an optimisation model and its effective beam search heuristics for floor-storage warehousing systems

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An optimisation model and its effectivebeam search heuristics for floor-storage warehousing systemsT. Nishi a & M. Konishi ba Graduate School of Engineering Science, Osaka University ,Osaka 560-8531, Japanb The Graduate School of Natural Science and Technology,Okayama University , Okayama 700-8530, JapanPublished online: 10 Feb 2009.

To cite this article: T. Nishi & M. Konishi (2010) An optimisation model and its effective beamsearch heuristics for floor-storage warehousing systems, International Journal of ProductionResearch, 48:7, 1947-1966, DOI: 10.1080/00207540802603767

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International Journal of Production ResearchVol. 48, No. 7, 1 April 2010, 1947–1966

An optimisation model and its effective beam search heuristics

for floor-storage warehousing systems

T. Nishia* and M. Konishib

aGraduate School of Engineering Science, Osaka University, Osaka560-8531, Japan; bThe Graduate School of Natural Science andTechnology, Okayama University, Okayama 700-8530, Japan

(Received 23 August 2007; final version received 1 November 2008)

This paper presents an optimisation model and its effective solution techniqueusing beam search heuristic for floor-storage warehousing systems. For a floor-storage system, storage can be accessed from the top of stacks only. The objectiveis to minimise the number of re-handling operations by optimally determining thestorage location and by grouping products for each customer that fit a givensequence for receiving and retrieving operations. An integer programming modelis formulated and an approximate solution technique based on the beam searchmethod is proposed to solve the problem by incorporating effective heuristicsto reduce the search space using future receiving and retrieving requests.The effectiveness of the proposed method is demonstrated for industrialwarehousing problems in a steel plant with 58 storage areas involving morethan 3000 retrieving operations. The proposed solution method is shown to bemore efficient than the traditional branch-and-bound method for solving integerprogramming problems.

Keywords: warehousing; floor-storage systems; storage allocation; beam search;optimisation; integer programming problem

1. Introduction

Warehousing systems are one of the crucial components of the manufacturing anddistribution companies in a supply chain. It is essential to improve the delivery speedreducing inventories in warehouses in order to achieve agile and lean manufacturing(Amato et al. 2002, Croom et al. 2002, Miller 2001). The modeling and optimisation ofwarehousing systems has received much attention with respect to reducing distributioncosts in many industries. An extensive review of warehousing design and control strategiesis provided by Rouwenhorst et al. (2000). Models for warehousing system have beenintroduced by Van den Berg et al. (1999) and Gu et al. (2007).

In floor-storage warehousing systems, the storages are stacked accumulatively on topof other products when the storage area is limited. It is therefore necessary to re-handlestorage efficiently within the floor-storage area in order to retrieve a product fromunderneath other products in the storage area when a product cannot be retrieved from thetop of the stack in the floor-storage area. In addition, products for a customer must beretrieved and gathered together before shipping.

*Corresponding author. Email: [email protected]

ISSN 0020–7543 print/ISSN 1366–588X online

� 2010 Taylor & Francis

DOI: 10.1080/00207540802603767

http://www.informaworld.com

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As a result, it is very difficult to generate a feasible storage allocation plan to efficientlyretrieve products in warehousing operations. A closely related problem is the containermarshalling operation for a container ship where one container is stacked on top of othersto save storage space in the container yard. Containers in the yard can be accessed onlyfrom the top of a stack. Re-handling is needed if the targeted container is not located at thetop. In a container yard, re-marshalling means re-positioning of the exporting containersbefore the loading starts so that the containers can be loaded as fast as possible.

However, receiving/retrieving operations are not considered in the re-marshallingprocess. The final layout is specified for the input pallet and the output pallet only forfloor-storage systems. The number of products handled in floor-storage warehousing ismuch larger than that in the marshaling process in container yards. The main importanceof re-handling for floor-storage warehousing systems is to aggregate storages for the samecustomer for each storage location to improve grouping performance before the shipmentof products.

The optimisation problem treated in this paper is to optimise the storage locationwithin the floor-storage system and reduce the number of operations for rearrangingproducts so that the total number of warehousing operations is minimised. Warehouseplanning will facilitate the efficient accumulation of products to be shipped to a customerwhen the sequence of requests for receiving and/or retrieving operations is given.

Computational simulation-based approaches and/or optimisation approaches havebeen addressed for warehousing models (Azadivar 1984, 1986). Optimisation models forwarehousing systems have been proposed to optimise the total quantities of products inwarehousing to reduce both the inventory and ordering costs for a limited storage area(Jolayemi and Olorunniwo 2004). One popular approach to warehouse storage allocationis to divide warehouse storage areas into a forward area and a reserved area (Van den Berget al. 1998). The forward area is used for order-picked products and the reserved area isprepared for storage of received products. The storage allocation assignment problem is toallocate incoming products to suitable storage locations. Class-based storage heuristics areoften used to determine storage location according to demand quantity. For marshallingprocesses in container yards, a dynamic programming model has been developed tooptimise storage location in order to minimise the number of relocation movements (Kimet al. 2000). A storage space allocation problem, including quay cranes, yard cranes,storage space, and internal trucks, to minimise the total distance to transport containers,was studied by Zhang et al. (2003). A mathematical model for the re-marshalling problem,to determine a plan to relocate containers in a yard to reduce the need of re-handling insubsequent loading operations, was developed by Lee and Hsu (2006). However, theconventional warehousing model was created simply to determine the allocation ofproducts to storage locations when the demand is known. Thus, it cannot be applied toindustrial warehousing operational storage planning problems, which should take intoaccount the sequencing of operations for the movement of each product and the batchingof products for each customer to make the warehousing operation more efficient (Nishiet al. 2004).

In this paper, an optimisation model for solving warehousing operational storageplanning problems is developed. The objective of warehousing activities is not only toreduce the number of operations for receiving and retrieving, but also to minimise theoperations required to aggregate into group products for each customer. The efficiency ofretrieving and grouping of products for a customer is expressed by a new performanceindex proposed in this paper. This grouping performance takes into account the

1948 T. Nishi and M. Konishi

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operations required to re-handle products stacked in storage areas so that the storagesfor a customer may be aggregated within the warehouse before shipment. Thewarehouse treated in this paper comprises several storage locations that are accessedand serviced by an automated overhead crane. The products received at the warehouse areallocated to storage locations, transported, and stacked on top of each other in eachstorage location up to the permissible height. In each storage location, the pickingoperation is last-in-first-out and only those products at the top of a stack can be retrievedfirst. If the products in the middle or at the bottom of the stack are demandedfrom a storage location, then products at the top of the stack must be removed andmoved to another location. The required products are then picked and transported to theshipping area.

The warehousing operational planning problem is formulated as an integerprogramming problem for optimisation. The size of the problem, in terms of thenumber of variables, increases exponentially with increasing number of products, numberof storage areas, and the planning horizon. To solve the problem efficiently, the beamsearch method is used. Beam search is a heuristic method for solving a wide range ofoptimisation problems. Many applications of this method have been reported in literature.Sabuncuogly and Bayiz (1999) studied its performance for solving job-shop schedulingproblems. However, the application of the beam search method to solve warehousingproblems has not been reported in the literature. This paper proposes an effective beamsearch method combined with a heuristic rule to solve warehousing storage allocationproblems of practical size for an industrial steel plant.

The paper is organised as follows. In Section 2, a warehouse operational storageplanning model is developed as an integer programming problem. In Section 3, an effectivebeam search method is proposed for solving storage allocation planning problems, tominimise both the number of operations and the grouping performance index. Theheuristics to reduce the search space for the proposed beam search method are alsoproposed. In Section 4, the performance of the proposed method is compared with that ofthe CPLEX solver. The proposed method is applied to solve a practical large-sizedwarehousing problem using actual data for an industrial steel plant. The effectiveness ofthe proposed method for large-sized problems is demonstrated. Section 5 concludes thepaper and mentions future work.

2. Warehousing storage allocation planning problem

In this section, the warehousing storage allocation planning problem is stated and anoptimisation model of an integer programming problem is developed.

2.1 Problem statement

Figure 1 shows a model of a floor-storage warehousing system. The warehouse consistsof several storage areas and an overhead crane. Different products for differentcustomers are stored in each storage area. Hereafter we call one item of a product fora customer a ‘product’. The warehousing operation is a dynamic situation where requestsfor receiving/retrieving are given dynamically. However, if the correct requestinformation is available in advance, the problem can be considered as a static problem.The static problem is solved each time when the request information is changed for

International Journal of Production Research 1949

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a dynamic situation. The following conditions are assumed for formulating the problem ofwarehouse operational planning as an integer programming problem.

(i) The product receiving and retrieving sequence of requests is specified for theplanning period. The initial allocation of products in the warehouse is known.Each product in the warehouse is associated with a customer for shipping fromthe warehouse.

(ii) The transport of a product can be performed only from the top of a stack ofproducts.

(iii) The total planning horizon is divided into multiple time periods. One time periodis defined by the time required for handling a product by an automated overheadcrane in the warehouse. However, multiple handling of products at a time can beperformed for the industrial problem presented in Section 4.2.

(iv) The crane can transport a product from a storage area to any other storage area.However, once the products are taken to the pallet for shipping, these productscannot be returned to the storage area.

For floor-storage systems handling large-sized products, the shipping time greatlydepends on the number of operations for all receiving and retrieving requests. The traveling,picking, and dropping time are sufficiently short with respect to the operational time for there-handling operation. Therefore, the objective of the warehousing operational planningproblem is to optimise storage allocation planning to minimise the number of operationsrequired to complete all receiving and retrieving requests and to maximise the groupingperformance. For practical warehousing, the products for a customer should be collected ina storage area even when the timing and allocation of receiving and/or retrieving productsare different. A product-group is a series of products for the same customer. The products ina stack at a storage area are reordered into a product-group so that products for the samecustomer can easily be checked and retrieved. In this problem formulation, we assume thatonly a product can be transported by an overhead crane, but a series of products for thesame customer can be retrieved in one operation.

2.2 Model development

The following notation is used for the formulation.

Parameters and sets

P set of products to be handled in warehousingM set of storage areas including input pallet and output pallet in the

warehouse

2 nd.3 rd.

1 st.

area 0 area 1 area 2 area 3 area M (output pallet)

(i) Transport a product from pallet to height 6 in area 2

(ii) Retrieve a product from height 4 in area M

(iii) Transport a product from height 3 in area 4 to height 2 in area M

Outputsequence

2 nd.3 rd.

1 st.

(input pallet)---

Inputsequence

Figure 1. Warehousing model.


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H set of locations in the storage areaGh,m product number at height h in the stack of storage area m in the final

layout

Decision variables

xm,h,t,p binary variable that takes the value 1 if product p ( p2P) is allocated

to storage area m (m2M) at height of location h (h2H ) in time period

t, and 0 otherwisezm,h,t binary variable that takes the value 1 if a product at height h of the

location is at the top of the stack in a storage area where the product

can be retrieved to the pallet, and 0 otherwisewm,h,t,p binary variable that takes the value 1 if product p is transported to

height h of the location in storage area m in time period t, and 0

otherwisevm,h,t,p binary variable that takes the value 1 if product p is retrieved from

height h of the location in storage area m in time period t, and 0

otherwise�t binary variable that takes the value 0 if the storage allocation is the

same as the specified final layout, and 1 otherwise

Constraint (1) states that at most one product can be allocated at height h of the

locations of storage area m in the same period:

Xp2P

xm,h,t,p � 1 ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ:ð1Þ

If the receiving/retrieving operation is regarded as a movement of products from

the input pallet to a storage area/from a storage area to the output pallet, the total

quantity of products for all storage areas including input and output pallets is

constant during all time periods. Constraint (2) states the products flow conservation

constraint:

Xp2P

Xm2M

Xh2H

xm,h,t�1,p ¼Xp2P

Xm2M

Xh2H

xm,h,t,p ð8t ¼ 1, . . . ,T Þ:ð2Þ

All storages must be placed at the bottom of a stack or on top of another area. Then

Xp

xm,h�1,t,p �Xp

xm,h,t,p ð8m 2M, 8h 2 Hnf1g, 8t ¼ 1, . . . ,T Þ: ð3Þ

The variable zm,h,t represents the situation where a product at height h in time

period t is at the top of the stack in storage area m. The variable zm,h,t satisfies the

following condition. Equation (A) implies that if there is a product p at height h and

there are no products at height hþ 1 in time period t, the variable zm,h,t¼ 1, and zm,h,t¼ 0

otherwise:

Xpxm,h,t,p ¼ 1

� �^

Xpxm,hþ1,t,p ¼ 0

� �) zm,h,t ¼ 1, otherwise zm,h,t ¼ 0: ðAÞ


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The logical constraint (A) can be rewritten as

�Xp2P

xm,h,t,p þ zm,h,t � 0 ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ,ð4Þ

Xp2P

xm,hþ1,t,p þ zm,h,t � 1 ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ,ð5Þ

Xp2P

xm,h,t,p �Xp2P

xm,hþ1,t,p � zm,h,t � 0 ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ:ð6Þ

The binary variable wm,h,t,p is introduced to represent the situation when product

p is transported to height h of the location in storage area m in time period t. wm,h,t,p

satisfies

ðxm,h,t,p � xm,h,t�1,p � 1Þ ) wm,h,t,p ¼ 1, and wm,h,t,p ¼ 0 otherwise: ðBÞ

The logical constraint (B) can also be rewritten as

xm,h,t,p � xm,h,t�1,p � wm,h,t,p � 0 ð8m 2M, 8h 2 H, 8p 2 P, 8t ¼ 1, . . . ,T Þ, ð7Þ

xm,h,t,p � xm,h,t�1,p � 2wm,h,t,p � �1 ð8m 2M, 8h 2 H, 8p 2 P, 8t ¼ 1, . . . ,T Þ: ð8Þ

The binary variable vm,h,t takes value 1 if product p is retrieved from height h in storage

area m and 0 otherwise. vm,h,t satisfies

ðxm,h,t�1,p � xm,h,t,p � 1Þ ) vm,h,t,p ¼ 1, otherwise vm,h,t,p ¼ 0: ðCÞ

Logical constraint (C) can be rewritten in the same manner:

xm,h,t,p � xm,h,t�1,p þ vm,h,t,p � 0 ð8m 2M, 8h 2 H, 8p 2 P, 8t ¼ 1, . . . ,T Þ, ð9Þ

xm,h,t,p � xm,h,t�1,p þ 2vm,h,t,p � 1 ð8m 2M, 8h 2 H, 8p 2 P, 8t ¼ 1, . . . ,T Þ: ð10Þ

Constraint (11) states that product storing at height hþ 1 and product retrieving at

height h cannot be executed simultaneously in the same storage area m in time period t:Xp

ðwm,hþ1,t,p þ vm,h,t,pÞ � 1 ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ: ð11Þ

Constraints (12) and (13) ensure that at most one product can be stored or retrieved in

a time period: Xm

Xh

Xp

wm,h,t,p � 1 ð8t ¼ 1, . . . ,T Þ, ð12Þ

Xm

Xh

Xp

vm,h,t,p � 1 ð8t ¼ 1, . . . ,T Þ: ð13Þ


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Logical constraints (D1), (D2), (D3), and (D4) state that the products can be stored at

or retrieved from the top of the stack in the storage area:

ðzm,h�1,t�1 ¼ 1Þ )X

pwm,h,t,p � 1

� �, ðD1Þ

ðzm, h�1, t�1 ¼ 0Þ )X

pwm,h,t,p ¼ 0

� �, ðD2Þ

ðzm,h,t�1 ¼ 1Þ )X

pvm,h,t,p � 1

� �, ðD3Þ

ðzm,h,t�1 ¼ 0Þ )X

pvm,h,t,p ¼ 0

� �: ðD4Þ

Equations (D1) and (D2) can simply be rewritten as

zm,h�1,t�1 �Xp2P

wm,h,t,p ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ,ð14Þ

zm,h,t�1 �Xp2P

vm,h,t,p ð8m 2M, 8h 2 H, 8t ¼ 1, . . . ,T Þ:ð15Þ

The final storage layout should be specified to satisfy the constraints of all receiving

and retrieving operations. These conditions can be described by the variable gh,t2 {1, 0},

which takes the value 0 if the product at height h on outpallet2M in time period t is the

same as the specified product Gh,outpallet, and 1 otherwise. xoutpallet,h,t,Gh,outpallettakes the value

1 if product Gh,outpallet is at height h on the output pallet in time period t, and 0 otherwise.

gh,t satisfies the constraint

gh,t þ xoutpallet, h, t,Gh,outpallet¼ 1 ð8h 2 H, 8t ¼ 1, . . . ,T Þ, ð16Þ

where �t2 {1, 0} is defined to represent the counting time required to complete all

operations. �t takes value 0 if the allocation of the pallet is the same as the specified

allocation, i.e.P

hgh,t¼ 1, and 1 otherwise. If all the products on a pallet in time period t

are the same as the pre-specified products Gh, the variable �t takes value 0 after time period

tþ 1. Only one product can be received/retrieved or re-handled. Therefore,P

t�t indicatesthe total number of operations required to complete all the demanded receiving/retrieving

warehousing operations. These conditions are represented by

gh,tþ1 � gh,t ð8h 2 H, 8t ¼ 1, . . . ,T Þ, ð17Þ

�t �Xh

gh,t � 0 ð8h 2 H, 8t ¼ 1, . . . ,T Þ, ð18Þ

K�t �Xh

gh,t � 0 ð8h 2 H, 8t ¼ 1, . . . ,T Þ: ð19Þ

Equation (17) states that gh,tþ1 cannot take value 1 if gh,t takes value 0. Equations (18)

and (19) state that �t can take value 0 only when gh,t becomes 0 for all h in time period t,

where K is a large non-negative integer that corresponds to the upper bound ofP

hgh,t.


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Let �(m, h, t) be an item of the product stored at height h of a location in storage area m

in time period t. The grouping performance index, JGroupm , for each allocation of products

in the stack in storage area m at the final time period T is defined as

JGroupm ¼

Xp2P

Xhj�ðm,h,T Þ6¼�ðm,hþ1,T Þ, h6¼H

xm,h,T,p: ð20Þ

Figure 2 shows an example of a grouping performance index. If the products for one

customer are subsequently located on stacked products for the same customer, these

products are considered to form a product-group. In Figure 2, the grouping performance

in areas 1, 2, and 3 is 1, 4, and 3, respectively. This is because the products in area 1 for

customer B can all be retrieved at once, then 1, 4, and 3 retrieval operations are required

for areas 1, 4, and 3, respectively, because the number of product-groups from the top of

the stack is 1 for area 1, 4 for area 2, and 3 for area 3. The total grouping performance

index for the layout is 8. A lower value of the index of product-groups indicates better

grouping performance. To obtain a smaller value for the grouping performance index,

a number of products must be transported to other storage areas. A number of operations

are required to rearrange the products. Intuitively, there is a trade-off between the

minimisation of the number of operations and the minimisation of the grouping

performance index. This relationship is demonstrated by numerical results in Section 4.3.Given an initial layout of products {xinpallet,h,0,p} (8h2H, 8p2P) and a final layout of

products {Gh,outpallet} (8h2H ), the problem can be formulated as an integer linear

programming problem:

MinimiseXTt¼1

�t þ �XMm¼1

JGroupm

!,

subject to (1)–(21),

xm,h,t,p 2 f1, 0g, zm,h,t 2 f1, 0g, vm,h,t,p 2 f1, 0g, wm,h,t,p 2 f1, 0g

ð8m 2M, 8h 2 H, 8t ¼ 1, 2, . . . ,T Þ, ð21Þ

where � is the weighting factor for the objective function. A suitable weighting factor is

chosen in an interactive way. For a real case, several weighting factors should be used for

area 2 area 3

Number of group product = 3

Customer B

Customer A



area 1

Figure 2. Grouping performance index.


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preliminary experiments. Practitioners can select suitable results by setting differentweighting factors.

A link between the targeted storage allocation and variable {xm,h,t,p} can be achievedusing Equation (16). The link between constraint (16) and the objective function can beachieved using Equations (17)–(19). The objective of planning warehousing storageallocation is to minimise the sum of the total number of operations and groupingperformance index for a given set of receiving and retrieving requests. The branch-and-bound method is applicable for solving this problem if the problem scale issufficiently small. Note that the planning horizon T is defined by the upper bound ofthe total completion time, i.e. (

Pp

Pm

Phxm,h,0,p)

2. If T is large, the model becomes largeand the computation time will increase. However, if T is too small, the problembecomes infeasible.

3. Beam search heuristics

The problem formulated in the previous section is to minimise the weighted sum of thetotal number of operations in warehousing and the grouping performance index.The branch-and-bound methods used to solve the integer-programming problem describedin the previous section require too much computational effort if the size of the problem islarge. The integer optimisation model also becomes extremely complex if the practicalwarehousing constraints imposed by the aggregation of plural products, separation ofproduct-groups, and grouping performance index are taken into account for theoptimisation model. In this section, the beam search method is applied to obtaina near-optimal solution in a reasonable amount of time.

3.1 Beam search method algorithm

An efficient beam search with heuristics is proposed to solve the problem of minimising theweighted sum of the number of operations and grouping performance index. Thealgorithm for the proposed beam search consists of the following steps. Hereafter, wedefine a node nk representing the storage allocation at a warehouse, and level t is defined asthe level of a search tree for the beam search, which corresponds to time period t in theoptimisation model of Section 2.

Algorithm for the beam search method

Step 1: Set beam width W and initialise the allocation of products to storage areas.The depth of the search tree for the beam search is called a level. The initial level t isset to 0. The tentative value of objective function z¼þ1, and the number ofcandidate nodes k is set to 0. The list of candidate nodes �¼ {n0}. n0 is the initialallocation. t tþ 1.

Step 2: Generate candidate nodes from level t� 1 to level t.

(i) Generate candidate nodes for the next receiving operation from an input palletand generate the candidates for all nodes for the retrieving operation into anoutput pallet. �¼ �[ {nkþ1, nkþ2, . . . , nkþN1

}. N1 is the number of nodes generatedin Step 2(i). k kþN1.


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(ii) Generate candidate nodes for re-handling operations, if necessary.

�¼ �[ {nkþ1, nkþ2, . . . , nkþN2}. N2 is the number of nodes generated in Step

2(ii). k kþN2.

The detailed step for the generation method (GEN) for candidate nodes is explained in

Section 3.2.

Step 3: Select W candidate nodes from level t. W is the beam width specified a priori.

(i) Calculate a heuristic function g(nk) (8nk2�) to estimate whether a candidate node

at level t has the possibility of generating a near-optimal solution at the final level.

The effective heuristic function for the problem is shown in Section 3.4.(ii) Delete certain candidate nodes according to the least value of the heuristic

function or objective function at level t.(iii) Select W candidate nodes from the list of candidate nodes �.

The detailed selection method at Step 3(iii) for candidate nodes is explained in

Section 3.4.

Step 4: Check the layout of the input pallet and output pallet. If all receiving and

retrieving requests have been generated for all levels, the algorithm is terminated.

Otherwise, t tþ 1 and return to Step 2.

The branching tree of the algorithm of the beam search method is illustrated in

Figure 3. The initial storage allocation, receiving and retrieving requests are specified in

the table on the left-hand side of the figure. Each level has a request for a receiving or

retrieving operation. For each level, all alternative operations for warehousing are

1 4 6

3 5

2

area A area B area C

6retrieving

7receiving

3retrieving

customerevent

Initial condition

3 2 1 1 2 3

2 3

2 3 3 3 4

2 3 2 3 1 1 2 3 132 1

A→B A→C B→A B→C C→A C→B

3→pallet 3→pallet

Initial condition

receiving and retrieving requests

6→pallet

7→A 7→B 7→C

Level 1

Level 2

Level 3

Level 4

A→B A→C B→A B→C C→A C→B A→B A→C B→A B→C C→A C→BA→B A→C B→A B→C C→A C→B

6→pallet

7→A 7→B2

7→C

1 4 6

3 5

2Initial condition

3

2

A→B A→C B→A B→C C→A C→B

3→pallet 3→pallet

6→pallet

7→A 7→B 7→C

Level 2

Level 3

A→B A→C B→A B→C C→A C→B A→B A→C B→A B→C C→A C→BA→B A→C B→A B→C C→A C→B

6→pallet

7→A 7→B 7→C

6→pallet

7→A 7→B 7→C

6→pallet

7→A 7→B

2

7→C

Figure 3. Branching example of the proposed beam search.


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generated as plural nodes. The nodes are selected by evaluating the heuristic function,

whether or not the generated nodes have the possibility of deriving a near-optimal solution

at the final level. The heuristic function is shown in Section 3.3. The value of the heuristic

function is computed as the number denoted at each node in Figure 3. At each iteration,W

candidate nodes (number of candidates) are selected for the next iteration. If level t is

reached for the completion of all requests, the algorithm is terminated. The derived nodes

are adopted as alternative solutions.

3.2 Method for generating alternative nodes

It is impractical to generate all of the alternative nodes for a search tree with an increase in

the number of combinations for evaluating the alternatives. For example, the number of

combinations for the movement of a product is 60C2¼ 1770 for a warehouse consisting of

58 storage areas. The nodes with less possibility of generating a near-optimal solution

should not be regenerated in advance to enable an efficient search for a beam search

method. The flowchart of the method for the generation of alternative nodes is shown in

Figure 4. The steps of the algorithm are as follows.

Method for the generation of candidate nodes: GEN procedure

Step 1: If the current request is a receiving operation, products for the same customer are

searched for among the products on top of the stack in the storage areas. If a product for

the same customer is found on top of a stack in a storage area, the product to be received is

allocated to that storage area in order to aggregate products for the same customer.

The nextoperation?

Y

N

receiving retrieving

Generation of a node of the space atthe same customer

There is a space where the top of product is the

next retrieving?

There is a spacewhich can retrieve from the

top of the space

Generation of a node of the space atthe next product is retrieving

End

Generation of a node of the space where there exists no future operations

There is a spacewhich can receive on the

same customer

Generation of a node of the spaceAt the top of product is retrieving

N

N

Y

Y

Start

Figure 4. Flowchart for generating alternative nodes.


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Otherwise, if a product for the same customer is not found, alternative nodes for selectingstorage areas are generated by the IMP procedure explained in Section 3.5.

Step 2: If the current request is a retrieving request, the product to be retrieved issearched for among the products on the top of a stack in the storage areas.

(i) If the product to be retrieved is found, the nodes for the retrieving operation aregenerated.

(ii) A product for the same customer is searched for among the products on the top ofa stack in the storage areas. If products for the same customer are found, they areaggregated on a stack in the same storage area. These procedures are based on thepractical operation of a warehouse in which products for the same customer aregathered at one location in order that they can be checked and retrieved in oneoperation. Otherwise, if products for the same customer are not found, thealternative nodes for selecting storage areas with fewer future receiving/retrievingrequests are selected by the IMP procedure (details are given in Section 3.5).

3.3 Computing the heuristic function

The heuristic function g(nk) for evaluating the node nk for the branching process of thebeam search algorithm is explained in this section. The candidate nodes are deleted, exceptfor the node having the minimum value of the heuristic function in the selection step.The heuristic function is defined as the lower bound of the number of operations forretrieving a product for the next retrieving request from the stacks in the storage areas ofa warehouse

gðnkÞ ¼XHm

h¼hk

Xp2P

xmk,h,t,p, ð22Þ

where the next retrieving product for the allocation of node nk at level t is stored at heighthk of the location in storage area mk.

Figure 5 shows the computation method for the heuristic function. When three types oflocation are given for retrieving product 6, as shown in Figure 5, the minimum number ofproducts that should be removed for the next retrieving operation is calculated as theheuristic function. The numbers are denoted in the box, for example ‘1(5)’ means that theproduct number is 1, and five series of products are stored at that height at that location.The minimum number of operations for retrieving product 6 in Figure 5(i) is 25 productsfor area C, which consists of 20 products for product 5, and five products for product 6.The total number for the other cases shown in Figures 5(ii) and (iii) is 30 and 25,respectively. The storage area exhibiting the minimum value of the heuristic function (forFigure 5(iii)) is determined to select the candidates for the beam search. Thus, the noderepresenting allocation (ii) is deleted in the selection step.

3.4 Method for selecting alternative nodes

There are four selection steps for the proposed beam search. In the first step, the candidatenodes representing the same movements compared with a previous level are deleted;


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for example, the movement from B to A after the movement from A to B, is deleted. In thesecond selection step, the heuristic function is calculated for all node candidates. Thenodes with the minimum value of the heuristic function remain. Then the total number ofoperations is calculated and the nodes with the minimum number of operations remainand the others are deleted. In the third step, the grouping performance is calculated andthe nodes with the minimum number of grouping performances remain. In the final step ofthe selection, the nodes with more than a pre-determined number of beam widths W aredeleted to reduce the computational burden for large-sized practical problems.

3.5 Improvement of the beam search heuristic using future request information

In this section, the proposed beam search is improved using future request information tosolve a large-sized problem. The improved procedure for generating alternative nodes isadded to Step 1 and Step 2(ii) of the original algorithm of Section 3.2. In the proposedprocedure, the alternative storage areas for a product are deleted so that the products arenot allocated to the storage area to receive or retrieve products in the near future. Figure 6shows the method for the generation of candidate nodes. For the current receiving producton top of the stack in the storage area, if there is no product for the same customer, thereare no candidates for the storage area for a product for receiving with no re-handling.Then the receiving product is allocated to a storage area where receiving and/or retrievingdoes not occur in future requests.

Consider a situation where a product is allocated to a storage area as illustrated inFigure 6. Here we assume that the retrieving requests are given for customers1! 2! 3! 4! 5. The improved procedure for determining the candidates for allocationof storage areas is given below.

Improved procedure for the beam search method (IMP procedure)

Step (i): Delete the nodes of allocation to storage areas where products cannot beallocated in size. In Figure 6, storage area 1 is assumed to be the storage area where theproduct cannot be allocated in size.

Step (ii): For each storage area, products in the stacks are searched for according to theretrieving requests. If the products that require retrieving are found, the index that has theminimum number of retrieving orders is denoted at each storage area. If there is no

The next retrieving product = ‘6’

1(5) 2(15) 6(5)3(10) 5(20)

4(5)

1(5) 2(15) 6(5)3(10) 5(20)4(10)

Minimum number of handling = 25



(i)

(ii)

(iii)

1(5) 2(15 6(5)3(10) 5(20)4(5)


6(5)5(20)4(5)5(20)4(5)

5(20)

1(5) 2(15) 6(5)5(20)4(5) 5(20)4(5)



Figure 5. Computation method for the heuristic function.


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retrieving request for a storage area, �1 is allocated to the storage area. Otherwise, if thereis no product in the stacks in the storage area, �2 is allocated to the storage area.

Step (iii): The candidate nodes are selected in descending order of the index for eachstorage area. The method is based on a heuristic in which there is no future request forstorage areas with a small index.

Step (iv): For each storage area, the customers with products on top of the stacks aresearched for among the receiving requests. If receiving products for the same customer arefound, the index that has the minimum number of receiving orders is allocated to eachstorage area. If there is no product with a receiving request from a customer for a storagearea, 0 is allocated to the storage area. The storage areas with 0 are selected as thecandidate nodes.

The IMP procedure was developed from an idea from human operators in an industrialwarehouse. The beam search method with the procedure described in this section willhereafter be called the beam search heuristic.

4. Computational experiments

4.1 Effectiveness of the proposed beam search heuristic

In order to evaluate the performance of the proposed beam search heuristic, five small-sized problems are solved using the proposed method and a commercial solver (CPLEX10.1 provided by iLOG). An Intel Pentium IV 3.4GHz processor with 1 GB memory wasused for computations. The initial conditions and the retrieving sequences for the exampleproblems are depicted in Figure 7.

In the example problems, there is no receiving product and all of the products areretrieved so that the CPLEX solver can derive an optimal solution. Therefore, thegrouping performance is 0 after all requests have been completed. The objective is to

12

cannot be allocated in sizeor length

23

110

2 4 6 7 8

2785

4

10

9149

4395

10

589

69

1

–10

–2

3

4–1

Retrieving requests: 1->2->3->4->5

1

03

2 4 6 7 8

0

22

–2

3

22

Receiving requests: 7->9->10

3

The printed number isorder of future retrieving

The printed number isorder of future receiving

439510

589

69

149

10

9

27854

2

110

6

3 5 9

3 5 9

Figure 6. Procedure for reducing the search space of the proposed beam search method.


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minimise the number of operations to retrieve all of the initially stacked products.To derive an optimal solution for the example problem, the integer programming model ofSection 2 is applied. The planning horizon is set to 20 time periods. The computationalresults for the example problems are summarised in Table 1. The value of the objectivefunction (Obj.), the computation time (CPU time) and the solutions are listed in Table 1.An ordered pair (a, b) is used to represent the movement of a product from area a to areab, and p denotes the output pallet. From the computational results, it is found that theproposed method can generate an optimal solution for Cases 1, 4, 5 and 6 within 0.01seconds for the six cases examined. The heuristic of the IMP procedure using futurerequest information can effectively reduce the number of movements for retrievingoperations. The computation time for the CPLEX solver increases exponentially with anincrease in the number of products treated in the model. For Case 6, the computation timefor CPLEX is more than 31 hours. From the results of the computations, it can be seenthat the proposed method can efficiently derive a near-optimal solution for small-sizedproblems. The beam search heuristic cannot ensure the generation of an optimal solutionfor each problem. However, if the size of the integer programming problem is increased,the computation time for the branch-and-bound method rapidly increases with respect toan increase in the number of storage areas, the height of storage, the number of products,and the number of time periods. Therefore, the integer programming method cannot beapplied to large-sized industrial problems.

4.2 Application of the beam search heuristic to an industrial-sized problem in a steel plant

The proposed method was applied to a large-sized practical problem. Products for thesame customer successively loaded in the same storage area are aggregated as a product-group to reduce the number of handling operations for the computation. A practicalproblem with 3165 receiving and 3412 retrieving requests over a period of one month for58 storage areas was solved using the proposed method. Details of the receiving andretrieving requests for the industrial steel plant are shown in Table 2. The initial allocationconsisted of 2706 products in 58 storage areas. For the initial allocation, series of productsfor the same customer located successively in the same storage area were aggregated intoa product-group. The total number of product-groups was reduced to 467 products,which, using our approach, can be computed in a reasonable time. The total computation

Retrieve A→B→C

A ACC

B A ACC

BAB

A A

CC

BAB

B

Case 1

Area1 A A

BB

CC A

B C

C EC

D

AB E

C DA

A CA

BAC B A

C

ACB

AB

Retrieve A→B→C Retrieve A→B→C

Retrieve A→B→C Retrieve A→B→C→D→Ε Retrieve A→B→C

Area2 Area3 Area1 Area2 Area3 Area1 Area2 Area3 Area4 Area5 Area6

Area1 Area2 Area3 Area1 Area2 Area3 Area1 Area2 Area3

Case 2 Case 3

Case 4 Case 5 Case 6

Figure 7. Initial allocation and retrieving requests for example problems.


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Table

1.Computationalresultsforthesm

all-sized

example

problems.

Proposedmethod

CPLEX

Case

Number

of

Products

Obj.

CPU

time(s)

Solution

Obj.CPU

time(s)

Solution

15

80.01

{(1,3),(1,p),(3,1),(3,1),(3,p),(2,p),(1,p),(1,p)}

826

{(1,2),(1,p),(2,1),(2,1),(3,p),(2,p),(1,p),(1,p)}

27

12

0.01

{(3,1),(3,p),(2,1),(2,p),(1,2),(1,3),(1,3),

(1,p),(2,p),(2,p),(3,p),(3,p)}

11

179

{(1,2),(1,p),(3,1),(3,p),(2,3),(2,1),(2,p),

(1,p),(2,p),(1,p),(3,p)}

38

14

0.01

{(3,1),(2,3),(2,p),(3,2),(3,2),

(3,p),(1,3),(1,3),(1,p),(2,p),(2,p),(2,p),

(3,p),(3,p)}

13

1742

{(1,3),(1,p),(3,1),(3,1),(2,1),(2,1),(2,p),

(3,p),(2,p),(1,p),(1,p),(1,p),(1,p)}

49

15

0.01

{(2,1),(2,p),(3,2),(1,2),(1,2),(3,2),

(3,p),(1,3),(1,p),(2,p),(2,p),(2,p),(2,p),

(2,p),(3,p)}

15

18,316

{(2,1),(2,p),(3,2),(3,2),

(3,p),(1,2),(1,2),(1,3),(1,p),(2,p),(2,p),

(2,p),(2,p),(2,p),(3,p)}

510

15

0.01

{(1,p),(2,3),(2,3),(2,p),(1,2),(1,p),

(1,p),(3,1),(3,p),(4,2),(3,p),(2,p),(2,p),

(1,p),(3,p)}

15

12,067

{(1,p),(2,3),(2,1),(2,p),(1,2),(1,2),(1,p),

(3,p),(3,2),(1,p),(3,p),(2,p),(2,p),(3,p),

(2,p)}

614

19

0.01

{(3,p),(5,p),(1,3),((1,p),(4,3),(4,p),(2,4),

(2,p),(6,4),(6,p),(5,3),(5,p),(4,p),(4,p),

(4,p),(3,p),(3,p),(3,p),(3,p)}

19

112,500

{(3,p),(6,3),(5,p),(6,p),(1,6),(1,p),(5,1),

(5,p),(2,6),(4,5),(2,p),(4,p),(3,p),(4,p),

(6,p),(1,p),(5,p),(3,p),(6,p)}


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Table 2. Requests data for the industrial steel plant.

Request ID OperationaNumber ofproducts Request ID Operationa

Number ofproducts

1 O 4 52 I 642 I 96 53 O 23 O 5 54 I 264 I 60 55 O 15 O 1 56 I 576 I 83 57 O 17 O 3 58 I 368 I 45 59 O 39 O 2 60 I 310 I 56 61 O 111 O 1 62 I 612 I 45 63 O 113 O 1 64 I 14814 I 70 65 O 715 O 3 66 I 2716 I 109 67 O 117 O 5 68 I 7518 I 1 69 O 119 O 1 70 I 720 I 65 71 O 121 O 4 72 I 5522 I 165 73 O 223 O 6 74 I 424 I 26 75 O 125 O 2 76 I 126 I 217 77 O 627 O 7 78 I 6228 I 91 79 O 329 O 1 80 I 6330 I 26 81 O 131 O 3 82 I 15232 I 14 83 O 233 O 5 84 I 11134 I 52 85 O 135 O 3 86 I 7936 I 179 87 O 137 O 2 88 I 12538 I 51 89 O 339 O 1 90 I 1640 I 118 91 O 141 O 2 92 I 742 I 48 93 O 343 O 1 94 I 1344 I 104 95 O 245 O 1 96 I 10746 I 33 97 O 347 O 2 98 I 6948 I 41 99 O 549 O 2 100 I 350 I 54 101 O 151 O 1

aI¼Receiving, O¼ retrieving.


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time is 517 seconds using a Pentium IV 2GHz processor with 512MB memory witha beam width of W¼ 10. A feasible allocation is obtained using the proposed method

without concentrating the receiving products in specific storage areas.The effect of the IMP procedure on the number of operations required to complete all

requests was investigated. Figure 8 shows the effect of the use of the number of futurerequests on the number of movements for the proposed method. The results show that the

number of operations is reduced if the available number of future requests is increased.The proposed heuristic for selecting candidate nodes taking into account future requests isvery effective in reducing the number of operations. This is due to the fact that the number

of re-handling operations is reduced by allocating the products to storage areas wherethere are few future requests. The heuristic for the IMP procedure was developed by

interviewing warehousing operators. The candidate nodes were also drastically reduced byincorporating the IMP procedure.

4.3 Effects of the beam width W on the solution

The effect of the beam width W on the solution was examined using the proposed method.

The computational results, including the computation time for the proposed method,are shown in Table 3 for beam widths W of 1, 10, 30, 50, and 100. The effect of theIMP heuristic is dominant when the beam width is equal to 1 for the proposed beam

9000

9500

10000

10500

11000

11500

12000

12500

10 30 60 200

Num

ber

of m

ovem

ents

Available number of future requests [-]

Figure 8. Effect of available number of future requests.

Table 3. Computational results for the practical problem.

Beam width W

1 10 30 50 100

Number of operations 9110 9292 9266 9389 9159Number of product-groups 235 211 213 213 201Computation timea (s) 134 517 1169 2041 4043

aPentium IV, 2GHz processor, 512MB memory used for computation.


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search heuristic. Therefore, the number of operations is minimised, whereas the grouping

performance for each customer is no better than that of the other cases. On the other hand,

when the beam width is increased, the total number of operations is increased, and the

grouping performance for each customer is improved. The total computation time for

a large beam width is greater than that for a small beam width. From these results it can be

observed that the number of operations is reduced if the beam width W is set to a small

value. On the other hand, the grouping performance is reduced if the beam width W is set

to a large value. An optimal solution can be derived for the problem of minimising the

total number of operations when the problem scale is small, as described in Section 4.1.

The weighted sum of the minimisation of the number of operations and the grouping

performance can be obtained by setting an appropriate value for the beam width, because

the IMP procedure is useful for obtaining a near-optimal solution for minimising the

number of operations. On the other hand, the beam search heuristic combines these

heuristics to minimise the weighted sum of the objective function. This demonstrates that

the proposed method is applicable to practical warehouse operational storage planning

problems by changing the beam width of the beam search heuristic.

5. Conclusion

In this paper, an optimisation model for practical warehousing operational storage

allocation planning has been developed. An efficient beam search method heuristic has

been proposed to minimise the weighted sum of the number of operations and grouping

performance index. In order to reduce the search space for the proposed beam search

method, a heuristic for restricting the generation of candidate nodes is embedded in the

proposed method. The IMP procedure is effective for a single objective function to

minimise the number of operations for re-handling products. The proposed method

combines several heuristics to minimise the total number of operations and grouping

performance by employing beam search heuristics. The IMP procedure was developed

from the heuristic of actual human operators used for practical warehousing. The

effectiveness of the proposed method is demonstrated by solving small-sized example

problems by comparing the performance with that of CPLEX. The computational results

show that the proposed method can derive a near-optimal solution with significantly less

computational effort than for CPLEX solving an integer programming problem.

The proposed method was applied to a large-scale problem of an industrial steel plant.

The results demonstrate that the proposed method can effectively generate a feasible

solution for more than 3000 retrieving/receiving requests in a reasonable computation

time. The proposed model will be applicable to the re-marshaling process in container

yards, and the handling of ship-building parts. The application of the proposed method to

the optimisation of container yards is one of our future works.

Acknowledgements

The authors would like to thank JFE Steel Corporation for providing actual warehousing data andtheir efforts and discussions. The authors would particularly like to thank the anonymous reviewersand the Associate Editor for their constructive comments that helped to improve the quality ofthis paper.


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an optimisation model and its effective beam search heuristics for floor-storage warehousing systems

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