the algorithm considersassigning product i to flow q and p to flow j.

5 Numerical ExampleWe use the model presented in section 3 to solve the following simple numericalexample. A warehouse handles six classes of products. The average annual demand,order cost, price and space required per unit load for each class of product is shown inTable 1. Assuming a carrying cost rate of 10%, total available space of 100,000 squarefeet, determine how the products should be allocated to each of the four areas. Assumethe handling and storage costs given in Table 2, an upper and lower bound of 75,000 and35,000 square feet for reserve and forward areas and a maximum of 15,000 square feetfor the cross docking area. Also, assume, pi =0.2 and 1 for products 3 and 6, 0 for allothers.

Table 1. Product specific input data for numerical exampleProducts 1 2 3 4 5 6

Annual Demand 10000 15000 25000 2000 1500 95000Order Cost 50 50 50 50 50 150

Price / Unit load 500 650 350 250 225 150Space required 10 15 25 10 12 13

pi 0 0 0.2 0 0 1Order Quantity 141.42 151.91 267.26 89.44 81.65 1378.41

Using the price, demand, order cost and carrying cost rate, it is easy to determine theorder quantity in table 1 using the EOQ formula. It is also easy to calculate the dwell timeper unit load of each product using the Ti formula in section 3. For a given product, weassume the dwell time is the same irrespective of the flow to which it is assigned. Afterall, the dwell time is a function of the product demand and order quantity. Based on theabove and the costs given in table 2, it is easy to setup the model developed in section 4.Solution of the model yields an assignment of products 2 and 5 to flow 1 (cross-docking),products 1 and 6 to flow 2 (reserve), products 3 and 4 to flows 3 and 4, respectively, at anannual handling and storage cost of $26,300. Because of the lower handling and storagecosts in the reserve area, 98.5 percent of the available space is assigned to this area.

Table 2. Handling and storage costs for each area-product combinationArea/Product 1 2 3 4 5 6

1 0.0707(20)

0.0203(15)

0.0267(4)

0.3354(5)

0.4083(15)

0.0726(20)

2 0.0849(5)

0.2023(5)

0.0428(20)

0.559(4)

0.6804(25)

0.0871(5)

3 0.1061(10)

0.2023(10)

0.0054(1)

1.0062(5)

1.2248(45)

0.1088(10)

4 0.0778(15)

0.2023(10)

0.0481(9)

0.0671(1)

0.8165(30)

0.0798(15)

7 Experimental ResultsTable 3 shows some solution details (CPU time, integer variables, pivots and branches)explored by the branch-and-bound algorithm in LINGO version 6 – a commercial branchand bound based, mixed, integer programming solver. LINGO was used to solve themodel directly for four problem instances. Table 4 provides further details concerningsolution quality. In particular, note that the gap between the lower bound (obtained bysolving an LP relaxation of the model in section 4) and the optimal value is insignificantindicating the power of the model. Also shown in table 4 are the α, β, γ values indicatingthe percentage of space allocated to cross-docking, reserve and forward areas,respectively.

Table 3. Solution details for four problemsNumber ofProducts

CPU Run time(Hrs:Min:Sec)

Number ofInteger variables Branch Pivots

100 0:0:0.24 400 0 142500 0:1:25.24 2000 373 20446

1000 0:0:32.24 4000 18 12243000 0:17:8.54 12000 422 19265

Table 4. Lower bound (LB) and optimal objective function values (OFVs) for fourproblems - 0.2 < α<=0.5, 0<β <1,0.1<γ<0.3

Number ofProducts LB Optimal

OFV α β γ

100 1,310,302 1,310,302 0.3254 0.418047 0.256553500 6,312,950 6,313,062 0.265674 0.434319 0.3000071000 12,904,700 12,904,750 0.282854 0.417169 0.2999773000 37,172,464 37,174,840 0.271169 0.428842 0.299989

Tables 5 and 6 include experimental results that demonstrate efficiency of theheuristic algorithm. They consider problems varying in size from 6 to 30 classes ofproducts, the total available warehouse space varies from 15,200 to 247,100 square feet.The α, β, γ values obtained in the optimal solution are also shown in the two tables. LB,Initial, Search and Optimal in tables 5 and 6 refer to the lower bound solution provided atthe end of step 1, the modified feasible solution available at the end of step 2, the solutionprovided by the search algorithm at the end of step 3 and the optimal solution obtained bysolving the model using LINGO.

Table 5. Performance of Heuristic Algorithm for 0.1<α < 0.2, 0.5 < β <1, 0.1< γ < 0.2Number ofProducts 6 10 15 20

α 0.1366 0.1050 0.1441 0.1362β 0.6799 0.7533 0.6954 0.6644γ 0.1835 0.1417 0.1605 0.1995

Total Space 15,200 64,700 119,900 130,200LB 26,335 52,722 150,153 167,673

Initial 27,587 60,029 154,241 179,193Search 27,587 60,029 154,241 174,796Optimal 27,587 58,324 154,241 173,012IP Gap 4.8% 10.6% 28.6% 3.2%

Heuristic Gap 0% 2.9% 0% 1%

The IP gap in the two tables corresponds to the gap between the lower bound andoptimal OFV and the heuristic gap corresponds to the difference in the OFVs of theoptimal solution and that obtained by the heuristic algorithm. Table 6 includes two more

problems in addition to those considered in table 5. The constraints on α, β, γ values aredifferent from those in table 5. For both data sets, the IP and heuristic gaps are relativelysmall (except for the third problem in table 5) confirming that the model and the heuristicare efficient.

Table 7 is included to show the power of the algorithm in step 1 of the heuristicalgorithm described in the previous section. Note that even for problems containing 50product classes, the lower bound is quite close to the optimal objective function value.

Table 6. Performance of Heuristic Algorithm for 0.2 <α < 0.3, 0.5 < β < 0.8, 0.1< γ < 0.2Number ofProducts 6 10 15 20 25 30

α 0.2947 0.2671 0.2096 0.2565 0.2219 0.2775β 0.5971 0.5718 0.6289 0.6429 0.6704 0.5502γ 0.1082 0.1611 0.1615 0.1006 0.1077 0.1724

Total Space 15,200 64,700 119,900 130,200 163,700 247,100LB 26,335 52,722 150,153 167,673 248,725 313,032

Initial 32,894 55,597 155,310 179,193 253,887 317,962Search 32,894 55,597 155,310 174,796 253,887 317,962Optimal 30,208 55,597 155,310 173,209 253,887 316,962IP Gap 14.7% 5.5% 3.4% 3.3% 2.1% 1.9%

HeuristicGap 8.9% 0.92% 0% 0.9% 0% 0.3%

Table 7. Comparison of LB with Optimal OFVs for four additional problemsNumber ofProducts 35 40 45 50

α 0.2721 0.2850 0.3000 0.2849β 0.5348 0.5256 0.5157 0.5278γ 0.1931 0.1895 0.1843 0.1874

Total Space 295,200 348,100 406,500 456,200LB 329,233 354,523 415,097 522,741

Optimal 333,162 358,452 418,800 529,196IP Gap 1.2% 1.1% 0.9% 1.2%

8 ConclusionsIn this paper, we have simultaneously considered the product allocation and functionalarea size determination problems in the design of a warehouse. We provide amathematical model and a heuristic algorithm to solve the two problems jointly so thatannual handing and storage costs can be minimized. The input data requirement for thismodel is readily available in most warehouses and the model considers realisticconstraints. We believe this is the only available model that considers the two problemssimultaneously and allows the user to solve them optimally.

9 References

1. Ashayeri, J. and Gelders, L.F., Warehouse design optimization, 1985, EuropeanJournal of Operational Research, 21, 285-294.

2. Bozer, Y.A., Optimizing throughput performance in designing order picking systems,1985, PhD thesis, Georgia Institute of Technology, Atlanta, GA, USA.

3. Frazelle, E.H., Hackman, S.T., Passy, U., and Platzman, L.K., The forward-reserveproblem, 1994, in Ciriani, T.A. and Leachman, R.C. (Eds), Optimization in Industry2, Wiley, 43-61.

4. Gray, A.E., Karmarkar, U.S. and Seidmann A., Design and operation of am orderconsolidation warehouse: Models and application, 1992, European Journal ofOperational Research, 58, 14-36.

5. Hackman, S.T. and Rosenblatt, M.J., Allocating items to an automated storage andretrieval system, 1990, IIE Transactions, 22(1), 7-14.

6. Hackman, S.T. and Platzman, L.K., Near-optimal solution of generalized resourceallocation problems with large capacities, 1990, Operations Research, 38(5), 902-910.

7. Heragu, S.S., Facilities Design, 1997, PWS Publihing Co., Boston, MA.8. Heragu, S.S., Huang, C-S., Mantel, R.M., and Schuur, P.M., A Mathematical Model

for Warehouse Design and Product Allocation, DSES Technical Report 38-03-502,2003, Rensselaer Polytechnic Institue, Troy, NY 12180.

9. Rouwenhorst, B. Reuter, B., Stockrahm, V., van Houtum, G.J., Mantel, R.J., andZijm, W.H.M., Warehouse design and control: Framework and literatures review,2000, European Journal of Operational Research, 122, 515-533.

10. van den Berg, J.P., Sharp, G.P., Gademann, A.J.R.M., and Pochet, Y., Forward-reserve allocation in a warehouse with unit load replenishment, 1998, EuropeanJournal of Operational Research, 111, 98-113.

10 AcknowledgementResearch presented in this paper has been supported in part by NSF grant #DMI-9900039. We gratefully acknowledge their support.