A warehouse location-routing problem

Vol. 19B, No. 5, pp. 381-396, 1985 Printed in the U.S.A.

0191-2615/85 $3.00+.00 © 1985PergamonPressLtd.

Transpn. Res.-B

A WAREHOUSE

LOCATION-ROUTING

PROBLEM

JOSSEF PERL Department of Civil Engineering, University of Maryland, College Park, MD 20742, U.S.A.

and M A R K S. DASKIN Department of Civil Engineering, Northwestern University, Evanston, IL 60201, U.S.A. A b s t r a e t I T h e interdependence between distribution center location and vehicle routing has been recognized by both academics and practitioners. However, only few attempts have been made to incorporate routing in location analysis. This paper defines the Warehouse Location-Routing Problem (WLRP) as one of simultaneously solving the DC location and vehicle routing problems. We present a mixed integer programming formulation of the WLRP. Based on this formulation, it can be seen that the WLRP is a generalization of well-known and difficult location and routing problems, such as the Location-Allocation Problem and the Multi-depot Vehicle Dispatch Problem. It is therefore a large and complex problem which cannot be solved using existing mixed-integer programming techniques. We present a heuristic solution method for the WLRP, based on decomposing the problem into three subproblems. The proposed method solves the subproblems in a sequential manner while accounting for the dependence between them. We discuss a large-scale application of the proposed method to a national distribution company at a regional level.

1. I N T R O D U C T I O N

The choice of locations for distribution centers (warehouses) is among the most critical management decisions. Both the cost of a distribution system and the level of customer service provided by the system are significantly affected by the number, size and locations of the distribution centers (DCs), as well as by the decision on which customers to serve from each center (i.e. the allocation of customers to DCs). Consequently, a significant amount of research has been devoted to the development of mathematical models for DC locations (Akine and Khumawala, 1977; Elson, 1972; Feldman and Lehrer, 1966; Geoffrion and Graves, 1974; Geoffrion and McBride, 1978). The DC location problem can be stated as follows: A company ships from one or more supply sources to a number of regional DCs. The shipping from the supply sources to the DC (trunking) is commonly done in Truck Loads (TLs) and/or Car Loads (CLs). The company then delivers goods from the DCs to its customers. The problem is to determine the number, size and locations of the DCs and the allocation of customers to them so as to minimize the total system cost. The problem is basically one of finding the optimal balance between warehousing costs and transportation costs. Warehousing costs include both fixed and variable costs, while the transportation costs consist of the trunking and delivery costs. The transportation cost is often the most significant cost component in a distribution system. Yet, existing models for DC location inaccurately represent the transportation cost, or, more specifically, the delivery cost. This paper presents a new methodology for DC location which represents delivery costs more accurately than do existing location models. Existing mathematical models for the DC location problem represents the delivery cost by the "Moment Sum" equation: M

N

2 2 c..x. j = l i=1

where Cli Xji M N

= = = =

the the the the

cost per unit weight (volume) of shipping from DC j to customer i quantity shipped from DC j to customer i number of DC sites number of customers. 381

(1)

382

J. PERLand M. S. DASKIN

The implicit assumption behind the "Moment Sum" representation of the delivery cost is that loads are delivered from the DCs to the customers on a straight-and-back basis and that the cost of delivering from the DC to any customer is independent of other deliveries made. Since delivery vehicles usually operate on routes which include multiple customers, the cost of delivering any given amount to a given customer may be marginal if the customer can be included on an existing route with an insignificant increase in travel distance. However, the cost of delivering the same amount to the customer may be significant if the delivery vehicle does not have the required excess capacity and a special trip is needed. Delivery operations are difficult to analyze since they involve a mixture of the line-haul and terminal-type activities (Shrock, 1974). Delivery operations may be divided into three major components (see Fig. 1): stem distance, stop time and variable running distance. Stop time is a function of shipment characteristics and is independent of the location of the DC or of the occurrence of other deliveries. The stem distance is the sum of the distances from the DC to the first customer on the route and from the last customer back to the DC. The stem distance represents the effect of the location of the DC on the delivery cost. Finally, the variable running distance is the source of dependence between the cost of delivering to any given customer and the occurrence of other deliveries. Both the stem distance and the variable running distance depend on vehicle routing. Therefore, the delivery cost in the D C location problem depends on vehicle routing. The interdependence between DC locations and vehicle routing has been recognized by both academics (Eilon et al., 1971; Webb, 1968; Wren and Holliday, 1972) and by practitioners, as stated by Rand (1976). "Many practitioners are aware of the danger of suboptimizing by separating depot location from vehicle routing." However, the computational complexity of the vehicle routing problem made it impractical to incorporate routing into location analysis (Rand, 1976). Little work has been done on integrated location-routing problems. The error from using the "Moment Sum" to represent the delivery cost in the DC location problem was first studied by Webb (1968). He concluded that while in many cases the "Moment Sum" and the actual routing distance might be highly correlated, their use resulted in the selection of different DC locations. Eilon et al. (1971), suggested an approach for estimating routing cost in the DC location problem, based on the probabilistic approximation of the Travelling Salesman Problem (TSP). This estimate is accurate only under some special conditions which rarely exist in practice (Eilon et al., 1971). Or (1976) defined a location problem in which the facilities were uncapacitated, the delivery costs include the cost of regular deliveries and the cost of direct shipments, and there is no fixed cost of establishing a facility. Having realized the complexity of the problem, Or (1976) made no attempt solve it. Laport and Nobert (1981) formulated an integer programming model for the problem of selecting an initial node in the Multi-Travelling Salesman Problem (MTSP). This problem is significantly simpler than the single DC location

~ STEM DISTANCE

~

VARIABLERUNNING

/

STEM DISTANCE

Fig. 1. Route components.

A warehouse location-routingproblem

383

problem. The ALLOCRO model developed by Federgruen and Lageweg (1980) considers a distribution system with two types of intermediate facilities--DCs and local depots. The major weakness of the ALLOCRO model is that it does not solve the location problem, but starts with some initial set of selected DC locations and evaluates incremental changes of that set. The remainder of this paper is organized as follows. Section 2 presents the definition and mathematical formulation of the combined DC location-routing problem. This problem is extremely large and complex due primarily to the embedded Travelling Salesman Problem, and needs to be solved heuristically. Section 3 discusses the heuristic model which is based on decomposing the location-routing problem into three subproblems. The subproblems are formulated mathematically and the relationship between each subproblem and the combined location-routing problem is discussed. Computational results on a test problem are presented in Section 4. Section 5 summarizes a large-scale application of the model to a national distribution company at a regional level. Section 6 contains the paper's conclusions. 2. THE WAREHOUSE LOCATION-ROUTING PROBLEM In this section we define the Warehouse Location-Routing Problem (WLRP) as the problem of solving simultaneously the DC location and vehicle routing problems. We then formulate the optimization model for the WLRP. As shown by Perl (1983), the WLRP is a generalization of some well-known and difficult location and routing problems, including the Transportation Location Problem (TLP), the General Transportation Problem (GTP), the Multi-Depot Vehicle Dispatch Problem (MDVDP), and the Travelling Salesman Problem (TSP). The WLRP is defined as follows. The location and expected requirements of a set of N customers are given. Each customer is to be assigned to a regional warehouse which will periodically supply the customer's expected requirement. The requirements are specified in units of a single representative commodity. Also given is a set of M potential sites for the warehouses and the warehousing costs at each potential site. The warehousing costs consist of a fixed cost for establishing the warehouses and a linear variable cost. Trunking cost is linear with respect to the quantity shipped. Deliveries are carried out by delivery vehicles dispatched from the warehouses and operating on routes which include multiple customers.The maximum number of delivery routes that can be operated by the fleet is given. The total cost of delivery operations is proportional to the distance traveled by the delivery vehicles. The problem is to determine the number, size and locations of the warehouses, the allocation of customers to warehouses, and the delivery routes, so as to minimize the sum of transportation costs and warehousing costs. The location of DCs is usually considered to be a long-term "strategic" decision. The WLRP represents the interdependence between this strategic decision and shorter-term "tactical" decisions regarding the type and size of delivery fleet. In this context, the size of the delivery fleet is not given but should be determined so as to minimize the total long-term cost. Consequently, the WLRP does not assume that the actual size of the fleet is given, but assumes only that there is some limit on the maximum number of routes that can be operated. This should not be viewed as a restriction but as an additional flexibility of the model which allows us to analyze the effect of limited fleet size. The solution under the assumption of no restriction on fleet size can be obtained using the following model by setting the maximum number of routes to a large number. In this case the actual number of routes will be determined by the model so as to minimize the total travel distance. The subscripts, sets, parameters and variables used in the model for the WLRP are as follows: (i) Subscripts: h, g = "point" index (customer or DC site)

(1 <=h<=N + M), i j k s

= = = =

(1 < - g < ~ N + M)

customer index (1 _-< i ---
384

J. PERLand M. S. DASKIN (ii) Sets: p=

{ph[1 ~ h _--
W=

{wj[N + 1 <=j <- N + M} set of all potential DC sites {wjlZj = 1} set of opened DC sites

L= (iii) Parameters: dij = distance between points i and j

qg = requirement of customer i FCi = fixed cost of establishing DC j

VCj = Tj = Crsj =

variable cost per unit throughput at DC j maximum throughput at DC j unit cost of trunking from supply source s to DC j

Ck = capacity of vehicle (or route) k Dk = maximum allowable length of route k CMk = cost per mile of delivery vehicle on route k (iv) Variables: if point g precedes h on route k otherwise if customer i is allocated to DC j otherwise if customer i is allocated to DC j otherwise f,j = quantity shipped from supply source s to DC j. The W L R P can be formulated as a mixed-integer programming model. N+M

Min C(X, Y, Z, f) =

S

E

N+M

FCj'Zj + E

E

N+M

CTsj'fsJ +

s=l j-N+I

j-N+I K

N+M N+M

+ E k-1

E

E

g=l

h=l

CM,'dg,,'Xghk

Subject to: K

N+M

(i) ~'~ ~'~ Xihk = l k-l

i=

1...N

h=l

N

N+M

(ii) ~

q, ~

i=1

Xihk -
k = 1 ...K

h-I

N+M N+M

(iii) ~

dghXghk < Dk =

~

g--I

h=l

k = 1... K h= 1 ...N+M

K

(iv) ~

~

~ Xgh, >- 1 V(V, V)

g ~ V h E V k= 1

where V is a proper subset of P containing W N+M

(v) ~

x~, -

g=l N+M

(vi)

E j=N=I S

(vii) ~ s=l

N+M

~

x~

g=l

= o

k = 1

K ""

h = 1...N

+ M

N

~ X a * -< 1

k = 1 . . .K

i=1 N

f,j - ~ qiYij = 0 j = N + 1 . . . N + M i=1

E j-N+I

385

A warehouse location-routing problem

s (viii) ~] f s j -

TiZj<--O

j = N + I . . .N

+ M

s=l

N+M

N+M

(ix) ~] Xih, + 2 h=l

(x) Xghk = O, 1 Yi/ = 0, 1 Zj = 0, 1 fsj >- 0

i = 1 • . •N

X j h k - Yij~-- l

j= k=

h=l g = 1...N i=l...N j=N+

N + I . . .N 1...K

+M

+ M

h = 1...N

+ M

k=

j=N+

1...N

+M

j = N+

1 ...N

+ M.

1

•

.

.

K

1...N+M

s=l...S

The objective function minimizes the sum of fixed warehousing cost, trunking cost, variable warehousing cost and delivery cost, respectively. Constraints (i) require that each customer be placed on a single route. This implies that the requirement of any single customer is less than the vehicle's capacity. If the requirement of some customer exceeds the capacity of the vehicle, it is first reduced by scheduling direct TL shipments from the DC to that customer. Constraints (ii) are the vehicle capacity constraints. Constraints (iii) specify that the total length of a route should not exceed the maximum allowable route length. This constraint set does not result from the statement of the WLRP, but is imposed in some cases by labor contracts which set a limit on the time drivers can spend on the road. Constraints (iv) require that every delivery route be connected to a DC. Constraints (v) stipulate that every point entered by the vehicle should be left by the vehicle. Constraints (vi) state that a route cannot be operated from multiple DCs. Constraints (vii) require that the flow out of a DC be equal to the flow into a DC. Constraints (viii) limit the flow through a DC to the DC capacity. Constraints (ix) link the allocation and routing components of the WLRP; they specify that a customer can be allocated to a DC only if there is a route from that DC going through that customer. This can be explained as follows. By constraints (i) each customer is assigned to exactly one route k. Constraints (i), together with constraints (iv), (v) and (vi), imply that there must be exactly one DC on each route K. Thus, consider any given customer i*, which by constraints (i) is assigned to exactly one route--say, k * - - a n d which by constraints (iv), (v) and (vi) is linked to a D C - - s a y , j * - - w e obtain the following: Xi*hk* = 1

and

h

Xj*hk* = 1.

~

(2)

h

Consequently, constraint set (ix) requires Yiv* = 1, thereby assigning customer i* to DC j*. If customer is not on a route from DC j, constraints (ix) are satisfied for either Yij = 0 or Yil --- 1, and since Yij has a positive cost coefficient, it is set to zero in any optimal solution to WLRP. Constraints (x) specify that the route design variables (Xuhk), the allocation variables (Y~j) and the location variables (Zj) can take on the values of 0 or 1, and that the flow from the supply sources to the DCs (fsj) be non-negative. For a small problem with one supply source, three potential DC locations, 14 customers, four routes and symmetric distances, the formulation (WLRP) includes 1201 integer variables and 8455 constraints. Increasing the number of customers to 100 and the number of routes to 25 would increase the number of integer variables to 265,528 and the number of constraints to over 6.33 × 1029. Even for small problems, the model for the WLRP cannot be solved using existing mixed-integer programming techniques. The next section discusses a heuristic solution method for the WLRP. 3. S O L U T I O N

METHOD

FOR

THE

WLRP

The proposed solution method is based on four assumptions. The first allows us to deal sequentially with some of the interdependencies within the WLRP. The remaining three are common assumptions in existing DC location and vehicle routing models, aimed at reducing the computational complexity of these problems. It should also be stated that the triangle inequality is assumed to hold throughout this work.

386

J. PERLand M. S. DASKIN

We first assume that a good approximation to the optimal solution of the M D V D P for any configuration of DCs can be obtained from a sequential minimization of the variable running distance and the stem distance. A simple geometric analysis can show that at least in the cases of " l o c a l route" (in which the variable running distance is large relative to the stem distance) and " o v e r - t h e - n i g h t " route (in which the variable running distance is small relative to the stem distance), the error from this assumption is small (Perl, 1983). Second, the trunking cost per unit is assumed to be constant. Under this assumption the total trunking cost is constant and the DCs may be viewed as supply rather than transshipment points. It is common in existing warehouse location models to ignore the trunking component and to consider warehouses as supply points (Akine and Khumawala, 1977; Geoffrion and McBride, 1978; Nauss, 1978). This assumption is justified by the low relative magnitude of the trunking component of the transportation costs. Practical experience indicates that the unit cost per mile in delivery operations is ten times higher than the unit cost in trunking (Geoffrion, 1978). Third, we assume that the delivery fleet consists of standard vehicles. This is a common assumption in the vehicle routing models which avoids the significant additional complexity that would otherwise result from the need to deal with the assignment of customers to vehicle types. Finally, we assume that there is no constraint on maximum route length. This assumption is strictly computational and can be eliminated at the expense of additional computation effort. Under the last three assumptions the following modified formulation of the WLRP (MWLRP) is obtained:

Min C ( X , Y, Z ) =

Z

FCj'Zj

+

Z

j =N+ I K

qi.rij

j =N+ 1

N+M N+M

+ ~'~ E

E

k~l

h=l

g=l

CM'dgh'Xxhk

Subject to: K N+M

(i) E Z k=l

l

i=l...N

h=l

N

N+M

qi Z

(ii) Z i=l

Xihk ~ C

k = 1 . . . K

h~l K

(iii) g~V h@V k = l

where V is a proper subset of P containing W N+M

(iv)

Z

N+M Xhg k --

g=l

E

X g hk = 0

k = 1.. h = 1 ..

K N + m

g=l

N+M

N

(v) Z

k= 1...K

j=N+I N

i~l

(vi) ~ ] q i Y u -

TjZj<= 0

j = N + 1 . .

N + M

i=l

N+M

(vii) ~

Xih, +

h=l (viii)

i=l...N

N+M

~ , Xjh~ -- Yij<= 1 h=l

0, 1 Yu = O, i Zj = 0, 1

Xghk =

j = N + 1 . . .N k

=

1

.

g = 1 . . .N + M i = 1...N j = N + 1 ...N

.

+ M

. K

h = 1 . . .N + M j = N + 1 ...N

k = 1 . . .K + M

+ M.

The formulation (MWLRP) is obtained from the original formulation (WLRP) as follows. By the second assumption, the trunking rate is the same between any supply source and any DC site. Consequently, the second component of the objective function of (WLRP) is constant

A warehouse location-routingproblem

387

and can be dropped from the formulation of the WLRE The second assumption implies that the DCs may be viewed as supply points, and therefore constraints (vii) of (WLRP) can be dropped. By the third assumption the fight-hand side of constraints (ii) of (WLRP) can be replaced by a standard vehicle capacity. Constraint set (iii) can be dropped from (WLRP) by the fourth assumption. The formulation (WLRP) then becomes (MWLRP). The heuristic solution method is based on decomposing the MWLRP into three components and solving the components (subproblems) either optimally or heuristically in a sequential manner, while accounting for the dependence between them. The MWLRP is decomposed into the following subproblems: (i) The Complete Mutli-Depot Vehicle-Dispatch Problem (Phase One) (ii) The Warehouse Location-Allocation Problem (Phase Two) (iii) The Multi-Depot Routing-Allocation Problem (Phase Three). Phase One assumes that all potential DC sites are used, and constructs an initial set of routes which minimizes the total delivery cost. As shown in Fig. 2, the warehousing cost and warehouse capacity are not considered in phase one. Phase Two locates the warehouses and allocates the routes constructed in Phase One (or Phase Three) to them. As Fig. 2 shows, in Phase Two only the variable running distance component of each route is held fixed, while the constraints on vehicle capacity and the number of routes are satisfied as part of the input. Phase Three simultaneously reallocates the customers to warehouses and solves the multi-depot routing problem for the warehouses selected in Phase Two, so as to minimize the variable warehousing cost and delivery cost. As shown in Fig. 2, the only component held fixed in Phase Three is the fixed warehousing cost. A formal statement and mathematical formulation of each subproblem is given below. The Complete MDVDP is defined as follows: N customers with given locations and expected requirements are to be supplied from M potential DC sites. The requirements are specified in units of a single representative commodity. Each customer should be placed on a single route which is operated by a delivery vehicle dispatched from a DC site. The maximum number of routes K that can be operated by the delivery fleet is given. The cost of delivery operations is linear in the distance travelled by the delivery vehicles. The problem is to design the delivery routes so as to minimize the total cost of delivery operations. The complete MDVDP can be formulated as a zero-one integer programming problem. K

Min

C(X)

=

N+M N+M

~

~

~,

k=l

g=l

h=l

dg h •

Xghk

Subject to: K

N+M

(i) ~

~

k=l

Xih~ = 1

i=

1...

N

h=l

N

(ii) ~

N+M

qi E

i=1

Xihk

~

C

k =

1 ...

K

h-I K

(iii) ~

~

~]Xg~k=> 1

V(V,V)

g E V hU_V k - I

where V is a proper subset of P containing W. N+M

(iv) ~

Xhg,-

g=l N+M

N

j=l

i=1

g=l

(v) (vi)

N+M ~] X,h, = 0

k = h=

1

K "'" 1...N+M

k=l.../( Xgh~ = O, 1

g =

1 . . .N

+ M

h =

1 . . .N

+ M

k=l...K.

388

J. PERLand M. S. DASKIN Warehousing Cost

Phase Fixed

Variable

Delivery Routes

Warehouse Capacity

Stem Distances

Variable Running

Distances

Constraints

Vehicle Canac~tv

Number o f Routes

One

N

N

X

X

x

x

Two

X

X

X

P

s

s

Three

F

X

X

X

x

x

F = Fixed as part of input. N = Not considered. S = Satisfied as part of input. X = Decision variable related--part of objective function.

Fig. 2. Fixed and variable elements in each of the three phases.

The formulation is the same as that presented by Golden et al. (1977) for the multi-depot routing problem except that a maximum route length constraint is not included in (MDVDP). In the Complete MDVDP, the set of depots is the entire set of potential DC locations. The formulation (MWLRP) reduces to (MDVDP) under the following conditions: (a) F C j = O,

VCj = VC

j = N + 1.....

N + M

N

(b) ~-" q,--
j = N + 1.....

N + M.

i=1

Condition (a) states that there is no fixed cost associated with opening of DC and that all the potential DC sites have the same variable cost. Under this condition, the first term of the objective function of (MWLRP) is zero, while the second is constant independent of the decision variables and may be eliminated. Since there is no fixed cost ( F C j = 0), the location variables (Zj) in (MWLRP) can be set to one for all DC sites. Since the capacities (~) are effectively unlimited by condition (b), constraints (vi) of (MWLRP) can be dropped. Finally, the allocation variables (Yij) can be omitted and constraints (vii) can be dropped from (MWLRP). The formulation (MWLRP) then becomes (MDVDP). The complete MDVDP is a distinct component of the (MWLRP). Given a feasible solution to (MWLRP), the set of delivery routes associated with it is a feasible solution to (MDVDP). This set of routes determines the delivery cost in the MWLRP. Since every set of routes associated with a feasible solution to (MWLRP) is feasible to (MDVDP), the optimal solution of (MDVDP) provides a lower bound on the delivery cost in the WLRP. The Complete M D V D P solution therefore minimizes the total routing distance and can be expected to provide a set of routes with "close to minimum" variable running distance. The second subproblem, the Warehouse Location-Allocation Problem (WLAP), can be stated as follows. A set of routes, each of which consists of an ordered sequence of customers without linkage to a warehouse is given. The total expected demand on each route is given. Each route is to be allocated to a regional warehouse which will periodically supply the expected demand on that route. The set of M potential sites for the warehouses and the capacity at each site are given. Also given are the warehousing costs at each site and the stem distances between each route and each warehouse site. The warehousing costs consists of a fixed cost for establishing the warehouse and a linear variable cost. The cost of delivery operations is linear in the distance traveled by the delivery vehicles. The problem is to determine the number, size, and locations of the warehouses, and the allocation of routes to them so as to minimize the sum of warehousing and delivery costs.

A warehouse location-routingproblem

389

The WLAP can be formulated as the following zero-one integer programming problem: MinC(Y,Z) =

~

FCj . Zj +

j=N+ 1 N+M

+

E j=N+I

~

VC~

Qk " Y~j

j =N+ 1 K

ECM'SDk,'Yk, k=l

Subject to: N+M

(i)

~]

rkj=

1

k=

1...K

j=N+I K

(ii) ~'~ Q~" Ykj - Tj. Zj-< 0 j = U + 1 . . . N + M k=l

(iii) Ykj = 0, 1 j = N + 1...N+M Zi = 0, 1 j = N + 1 . . . N + M. In addition to the notation defined in Section 2, the (WLAP) uses the following terms: if route k is allocated to DC j otherwise Qk = total expected requirement on route k

X,j

= I1

(0

SDkj = stem distance between route k and DC site j The objective function of (WLAP) minimizes the sum of the fixed cost, variable warehousing cost and stem distance cost. Constraints (i) specify that each route should be operated from a single DC. Constraints (ii) specify that both the allocation varibles (Ykj) and the location variables (Zj) are binary variables. The formulation (MWLRP) reduces to (WLAP) if the variable running distance component of the WLRP is prespecified (Perl, 1983). In other words, if the solution to the variable running distance component of (MWLRP) is given, the optimal solution to (MWLRP) can be obtained from the optimal solution to (WLAP) (Perl, 1983). The third subproblem, the Multi-Depot Routing Allocation Problem (MDRAP), is defined as follows: N customers with given locations and expected requirements are to be supplied from R supply points with given locations. Also given are the variable cost and the capacity at each supply point. Customers' requirements are specified in units of a single representative commodity. Each customer should be placed on a single route which is operated by a delivery vehicle dispatched from a supply point. The maximum number of routes that can be operated by the delivery fleet is given. The cost of delivery operations is linear to the distance traveled by the delivery vehicles. The problem is to allocate customers to supply points and to design delivery routes so as to minimize the sum of variable warehousing costs and delivery costs. The MDRAP is a generalization of the MDVDP in that it accounts for the variable costs and the capacities at the supply points. The MDRAP can be formulated as a zero-one integer programming model. K

Min C(X, Y) =

~ j=N+I

Subject to: K

(i) ~

N+R

~

k=l N

Xihk = 1 i = 1 . . . N

h=l N+R

(ii) ~ qi ~, Xihk <- C i=1

k = 1 ...K

h=l K

(iii) ~', ~] ~ Xghk ~ 1 '¢(V, V¢,) gEV hEV k= I

qi

"

+ E k=l

N+R N+R

E

E CM'dgn'Xgh,

g=l

h=l

390

J. l~RL and M. S. DASKIN

where V is a proper subset of P containing L. N+R N+R (iv) ~] X,,g, - ~ X~hk = 0 g=l

g=l

N+R

(v)

K

"'" I...N+R

h=

N

~

~X~jk_< 1

j=N+I N

(vi) ~

k = 1

k = 1 . . .K

i=1

qsY~s -

T;<=O

j = N + 1 . . .N

+ R

i=1

N+R

N+R

i = 1 . . .N

(vii) ~ Xihk + ~ Xjhk -- Yij<= 1 j = N + 1 . . . N h=l h=l k = 1 . . .K (viii) X~hk = 0, 1 Yij = 0, 1

g

=

1 . . .N

i = 1...N

+

R

h

=

j =N

1 . . .N

+ e +

+ 1...N

R

k

=

1 . . .K

+R.

The above formulation implicitly assumes that, if necessary, the potential warehouse sites in (MWLRP) are renumbered so that the first R sites are open. Looking at ( M W L R P ) and (MDRAP), one can see that the M D R A P is a distinct component of the MWLRP. Under the assumptions stated earlier, the M D R A P differs from the M W L R P only in that in the M D R A P the locations of the DCs are given. Consequently, (MWLRP) reduces to (MDRAP) if {~

(a) Z~ =

j = N + 1...N + R j = N + R + 1 . . .N + M.

This condition states that a given subset o f R DC sites is open while the remaining M - R DC sites are closed. Consequently, it is unnecessary to include in the formulation of the M D R A P those variables associated with the M - R closed sites. Under this condition, the first term of the objective function of (MWLRP) is constant, and can therefore be dropped from the formulation. The objective function of (MWLRP) then becomes the same as that of (MDRAP). By condition (a), the first R constraints in constraints (vi) of (MWLRP) may be reduced to constraint set (vi) of (MDRAP), while the remaining M - R constraints can be dropped. The formulation (MWLRP) then becomes (MDRAP). We now proceed to discuss how the above subproblems are combined to form a solution method for the WLRP. A flow chart of the solution method is shown in Fig. 3. Phase one solves the complete M D V D P and provides an initial set of routes. This set of routes (suppressing the linkages to DCs) is used as input to phase two, which solves the WLAP. The W L A P solution provides the optimal subset of DC locations, given the variable running distance associated with the input routes. The selected DC locations are used as input to phase three, which solves the MDRAP. Phase three attempts to improve both the allocation of customers to the open DCs, and the routing of delivery vehicles from these DCs. The output from phase three provides a new set of delivery routes, which are used as input to a new iteration through phase two. Two important features should be noted about the proposed method: (i) the s o l u t i o n s o b t a i n e d in both p h a s e t w o a n d p h a s e three a r e f e a s i b l e s o l u t i o n s to the W L R P ; (ii) the s o l u t i o n o b t a i n e d in p h a s e t w o is f e a s i b l e to p h a s e three w h i l e the solution f r o m p h a s e three is f e a s i b l e to the next iteration o f p h a s e two. Thus each iteration through phases two and three results in

the same or an improved solution for the WLRP. The procedure terminates when the improvement at some phase is less than a given • (e > 0). The values of the total system cost obtained by the method represent a monotonically decreasing sequence. The method is finite since there is a lower bound on the total system cost in the WLRP, which is the value of the objective function in the optimal solution to (MWLRP). The complete M D V D P and the M D R A P are routing-type problems which cannot be solved optimally using existing optimization methods. We developed heuristic methods for the complete M D V D P and the MDRAP. The literature on multi-depot routing algorithms is sparse. A common approach is the so called "allocation-first routing-second" which first allocates customers to

A warehouse

Complete

1

Solv:

location-routing

problem

391

MDVDP

MDRAP

1

Fig. 3. Flow chart of the solution method for the WLRP.

DCs and solves a sequence of single depot problems (Gillett and Johnson, 1976; Or, 1976). This approach is inherently suboptimal since the allocation of customers to DCs and the design of delivery routes are interdependent. Existing aglorithms which attempt to perform simultaneously the allocation and routing are limited by their storage requirement (Golden et al., 1977). The heuristic methods for the complete MDVDP and the MDRAP have similar structure; they consist of a basic algorithm and three improvement procedures. The basic algorithm generalizes the sequential savings method of Mole and Jameson (1976) to the multi-depot case. A key feature of the improvement procedures is that they include screening techniques to identify good improvement options. The proposed heuristic methods perform simultaneously the allocation and routing, yet are capable of solving efficiently relatively large routing problems. A detailed description of these heuristics, which include a comparative computational analysis showing their efficiency relative to some existing methods, is given in Per1 (1984b). The WLAP is a location-type problem solved using a specialized exact method for O-1 integer programming. In the current computer implementation of the methodology, the WLAP is solved by the Implicit Enumeration algorithm of Lemke and Spielberg (1967). An initial test of the efficacy of the proposed solution method for the WLRP was conducted by comparing the objective function value of the heuristic with a lower bound on the optimal solution to a small problem. The lower bound was obtained from an LP relaxation of (MWLRP) (Perl, 1983). For a test problem with 12 customers and two DC sites, the LP relaxation model included 104 constraints and 418 variables. The LP model was solved using subroutine ZX3LP of the IMSL computer package. The LP solution was obtained in 221.2 seconds on CDC Cyber 170/730. The objective function value in the solution obtained from the heuristic was 5.2% higher than the lower bound. The heuristic method required 0.56 seconds of CPU time. Additional testing of the heuristic procedure indicates that it can efficiently solve relatively large problems and can be used in practice to analyze the effects of a large variety of scenarios which

392

J. PERL and M. S. DASKIN Table 1. Data for example problem i.

Customer's

Demand

(Qi) = 20 units i = l...S5

2.

Vehicle Capacity

3.

Fixed Warehousing

4.

Variable Warehousin g Cost

5.

Warehouse Capacity

6.

Cost Per Vehicle Mile

7.

M a x i m u m Number of Routes

(C) = 120 units Cost

(FC) = $240 j = I...15 (VCj) = 0.74 S/unit j = i...15

(Tj) 550 units j = I...15 (CM) = $i.0 (K) = II

include changes in both company policies and in the environment in which it operates (Perl and Daskin, 1984).

4. C O M P U T A T I O N A L E X A M P L E

This section illustrates the proposed method for the WLRP on a test problem with 55 customers and 15 potential DC sites. In addition to the data summarized in Table 1, the locations of the customers and the DC sites were specified by their planar coordinates. The solution for the test problem is shown in Fig. 4. The computational results are summarized in Table 2. The complete MDVDP solution was obtained in 5.30 seconds of CPU time on a CDC Cyber 170/730. As expected, the complete MDVDP solution is characterized by a large number of DCs and relatively short travel distance (low delivery cost). The solution to the WLAP was obtained in 236.94 seconds of CPU time, with a limit of 5000 iterations in the Implicit Enumeration algorithm. This solution was obtained at iteration 3425 and could not be improved by the algorithm when the limit on the number of iterations was increased to 10,000. One possible reason for the slow convergence of the Implicit Enumeration algorithm may be differences in magnitude between the cost coefficients of the allocation variables (Yij) and the location variables (Zj). Differences in the magnitude of matrix coefficients and cost coefficients have been found to affect the convergence of Implicit Enumeration algorithms. The WLAP solution significantly reduced the total cost. As expected, this reduction was obtained from a substantial reduction in fixed cost, which resulted from a reduction in the number of open DCs, at the expense of a smaller increase in the delivery cost. Note that the computation time required by the Implicit Enumeration algorithm depends largely on the number of variables in the WLAP. The number of variables is related to the product of the number of routes and DC sites. In this case, the relatively large computational effort required for solving the WLAP is due to the large ratio of DC sites to customers. Our computational experience indicates that larger problems can be solved at lower computational effort when the number of potential DC sites is reduced. Solving the MDRAP resulted in a significant total cost reduction over the WLAP solution, and required 1.8 seconds of additional CPU time. The cost improvement resulted from a substantial reduction in delivery cost. In this case, the MDRAP solution is the final solution since the next iteration through phase two did not result in improvement.

5. C A S E S T U D Y

This section presents a large-scale application of the model to a regional distribution system of an intemational manufacturer and distributor. As of 1980, the company distributed roughly 133,000 different products, classified into five major product groups. The domestic operations of the company are decentralized through 20 divisions organized into three distinct business groups and serving 116,000 customers. Each business group has a number of divisions within

A warehouse

I3

Warehouse

0

Customer

location-routing

problem

393

Site

Fig. 4. Test problem solution.

each of the three functions-manufacturing, marketing and distribution. Most of the distribution decisions are made at the division level. The company’s distribution function is broken down into five autonomous distribution divisions. Four of these divisions distribute specialty items, shipped on demand. The largest distribution division handles items that are shipped regularly to customers via delivery-type operations. Our study conducted a DC location analysis for this last division, at the regional level. Roughly 75% of the division’s DCs are used jointly with other divisions. In the case of a joint DC, it is physically split and approximately 90% of the space is allocated to the division under study. The study area includes Missouri, Oklahoma and Western Kansas. In this area, the division has roughly 320 regular customers, currently served from two DCs located in Kansas City and Oklahoma City. The purpose of the study was to evaluate the design of the division’s distribution system, with regard to the number, size and locations of the DCs in the study area, and the allocation of customers to DCs in that area. The data required for this study were obtained from three sources: (1) a sequence of interviews with distribution managers, (2) the division’s data base and (3) DC managers. A detailed description of the data collection and compilation process is given in (Perl, 1983). In addition to the two existing DCs, two other potential DC locations were identified by the division’s management in Wichita and Tulsa. The company recently opened a new DC in North Carolina, from which estimates of the fixed and variable warehousing costs for the potential DC locations were obtained. The study compared three alternative distribution system designs-the “current system,” the “best system with the existing DCs” and the “‘proposed system.” Routing and scheduling

394

I.

PERL

and M. S. DASKIN

Table 2. Summary of test problem results

I

1 Allocation of Customers System

1. current system

226

92

2. Best System for Existing DC5

133

185

3. Proposed system

T

Okla. City

IKansas

0

Excess Capacity (ft3)

Allocation of Demand (lb) Kansas City

318

1 Okla.

City fKansas

City

1 Okla.

City

118032

69703.5

957061

224882

81209.5

106526

986640

197088

o

1

187735.5

)

--

1

135796

of delivery vehicles is currently done at the DCs. Since data on the routing and scheduling in the “current system” were not available, the total cost of the “current system” could not be estimated. The “best system with the existing DCs” is one in which the current DCs are maintained and the optimization is done only with respect to customer allocation and vehicle routing. The “proposed system ” is the one obtained from the model with all the four locations being considered as potential DC sites. The problem of this study included 3 18 customers and four DC locations. The output from the model (“proposed system”) included a single DC in Oklahoma City. The “proposed system” included eight routes with total travel distance of 9290.41 miles. The total weekly cost of the “proposed system” was $14,639. The “best system with the existing DCs” was obtained by solving the MDRAP with the existing DCs opened. This system included nine routes with total travel distance of 6782.2 miles and total weekly cost of $14,660. The total cost of the “proposed system” was marginally lower than that of the “best system with existing DCs.” However, the total travel distance in the “best system with existing DCs” was 27% lower than that of the “proposed system.” One can argue that the average travel distance between a customer and a DC is a proxy for the level of customer service that can be provided by a distribution system, where shorter average travel distance means a capability to provide faster, more reliable service. In view of the insignificant cost difference, the “best system with the existing DCs” may be preferred in this case over the “proposed system”. This illustrates a shortcoming of DC location models; namely, they do not represent the level of customer service. This is an important area for future research. Table 3 shows the aggregate allocation of customers in each of the three systems. The most important result shown in Table 3 is the difference between the “current system” and the “best system with the existing DCs. ” The allocation of customers to DCs in the “current system” is significantly different from that of the “best system with the existing DCs.” One hundred one customers (31.8%) and 3824.5 lbs. (20.4%) were allocated differently in these

Table 3. Summary

of allocations

No. of Routes

Total Cost ($)

CPU Time' (Seconds)

3667.17

6881.17

5.31

11

4581.84

6385.84

241.84*

10

4261.62

5795.62

243.71

Component

Sites Used

I

Complete MDVDP

1,2,4.5,7, 10,11,12,13, 14

11

II

WLAP

2,10,12

III

MDRAP3

2,10,12

Phase

in various systems

Total Travel Distance (Miles)

2. Time to complete 5,000 iterations of the implicit enumeration algorithm.

3 In this problem, the solution obtained from the MDRAP is the final solution.

A warehouse location-routing problem

395

two systems. In the "best system with the existing DCs" more customers and more demand are allocated to the DC in Oklahoma City than are currently allocated to that center. In both of the above two systems there is significant excess capacity at the two DCs.

6. S U M M A R Y

AND CONCLUSION

The interdependence between DC location and vehicle routing has been recognized by both academics and practitioners. However, few attempts have been made to develop integrated location-routing models. This paper defined the Warehouse Location-Routing Problem (WLRP) as the problem of simultaneously solving the DC location and vehicle routing problems. The WLRP was formulated as a mixed-integer programming problem. Both the Multi-Depot Vehicle Dispatch Problem and the Travelling Salesman Problem are subproblems of the WLRP. Consequently, the WLRP is a large and complex problem which cannot be solved directly using existing mixed-integer programming techniques. We presented a heuristic method for solving the WLRP, based on decomposing the problem into three subproblems and solving the subproblems either optimally or heuristically in a sequential manner, while dealing with the dependence between them. In a test on a small problem, the solution provided by the heuristic method was 5.2% higher than a lower bound obtained by an LP relaxation of the mixed-integer programming model. Based on this limited analysis on a small test problem, it seems that the proposed heuristic method can provide good solutions to the WLRP. A detailed step-by-step test of the proposed method for the WLRP was conducted on a problem with 55 customers and 15 potential DC sites. The solution to phase one (Complete MDVDP) was obtained in 5.31 seconds of CPU time and, as expected, was characterized by a large number of open DCs and low delivery cost. The second phase (WLAP) significantly improved the solution by reducing the fixed warehousing cost. The solution to phase two required 236.94 seconds of CPU time. The relatively large computational effort required for phase two in this case was due to the large ratio of DC sites to customers which resulted in a large number of variables in the WLAP. Our computational experience indicates that problems with significantly larger numbers of customers can be solved at lower computational effort when the number of potential sites is reduced. Phase three (MDRAP) achieved significant additional improvement and required only 1.8 seconds of additional CPU time. The case study presented in this paper demonstrates the ability of the model to solve largescale problems. The distribution system designed by the model ("proposed system") includes only one of the two DCs currently operated by the company in the study area. The difference in total cost between the "proposed system" and the "best system with the existing DCs," obtained by considering the current DC locations as given, was found to be insignificant. We might therefore conclude that with regard to site selection, the system currently operated by the company is a good one. However, the allocation of customers to DCs in the "current system" differs significantly from the "best allocation" as obtained from the model. It seems that the company can reduce the cost of the current system by shifting demand from the DC in Kansas to that located in Oklahoma City and restructuring the delivery routes accordingly. Acknowledgements--Partial support for the research presented in this paper was provided by grants from the National Council of Physical Distribution Management and the Transportation Center at Northwestern University. The authors remain, of course, solely responsible for the contents of the paper.

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