A model for warehouse order picking

Order picking in conventional warehouse environments involves determining a sequence in which to visit the unique locations where each part in the order is stored, and thus is often modeled as a traveling salesman problem. With computer tracking of inventories, parts may now be stored in multiple locations, simplifying replenishment of inventory and eliminating the need to reserve space for each item. In this environment, order picking requires choosing a subset of the locations that store an item to collect the required quantity. Thus, both the assignment of inventory to an order and the associated sequence in which the selected locations are visited affect the cost of satisfying an order. We formulate a model for simultaneously determining the assignment and sequencing decisions, and compare it to previous models for order picking. The complexity of the order picking problem is discussed, and an upper bound on the number of feasible assignments is established. Several extensions of TSP heuristics to the new problem setting and a tabu search algorithm are presented and experimentally tested. (~) 1998 Elsevier Science B.V.

E L S E V I E R

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH European Journal of Operational Research 105 (1998) 1-17

Theory and Methodology

A model for warehouse order picking

Richard L Daniels a'*, Jeffrey L Rummel b, Robert Schantz c

a School of Management, Georgia Institute of Technology, Atlanta, GA 30332-0520, USA

b School of Business Administration, University of Connecticut, Storrs, CT 06269-2041, USA

c IBM Corporation, Research Triangle Park, NC 27713, USA

Received ! September 1995; accepted 1 September 1996

Abstract

Order picking in conventional warehouse environments involves determining a sequence in which to visit the unique locations where each part in the order is stored, and thus is often modeled as a traveling salesman problem With computer tracking of inventories, parts may now be stored in multiple locations, simplifying replenishment of inventory and eliminating the need to reserve space for each item In this environment, order picking requires choosing a subset of the locations that store an item to collect the required quantity Thus, both the assignment of inventory to an order and the associated sequence

in which the selected locations are visited affect the cost of satisfying an order We formulate a model for simultaneously determining the assignment and sequencing decisions, and compare it to previous models for order picking The complexity

of the order picking problem is discussed, and an upper bound on the number of feasible assignments is established Several extensions of TSP heuristics to the new problem setting and a tabu search algorithm are presented and experimentally tested (~) 1998 Elsevier Science B.V

Keywords: Warehouse operations; Traveling salesman

1 Introduction

The trend towards just-in-time (JIT) inventory

systems removes some warehouse points from the

manufacturing chain, especially those used for stor-

age of work-in-process inventories, but the number

of warehouse transactions in many firms is still quite

large These transactions typically decouple use from

replenishment, or take advantage of ordering or trans-

portation economies At the plant level, warehouse

operations often assemble kits of parts to supply fab-

rication and assembly processes on the factory floor

In firms such as wholesale distributors, the task of

* Corresponding author Tel.: +1-404 894 8713; fax: +1-404 894

6030

processing warehouse orders is at the core of the business Since these transactions occur frequently, small savings on each can result in significant savings for the firm (see, e.g., Matson and White, 1982, Witt,

1987, and Gray et al., 1990)

Previous models of the order picking problem as- sume that all stock of a particular part is stored in one location in the warehouse, since tracking inventory ne- cessitated that space be permanently allocated for each part (see, e.g., Ratliffand Rosenthal, 1983, Goetschal- ckx and Ratliff, 1988, and Bozer et al., 1990) The problem of picking an order is then one of determin- ing the sequence in which locations should be visited

to minimize total cost (or time), which leads to the traveling salesman problem (TSP) This structure al- lows for implementation of a zone-picking strategy,

0377-2217/98/$19.00 (~) 1998 Elsevier Science B.V All rights reserved

PII S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 0 4 3 - X

where orders are assembled by multiple pickers, since

smaller TSPs are much easier to solve

Frequent changes in the set of parts warehoused,

along with the enhanced capability of computer sys-

tems to track inventories within the warehouse, adds

a dimension to the problem such a formulation of

warehouse operations does not include As the de-

mand for individual parts grows and shrinks, allocat-

ing warehouse space to parts becomes more complex

and costly Changes in demand, and the consequent

reallocation of warehouse space, often require move-

ment of stock that can cause severe disruptions to

warehouse operations, especially when the warehouse

is highly utilized Bar coding of stock promotes the

maintenance of perpetual inventory records, allowing

parts to be stored in any location in the warehouse The

assignment of an individual part to multiple locations

throughout the warehouse reduces the importance of

the stock allocation problem

Previous models required only the quantity of each

part required, but now we need the set of locations

in which each part is stored, and the amount of stock

contained at each location The order picking problem

now must determine both an assignment of inventory

to the current order and the determination of an appro-

priate picking sequence, Clearly interactions exist be-

tween inventory assignment and location sequencing

decisions, and this paper presents a model for solving

this problem

This generalization of the travelling salesman prob-

lem constructs TSP tours to visit a subset of the possi-

ble locations, such that the selected locations together

satisfy a set of portfolio constraints In these con-

straints, each location has associated with it a weight

that reflects the contribution of that location towards

satisfying the order requirement for the associated

part In this setting, the weights reflect the amount of

inventory at each location, but this construction could

easily be generalized to other situations, such as where

a salesman must sample a territory by calling on a

given mix of customers

This paper does not address the question of whether

storing parts in multiple locations is better than only

in one But in order to make the comparison, a method

for solving the picking problem is required In many

systems that use multiple locations, the computer sys-

tem prints a list of the locations and the person picking

the order must decide where to visit The technology

exists for randomly storing inventory and the question answered in this paper is how to use that technology in the most cost effective way With these tools, a further study could be constructed to find the expected benefit

of allowing storage of a part in multiple locations The paper is organized as follows The next section presents a formulation of the problem and compares

it to previous models for order picking Section 3 dis- cusses problem complexity and our experience gener- ating lower bounds and optimal solutions Section 4 examines several heuristic approaches to the problem, both by extending TSP heuristics to the new problem setting, and by constructing a tabu search algorithm for the problem Section 5 describes a set of experiments conducted to test the computational performance of the heuristics The paper concludes with a summary and suggestions for future research

2 Problem formulation

Consider the following warehouse environment When a shipment of an individual part arrives at the warehouse for storage, the part is assigned to an open location in the warehouse Given the ability of the computer system to retain the location, the part can

be stored anywhere in the warehouse; over time a particular part type may be stored in more than one location Orders (from customers or from the factory floor) arrive and specify a set of parts and quantities for each The computer system can produce a picking list that shows the requirements of the order and also lists the locations for each part and the quantity stored

at each location The problem we formulate is the de- cision determining how to match the inventory listed with the order requirements Although there are other considerations in choosing which inventory locations

to use (e.g., age of the inventory) we consider the problem of minimizing the cost (time) required to meet the order requirements

First, we define some notation for the problem

M denotes the set of different part types that must be picked for the current order We index the set of parts by k

N denotes the set of locations containing inventory

We index the set of locations by i and j Dummy locations for the starting and drop-off points of the tour can also be added to the formulation

R.L Daniels et al./ European Journal of Operational Research 105 (1998) 1-17

Pk denotes the set of locations containing part type k

These sets are assumed to be disjoint and cover N

cij denotes the cost of moving from location i to lo-

cation j These costs can be arbitrary, but in many

cases will satisfy the triangle inequality, a prop-

erty that can be exploited to reduce the size of the

solution space

qi denotes the amount of inventory stored at location

i

Ik denotes the total stock of part type k in the ware-

house, with:

jEek

rt denotes the required quantity of part k in the order

Without loss of generality, we can scale the qi

and assume that rt = 1 for all k In this case, a

feasible stock position for the order implies that

It ~> 1 for all k If the total amount of available

inventory is insufficient and a partial shipment is

desired, a dummy location i with qi = 1 - lk units

and zero costs in and out of the location may be

added; otherwise, part k should be deleted from

the problem

xij denotes the decision variable that defines the tour,

equalling 1 if location i immediately precedes lo-

cation j in the tour, and 0 otherwise Although

most TSP formulations do not use the notation xii,

we define:

1 if location i is not on the tour,

Xii 0 otherwise, (2.2)

with this structure requiring cii = O

L denotes the set of locations in the tour

The formulation of this problem can then be ex-

pressed as:

M i n ~ ~ CijXij (2.3)

iEN jEN

subject to

iEN

~ X~j = 1 gi E N,

jEN

<~ It

iEPk

(2.5)

qixii q- lkXjj ieP&Ij}

>/ Ik rk qj Vk E M, V j E P k , (2.7) {subtour elimination constraints}, (2.8)

The assignment constraints (2.4) and (2.5) require

that each location be either part of the tour (xij = 1 )

or excluded from the tour (xii = 1) The portfolio

constraints (2.6) and (2.7) ensure that sufficient in- ventory of each part type is collected Note that since

xii indicates whether a location is included in the tour:

qi( 1 - - X i i ) ) rk (2.10)

iE&

Rearranging terms and using the definition for lk yields (2.6) Expression (2.6) can be interpreted

as requiring that the amount of inventory left in the warehouse be sufficiently small to ensure that enough has been removed to satisfy the requirement This constraint must be written as an inequality since the sum of the quantities stored at various locations need not sum exactly to the requirement This formulation does not unambiguously specify which locations will

be emptied and which will retain inventory, hut in practice, some decision rule that considers inventory age or other characteristic could he used

A related issue concerns feasible solutions that se- lect many locations containing part type k to visit when

fewer would satisfy rk This situation can arise when

the cost matrix does not satisfy the triangle inequal- ity, and visiting extra locations reduces the tour cost

In this case, implementation of the solution requires 'grazing' of inventories (taking a small quantity from each location), or visits to locations where zero in- ventory is taken Our formulation restricts solutions

to visit the right number of locations, i.e., the solution contains no location such that the solution remains fea- sible if that location is deleted from the solution The constraint set to enforce this condition for location j

of part type k can be expressed as:

~-~qi(1 - x i i ) <~ r k + q j ( 1 - - X j j ) + I k X j j (2.11)

iE&

Rearranging terms as before yields (2.7) When the cost matrix does satisfy the triangle inequality, the ob- jective function makes (2.7) unnecessary since extra

locations always increase total cost However, con-

straints (2.7) reduce the size of the feasible region,

and therefore may prove helpful in solving the integer

programming problem

Subtour elimination contraints for the standard TSP

can be written in a number of ways (see e.g., Lawler

et al., 1985) Here the constraints are more difficult to

write down compactly, since we must prevent subtours

among the locations chosen to be visited (Xii = O)

while ignoring the rest of the locations (where xii -~

1 ) Since the focus of the paper is on specifying the

problem and exploring heuristic solution techniques,

a further discussion of these constraints is omitted

We now demonstrate that the order picking prob-

lem generalizes previous formulations The first spe-

cial case occurs when IPkl = 1 for all k If suffi-

cient inventory is available, then constraints (2.6) and

(2.7) are satisfied Since every location must be vis-

ited, xii = 0 for all i, and the standard TSP is obtained

Note that in this case the number of part types, the

tour length and the number of locations are the same,

or INI = ILl = IMI

Next, allow the number of locations to increase, but

require that qj >/rk for each j E Pk- This corresponds

to the case where a computer system only reports loca-

tions with sufficient inventory for the order The port-

folio constraints can now be satisfied by visiting any

one location for each part type, i.e., a feasible solution

can be constructed that visits each set Pk exactly once

For each part type, all but one xii = 1, and therefore

(2.6) and (2.7) are satisfied For the single location

i of the part type where xii ~" O, the assignment con-

straints force one of the xij = 1 Notice that the defini-

tion of xii a l l o w s US to write the assignment constraints

in standard form, rather than the form typically used in

the generalized traveling salesman problem (GTSP):

iEPk j~Pk

Further discussion of this formulation and its solution

can be found in Noon and Bean ( 1991 ) In this case,

the number of locations increases, but the tour length

and the number of part types remain equal, or INI />

ILl = I,v/I

In the order picking problem, the requirements for

some part types may be satisfied by visiting as few

as one location, while other parts require inventory

from multiple locations in order to satisfy (2.6) and (2.7) Thus, the number of locations visited is not necessarily determined by the problem parameters, or IN[/> ILl/> IMI

In the GTSP, (2.12) can be relaxed to an inequal- ity to allow the tour to become longer, hut less costly tours are only obtained when the cost matrix does not satisfy the triangle inequality Since the number of lo- cations is not known a priori, there is not a straight- forward transformation of the GTSP into the order picking problem The order picking problem also gen- eralizes the prize collecting TSP problem studied by Balas (1989), where only one part type (i.e., one side constraint) is considered

3 Problem complexity and bounds

The relationship between our problem and the stan- dard TSP makes it easy to show our problem to be NP-complete A potential solution can be described as the quantity removed from each location (inventory assigned to this order) and a tour connecting those locations where the quantity in nonzero Given such

a solution, checking for feasible quantities is polyno- mial and checking the tour constraints is the same as for the TSP, so our problem is clearly in NP As noted before, when IPkl = 1 for all k, the resulting problem

is the standard TSP, and this is sufficient to prove by restriction that our problem is NP-complete (Garey and Johnson, 1979, p 63)

In practice, the time required to solve our problem

is going to be a function of the number of potential inventory assignments over which tours might be con- structed, and the length of those tours The time re- quired for TSP solution methods as the tour length increases is well known Note that if there is special structure in the cost data, problems can sometimes be solved more quickly (see, e.g., Ratliff and Rosenthal, 1983), but we make no special assumptions about the costs in this paper

It is more difficult to see how the problem param- eters affect the number of potential inventory assign- ments Consider a problem where there is only one location per part type This problem instance requires solving one TSP among the [M[ locations Similarly, suppose that the available inventory for each part type just satisfies the requirements (Ik = 1 for all k) This

R.L Daniels et al./European Journal of Operational Research 105 (1998) 1-17

problem instance requires solving one TSP among all

IN[ locations Therefore, it is in the intervening cases

that the number of possible assignments grows The

following proposition allows us to bound the number

of feasible assignments

Proposition 1 For a given part type k stored in n

locations, scale all the quantities so that rk = 1, and

qj = 1 i f qj >1 rk Let the average inventory at each

location be given by Q = min(~-~qj/n, 1) and define

x = r l / Q ] Then S, the number of feasible assignment

vectors that satisfy the requirement of part type k, is

bounded by:

ments For each set of locations, a Lagrangean relax- ation of the associated TSP was formulated and solved Among all of the relaxations solved in this manner, the minimum cost was retained as a lower bound For small problems, this bound proved to be relatively easy to compute and adequately close to the corre- sponding optimal solution However, as problem size increased, the number of feasible assignments grew rapidly, increasing the computational cost of comput- ing the bound and weakening its quality

4 Heuristic solution approaches

The proof of Proposition 1 is given in Appendix A

Note that this factorial bound is for only one part type;

thus, in a problem with IMI part types, the total num-

ber of assignments is found by multiplying together

the individual bounds Even for small problems, the

worst case number of assignments grows quickly, e.g.,

an order for fifteen parts where each part has three

locations results in a worst case bound of over four-

teen million possible assignments Similarly, an order

for only five parts where each part has eight locations

results in a worst case bound of over 1.6 billion as-

signments

Constructing a lower bound on the optimal solution

proved to be difficult Since the problem is related to

the TSP, we looked at relaxations used for that prob-

lem to see if they could be extended to accommo-

date the additional side constraints (2.6) and (2.7)

Dualizing a sufficient number of constraints to obtain

a problem that could be solved efficiently, very poor

lower bounds resulted This observation is consistent

with the results reported by Tang and Denardo (1988)

for a problem of similar difficulty We also considered

constructing minimum spanning trees instead of tours,

but the inventory side constraints result in a problem

that could not be solved easily Finally, we considered

various methods for implicitly enumerating feasible

assignments, but dominance rules for reducing the size

of the solution space were ineffective when compared

with additional computational effort

In order to obtain at least one bound for evaluating

the performance of heuristics, we developed an algo-

rithm to generate all of the feasible inventory assign-

Given the difficulty involved with even obtaining

a lower bound for the order picking problem, we fo- cused primarily on constructing heuristic solution ap- proaches for the problem Standard nearest neighbor and shortest arc TSP heuristics (see, e.g., Rosenkrantz

et al., 1977, and Golden et al., 1980) were first mod- ified to conform with the new problem setting Since these myopic techniques make inventory assignment and location sequencing decisions simultaneously, their performance in approximating optimal solutions for the order picking problem should be poorer than for the TSP Therefore, search methods for sampling promising areas of the feasible region, including ran- domized versions of the modified nearest neighbor and shortest arc heuristics, as well as a tabu search approach for the order picking problem, were also developed

4.1 Nearest neighbor

In the standard TSP environment, the nearest neigh- bor heuristic starts at a given location with a list of all

of the unvisited nodes in the graph The cost to travel

to each unvisited node is examined, and the node rep- resenting the lowest cost is selected and removed from the set of unvisited nodes This process is repeated un- til the unvisited node set is empty Since arc selection

at any step depends on the previous nodes selected, the heuristic can be repeated from alternative initial nodes, with the best tour consistently retained Similar rationale was utilized to modify the near- est neighbor heuristic for the order picking problem The procedure first designates the home location as in- cumbent location i, and initializes the set of scheduled

locations S = ~ A comparison of the costs of mov-

ing from the incumbent location to each unscheduled

location j E N, j ~ S, yields the minimum-cost selec-

tion of location j* for the next position in the schedule

such that j* ~ S, j* E Pk* for k* E M, rk > 0, and

ci,j <~ ci,j for all locations j ~ S, j E Pk for k E M,

and r~ > 0 The set of scheduled locations is updated

to reflect the inclusion of location j*, S = S + {j*},

and the picking requirement for part type k* is modi-

fied by the amount of inventory stored at location j*,

rk* = r/~ - min{qj., rk* }

By designating location j* as the incumbent location

(location i above), the greedy process is repeated until

the picking requirement for all part types is satisfied

(rk = 0 for all k), or until the problem is found to be

infeasible

Thus, the quantity retrieved of each part type is

tracked as the modified nearest neighbor tour is con-

structed When the quantity picked of some part type

meets the associated requirement, both the current

node and all remaining locations containing that part

type are effectively removed from the unvisited set

The final tour generated by this process may contain

locations that can be removed without affecting the

feasibility of the solution One way this condition oc-

curs is when a set of locations is just short of meeting

the requirement for one part type, and the next location

added to the tour satisfies the requirement completely

In this case, all locations previously selected to sat-

isfy the requirement for this part type could be deleted

from the tour without affecting its feasibility More

generally, an added location may not contain sufficient

inventory to satisfy the requirement by itself, but can

be used in combination with more than one subset of

the previously chosen locations A further complica-

tion is that overpicking can occur for more than one

part type In this case, determining which locations of

one part type to remove from the tour depends on the

mix of locations retained for all other part types

Thus, while the modified nearest neighbor heuristic

can detect overpicking as the tour is constructed, the

status of unrequired locations is not resolved until the

entire tour is constructed (i.e., until the picking re-

quirement for all part types is satisfied) Given the lo-

cations in the final tour, all of the combinations of sub-

sets of these locations that meet the picking require-

ment of the associated part type without overpicking

can be identified and evaluated, with the cost of each alternative computed by assuming that the relative po- sition of retained locations remains consistent with the sequence defined by the initial tour The combination

of locations that yields the lowest cost then becomes the heuristic solution Note that the heuristic could be further modified by any procedure that attempts to im- prove the sequence for each alternative combination

of locations (see, e.g., Lin and Kernighan, 1973)

4.2 Shortest arc

Another greedy approach to the TSP is to iteratively select the shortest arcs among the nodes in the graph, and then construct a final tour from these arcs The se- lection process is constrained to maintain each node's degree below two, and to avoid subtours with the ad- dition of each new arc As a result, chosen arcs appear

in the final tour, and all nodes are connected

In the order picking problem, short arcs may con- nect two locations which are both not required in the final tour, since as other arcs are chosen, the quan- tity requirements for individual part types become sat- isfied Therefore, the modified shortest arc heuristic must check not only for subtours, but also for chosen arcs that should be deleted from the solution

To overcome this problem, selected arcs are added

to a candidate list of arcs that may be in the final so- lution A separate tour list maintains those arcs that form the heuristic solution Whenever an arc is added

to the candidate list, the list is examined for arcs that can be added to the tour list This is accomplished by searching for arcs that satisfy all of the following con- ditions: (i) either location joined by the arc contains

a part type whose requirement is not satified by the current tour; (ii) adding the arc does not cause a node

on the tour to exceed degree two; (iii) at least one

of the nodes will have degree two if the arc is added

to the tour; and (iv) adding the arc does not result

in a subtour Arcs are added to the candidate list and moved to the tour list in this manner until the tour list satisfies the picking requirement for all part types

As with the modified nearest neighbor heuristic, the final tour may contain locations that can be eliminated without affecting the feasibility of the solution In this case, an identical process is used to check for over- picking, and to determine which locations should be removed from the solution

R.L Daniels et al./European Journal of Operational Research 105 (1998) 1-17 4.3 Randomized construction

In each of the greedy heuristics, the best available

arc is consistently selected for inclusion in the tour

This myopic process can result in poor solutions, such

as when the arcs available at the end of the heuristic

tour construction all involve high cost In these cases,

choosing arcs that incur slightly higher cost earlier in

the heuristic can yield a more attractive set of avail-

able arcs late in the heuristic, often leading to signif-

icant reductions in total cost Modifying the greedy

heuristics so that arc selection is randomized repre-

sents one approach for allowing multiple solutions to

be explored while still providing some direction for

tour construction Feo et al ( 1991 ) describe a similar

approach, labeled greedy randomized adaptive search

procedures, to a machine scheduling problem

To implement this approach, a short list of alterna-

tives are generated whenever the heuristic must choose

the next arc in the tour One of the alternatives is then

chosen at random, with equal probabilities assigned

to each outcome A set number of random solutions

is generated in this manner, and the best solution re-

tained as the heuristic solution A randomized version

of each greedy heuristic is thus defined by the num-

ber of alternatives considered at each decision point,

and by the number of random solutions generated be-

fore the search is completed Clearly, if only one alter-

native is considered at each step, the original greedy

heuristic results For a given number of random itera-

tions, retaining a large list of alternatives may reduce

solution quality, as a significant proportion of the total

computational effort is directed towards unpromising

areas of the solution space To strike a balance between

search breadth and depth, the size of the candidate list

was set at four for the nearest neighbor heuristic, and

five for the shortest arc heuristic, with 5000 iterations

performed for each procedure

4.4 Tabu search

Tabu search is a strategic approach for generating

approximate solutions to combinatorial optimization

problems The general tabu-search methodology, de-

scribed by Glover (1989, 1990a), is characterized at

each stage of the process by a set of moves that can

be made to transform an incumbent solution into a

new solution, and by an evaluation function that mea-

sures the attractiveness of each move By consistently implementing the move in the local search neighbor- hood with the highest evaluation, the process avoids becoming trapped at a locally optimal solution while maintaining promising search trajectories Since suc- cessive moves need not yield strict improvement in the global objective, a list of forbidden, or tabu, moves is maintained to avoid cycling in the search, and a set

of aspiration criteria is typically defined to indicate when a move's tabu status can be temporarily over- ridden Creative manipulation of short-term and long- term memory structures allows the level of search in- tensification and diversification to be tailored to the problem of interest Glover (1990b) provides a partial list of difficult optimization problems for which tabu- search heuristics have yielded approximations of im- pressive quality In particular, tabu-search techniques have been successfully applied to the travelling sales- man problem by Knox (1989) and by Malek et al (1989a,b)

The tabu-search heuristic developed for the order- picking problem employs a nested-search strategy that decomposes the problem into inventory assignment and location sequencing subproblems At the highest level, alternative subsets of locations whose inventory

in total satisfies the picking requirement for each part are generated within a tabu-search framework (sub- procedure ATABU) Feasible assignments are then evaluated using a second tabu-search approach to de- termine the sequence in which the given locations should be visited to minimize total costs (subproce- dure STABU)

The tabu-search heuristic starts by constructing

an initial assignment and sequence of locations that together satisfy the picking requirement for all part types This is achieved by using the modified near- est neighbor heuristic previously described to yield initial assignment S H = S and associated sequence

~'H(sH) By totalling the costs implied by crH(S H) (including the cost of moving from the last location in

~.H (S H) back to the home location), an upper bound,

C (S H, zrH (SH)), on the optimal solution is obtained The greedy solution provides an assignment of lo- cations that is used to initialize the inventory assign- ment tabu search, i.e., S is the initial incumbent solu- tion for subprocedure ATABU Solution S also initial- izes the assignment tabu list T A - {S}, which identi- fies the set of forbidden moves, and thus prohibits the

search from cycling back to a previously-encountered

state Due to the potentially large computational cost

associated with accessing and maintaining a list of all

previously-encountered solutions, the assignment tabu

list only maintains a record of the 7~Aax solutions most

recently encountered in the search, with 7~Aax set equal

to the total number of locations in the warehouse, [N]

Since subprocedure ATABU must search through

the possible combinations of locations whose inven-

tory in aggregate satisfies the picking requirement of

all part types, it is necessary to define an appropri-

ate neighborhood N B ( S ) of incumbent assignment S

Note that:

s {s, slMi},

where Sk represents the set of locations in S that con-

tain inventory of part type k, i.e., j E Sk if and only if

j E S and j E Pk Subprocedure ATABU exploits this

structure to generate three distinct and exclusive neigh-

borhoods for incumbent assignment S, corresponding

to solutions in which the number of locations in the as-

sociated tour remains constant, increases, or decreases

from the number contained in S The first neighbor-

hood, NB1 (S), is defined by the set of all assignments

that can be constructed from S by exchanging a single

location contained in the incumbent solution j c Sk

with a single location f ~ Sk that contains the same

part type, f E Pk, where only exchanges in which the

picking requirement for part type k remains satisfied,

but not oversupplied (i.e., all remaining locations in Sk

are required to meet the picking requirement for part

type k), are permitted Since neighborhood NB1 (S)

is defined over all k E M and over all j E Sk and

j ' ¢ Sk, [NB~(S)I is bounded by:

{is, i × (IP l- is l)

kEM

The second neighborhood, NB2 (S), is defined by the

set of all assignments that can be constructed from S

by exchanging a single location contained in the in-

cumbent solution j E $1¢ with two locations f , j~

S~ that contain the same part type, f , f f E Pk, where

only exchanges in which (i) the picking requirement

for part type k remains satisfied (but not oversup-

plied), and (ii) an upper limit on the total number

of locations for part type k (easily computed by sort-

ing locations j E Pk in nonincreasing order of q j, and

adding the qj in this order until the total just meets

or exceeds the picking requirement for part type k)

is not violated, are permitted Again, since neighbor- hood NB2 (S) is defined over all k E M and over all

j E Sk and j ' , f ' ~ Sk, INB2(S)I is bounded by:

The final neighborhood, NB3 (S), is defined by the set

of all assignments that can be constructed from S by exchanging two locations contained in the incumbent solution j, f C Sk with a single location j " ~ Sk that contains the same part type, jpt C P~, where only ex- changes in which (i) the picking requirement for part type k remains satisfied (but not oversupplied), and (ii) the lower limit on the total number of locations for part type k (easily computed by sorting locations

j E Pk in nondecreasing order of q j, and adding the

qj in this order until the total just meets or exceeds the

picking requirement for part type k) is not violated, are permitted Since neighborhood NB3 (S) is defined over all k E M and over all j , f C Sk and j " ~ Sk, INB3(S) I is bounded by:

Note that the structure used to construct the neigh- borhood of incumbent solution S can exclude feasi- ble solutions and it is not always possible to reach

an optimal solution from any initial solution, espe- cially when there is a part type k with Pk consisting

of many locations with quantifies much smaller than picking requirements rk, and few locations with quan- tities exceeding rk More typically, the neighborhood allows the number of locations containing part type k

to systematically increase or decrease However, ex- periments with neighborhoods that make it possible to reach an optimal solution from any feasible solution greatly increases the computational requirements with virtually no improvement in heuristic solution quality for the problem instances considered in this paper

At each iteration of the subprocedure ATABU, the neighborhood of incumbent solution S:

NB(S) = NB1(S) t_J NB2(S) UNB3(S),

is generated as described above Each neighbor S ~ E NB(S) of incumbent solution S is evaluated by pass-

R.L Daniels et al./European Journal of Operational Research 105 (1998) 1-17

ing the associated assignment to the location sequenc-

ing tabu-search subprocedure, STABU, provided that

S ~ is not contained in the current assignment tabu list

Subprocedure STABU searches through the space of

possible sequences in which the given locations can be

visited, and returns a sequence, ¢r H (S I), whose cost,

C (S', ~ (S I)), approximates the cost of the associ-

ated optimal tour The minimum-cost solution encoun-

tered in the current iteration of subprocedure ATABU

is consistently retained, so that at the end of each it-

eration the minimum-cost neighbor, S*, of incumbent

solution S can be identified such that:

C(S*,TrH(S*))= min {C(S',~'H(S'))}

S'ENB(S)

I f C (S*,~rn(S*)) < C (Sn,~rH(Sn)), then S* is re-

tained as the best solution thus far encountered by the

search, S n = S*, and the value of C (sH,~rr~(sH)) is

set = C (S*, "n~(S * ) ) The search then moves to so-

lution S* by setting S = S* and T A = T A + {S* } (with

the oldest element of T A dropped from the tabu list if

ITAI > TmAax)

The process of generating the neighborhood of the

incumbent solution, evaluating each alternative assign-

ment using subprocedure STABU, and advancing the

search to the best solution encountered in the cur-

rent iteration, is repeated until a limit on either the

total number of iterations allowed for subprocedure

ATABU, [mAax, or the total number of consecutive iter-

ations tolerated without achieving an improvement in

global performance, GAma x, is reached While other TSP

heuristics (e.g., Lin and Kernighan, 1973) can be used

to identify an appropriate picking sequence, subproce-

dure STABU has proven in our computational experi-

ence to be an effective means for obtaining close ap-

proximations of the optimal sequence with little com-

putational effort

Subprocedure STABU evaluates each element of

the neighborhood of incumbent assignment S by em-

ploying a tabu-search approach to determine an ap-

proximation of the sequence in which the input loca-

tions should be visited to satisfy the picking require-

ment of all part types at minimum cost Given assign-

ment S t E NB(S), an initial sequence, ¢r(S~), is con-

structed using the nearest neighbor heuristic for the

associated TSP By totalling the costs implied by this

initial sequence, rrrt(S ~) = ¢r(S~), an upper bound,

C (S ~, ¢rn(SP)), on the optimal solution is obtained

Greedy solution or( S ~) represents not only an initial sequence for subprocedure STABU, but also initial- izes the sequence tabu list T s ~ {¢r(S ~) }, which again prevents the search from cycling back to a previously- encountered sequence In the interest of computational efficiency, the sequence tabu list only maintains a record of the TSax solutions most recently encountered

in the search, with 7~Sax set equal to the number of locations in the warehouse, INI

The neighborhood, NB (¢r(S ~)), of incumbent se- quence 7r(S t) is defined by the set of all pairwise ad- jacent interchanges that can be made among the loca- tions in 7r(S~); hence:

INB (¢~(8'))I = I S ' l - 1

Each alternative sequence ~r'(S ~) E NB (~r(S')) is evaluated by computing its cost, C (S',~a(S')), pro- vided that 7r(S ~) is not contained in the current se- quence tabu list The minimum-cost solution encoun- tered in the current iteration of subprocedure STABU

is consistently retained, so that at the end of each iter- ation the minimum cost neighbor, 7r* (S ~), of incum- bent sequence ~r(S') can be identified such that:

C(S',Tr*(S'))= min {C (¢F(S'))}

~r' ( S~)ENB(~r( St) )

I f C (S',Tr*(S')) < C (S',rrH(S')), then 7r*(S') is retained as the best sequence thus far encountered

in the search, "a'H(s ') = ~-*(S~), and the value of

C (S',~I(S')) is set equal to C (S',¢r*(S')) The

search then moves to sequence 7r*(S ~) by setting rr(S') = ¢r*(S') and T s = T s + {~-*(S')} (with the oldest element of T s dropped from the tabu list if ITSl >/~S~x)

The process of generating the neighborhood of the incumbent solution, calculating the cost of each al- ternative sequence, and advancing the search to the best solution encountered in the current iteration, is repeated until a limit on either the total number of iter- ations allowed for subprocedure STABU,/mSax, or the total number of consecutive iterations tolerated with- out achieving an improvement in global performance, GSrn ~, is reached

The nested-search strategy outlined in this section exploits the problem decomposition to search over the space spanning the sets of locations that together sat- isfy the picking requirement for all part types, and the possible sequences in which these locations can be

visited to minimize total cost The computational per-

formance of the tabu-search heuristic over a series of

test problems is discussed in the next section

Table 1 Experimental design: combinations of the number of locations,

INI, the number of part types, IMI, and the minimum number of locations per part type, P

5 Computational experience

To explore the computational behavior of the solu-

tion approaches developed in this paper, procedures

were coded and implemented on a 486 personal com-

puter As shown in Table 1, the experimental design

adopted for this study consisted of test problems in-

volving IN I 10, 12, 20, 30, 40, 60, and 80 loca-

tions and IM[ = 3, 6, 9, and 12 part types The min-

imum number of locations associated with each part

type, P = mink IPk], and the amount of inventory,

q j, contained at each location j = 1,2 IN[, were

also controlled, thus providing insight into the influ-

ence of these elements of the environment on problem

tractability and solution quality

Test problems were constructed by first randomly

generating coordinates for the [gl locations on a 100 ×

100 grid, with the home location assigned coordinates

(0, 0) The first IMI x P locations were used to satisfy

the specified lower limit on the number of locations

per part type, with locations (k - 1 )P + 1 through kP

assigned part type k The remaining I N I - ( [MI x P ) lo-

cations were then randomly assigned a part type With

the requirement for each part type set at rk = 1.0 for

k = 1,2 IMI, the amount of inventory contained

at location j, j = 1,2 INI, was determined us-

ing one of four distinct methods: (i) qj = 1.0, so that

any single location can satisfy the entire requirement

for the associated part type, (ii) qj = U[0.6, 1.1],

implying that either one or two locations containing

each part type must be visited in any feasible solution,

(iii) qj = 0.5, SO that exactly two locations must be

visited for each part type, and (iv) qj = U[0.0,0.5],

requiring feasible solutions to visit more than two lo-

cations per part type After generating inventory quan-

tities for each location, the total amount of available

inventory was calculated for each part type k, k =

1,2 IMI, when methods (ii), (iii), or (iv) were

used; if ~ j c e k qJ < 1.0, the associated quantities

were normalized so that ~ j s e k qJ = 1.0, thus ensur-

ing problem feasibilty Ten replications were gener-

ated for each combination of [NI, IMI, and P specified

in Table 1, and for each inventory assignment method,

resulting in a total of 1520 test problems

Number of Number of part types (IM[) locations (IN[) 3 6 9 12

30 4,6 2,4 1,2 1,2

40 6 4 , 6 2,4 1,2

For each test problem, a lower bound on the opti- mal cost was obtained by generating all combinations

of locations whose inventory in total satisfies without oversupplying the picking requirement for all part types, so long as the total number of assignments did not exceed 50 000 For each feasible assignment,

a Lagrangean relaxation of the resulting travelling salesman problem was solved, and the lowest cost over all assignments retained The test problems were also solved using the randomized nearest neighbor, randomized shortest arc, and tabu search heuristics described in Section 4 In addition, solutions were obtained for the modified single-pass versions of the nearest neighbor and shortest arc heuristics, as well

as for a part-by-part (PBP) method commonly en- countered in practice The PBP heuristic first focuses

on satisfying the picking requirement of part type 1

by computing the cost, co,j, of travelling between the

home location and each location j E P1 containing part type 1 Locations are then sorted in nondecreasing order of co,j, and a solution constructed by selecting

locations in this order until the corresponding total amount of inventory exceeds the picking requirement for part type l Repeating the process sequentially for part types 2, 3 IMI yields the final solution The computational results are presented in Tables

2 and 3 Table 2 shows that a lower bound on the optimal cost could be determined for all problems in which INI = 10, 12, and 20, and for all of the 30- location problems involving 12 part types For these problems, the average number of feasible assignments

is reported, along with the average standardized dif- ference between the cost of solutions generated by