Department of Industrial Engineering and ...

Department of Industrial Engineering and Management Sciences Northwestern University, Evanston, Illinois 60208-3119, U.S.A. Working Paper No. 03-012

Tradeoffs Between Customer Service and Cost in an Integrated Supply Chain Design Framework

Zuo-Jun “Max” Shen Department of Industrial and Systems Engineering University of Florida 303 Weil Hall P.O. Box 116595 Gainesville, FL 32611-6595

Mark S. Daskin Department of Industrial Engineering and Management Sciences Northwestern University Evanston, IL 60208

October 2003

This research has been supported in part by the National Science Foundation under grant DMI-0223277.

Tradeoffs Between Customer Service and Cost in an Integrated Supply Chain Design Framework Zuo-Jun “Max” Shen Dept. of Industrial & Systems Engr. University of Florida 303 Weil Hall P.O.Box 116595 Gainesville, FL 32611-6595 e-mail: [email protected]

Mark S. Daskin Department of Industrial Engineering and Management Sciences Northwestern University Evanston, IL 60208 e-mail: [email protected]

Abstract In designing a supply chain, firms are often faced with the competing demands of improved customer service and reduced cost. We develop a model that incorporates fixed distribution center location costs, working and safety stock inventory costs at the distribution centers, fixed and variable transport costs from a plant to the distribution centers and delivery costs to customers. Service is measured by the fraction of all demands that are located within an exogenously specified service standard. The nonlinear model simultaneously determines distribution center locations, the assignment of demands to distribution centers and the inventory policy at the distribution centers to optimize the cost and service objectives. We use the weighting method to find all supported points on the tradeoff curve. Our results suggest that significant service improvements can be achieved relative to the minimum cost solution at a relatively incremental cost.

1.

Introduction

Strategic supply chain design and redesign has become a major challenge for firms as they simultaneously try to reduce the related operating costs and improve customer service. In recent years, many companies have realized that important cost savings can be achieved by optimizing the supply chain as a whole. There are three major cost factors associated with designing and managing a supply chain: the facility location costs, the inventory management costs, and the distribution costs. These three key cost elements of a supply chain are highly related. For example, an effective distribution scheme depends on the location of facilities, and a good inventory management scheme depends on effective shipment plans. Thus, these three costs should be considered jointly in the same model. Being able to build a decision support system which integrates these cost elements of a supply chain network is a major challenge and can provide a company with a tremendous competition advantage in the market.

A crucial question in the design of effective supply chain networks is the identification of the number and locations of distribution centers (DCs). There exist extensive literature in location theory that addresses this problem; however, the majority of these location models do not consider the inventory related costs. Only recently, Shen (2000), Shen et al. (2003), and Daskin et al. (2002) introduce a joint location-inventory model which incorporates working inventory and safety stock inventory costs at the DCs and the distribution costs in the location model. The problem is to find the number and location of the DCs, the assignment of customers to DCs, so as to minimize facility location costs, shipment costs, and inventory costs. The key difficulty in the model is that both the inventory costs at DC and the shipment costs from plant to DC dependent nonlinearly on

customer assignments, which is endogenously determined and is not known a priori. By exploiting the special structure of the problem, they can solve this nonlinear problem efficiently. One key observation they derive from the model is that the traditional uncapacitated fixed charge (UFC) location model, which does not include any inventory costs, always underestimates the total cost significantly and locates more DCs than does the location-inventory model.

However, customer responsiveness, which is another important consideration when designing a supply chain, has been ignored in these papers. A retailer may be assigned to a DC which is very far away if doing so can reduce the total costs. This may not be desirable in a highly competitive business environment. Success in this highly competitive environment requires not only producing the best product at the lowest price, but also providing excellent customer service (Vijayan, 2000). A measure of the service quality is the service time, defined as how long it takes to transport the products (services) to the customer site when they are needed. For example, General Motors Corporation developed a program in Florida to reduce the amount of time Cadillac buyers must wait for new cars. New cars would be held at DCs and would be made available to the dealers on order. A 24-hour delivery standard was used to deliver new cars from DCs to dealers (Wall Street Journal, 1995). This program has since been expanded to include areas in Maryland and California. Another example comes from the PC industry. To provide high quality service, IBM has implemented a parts stock plan to support timebased service strategy. Specifically, IBM wants to keep the response time within a

threshold level, say 2 hours to one set of customers, 8 hours to another set of customers, and 24 hours for the rest of the customers (Ma and Wilson, 2002).

In both of the above problems, the locations of the DCs which will hold the inventory (vehicles, parts) is a critical set of decisions, since it will impact both the total costs (facility location costs, inventory costs, and transportation costs) and the customer responsiveness. There is a clear need to evaluate trade-offs among the total costs and customer service. To accurately capture the trade-offs, we build a multi-objective optimization model that balances all the relevant costs in a service network.

2.

Literature Review

There exists an extensive literature on the multi-objective analysis of location problems; however, most of these papers focus on continuous and network models. We only review multi-objective discrete location models in this paper.

Ross and Soland (1980) is one of the earliest papers on multi-objective location problems. They use an interactive approach that involves solving a finite sequence of the generalized assignment problems. Lee et al. (1981) apply integer goal programming technique to a facility location problem with competing objectives. The same technique has been applied to a multi-objective fire-station location problem by Badri, Mortagy, and Alsayed (1998), who consider objectives that incorporate costs, travel times, and travel distances from stations to demand sites. Another technique, hierarchical programming, has also been applied to multiple-objective location models where a clear

ranking or hierarchy of the different objectives exists. Daskin and Stern (1981) address two hierarchical objectives in their paper on emergency medical service vehicle deployment. The primary objective is to minimize the number of service vehicles needed to satisfy a certain service requirement; and the secondary objective is to maximize the extent of multiple coverage of service zones. There also exist works that apply general multi-objective mixed integer linear programming techniques to location problems. For example, Solanki (1991) develops an algorithm for solving bi-objective mixed integer linear programming problems and then apply it to bi-objective location problems. This algorithm uses an approximate scheme to represent the noninferior solutions. Finally, simulation has also been used to validate multiple-objective location models (Heller, Cohon, and Revelle, 1989).

Multiple-objective programming has also been applied to dynamic location problems (Schilling, 1980) and location problems with uncertainty (Belardo et al., 1984). Belardo et al. study the problem of locating oil spill response equipment using a partial set covering model.

With the existing equipment, their objective is to attain the best

protection while minimizing the risk of being unprepared for events like oil spills. Other earlier works on multiple-objective location models include a decision support computer system (Hultz, et al., 1981) and surveys (Revelle, Cohon, and Shobrys, 1981; Current, Min, and Schilling, 1990).

In a more general model, Jayaraman (1999) studies a multi-objective model for a service facility location problem with multiple products. Three objectives are considered in his

model: minimizing the fixed cost of opening facilities; minimizing the total variable costs of serving customer demands; and minimizing the average response time for serving customers. A mixed integer program is formulated and solved on a 30-node problem.

Recently, Fernandez and Puerto (2003) consider the multi-objective uncapacitated facility location problem. Each objective corresponds to a different scenario proposed by a different decision maker on the realizations of facility set-up costs and the allocation costs of customers to facilities. They also provide a review of other multi-objective location models, including continuous and network location models.

There are several papers that apply a multi-objective approach to supply chain planning models. For example, Sabri and Beamon (2000) propose an integrated multi-objective supply chain model for use in simultaneous strategic and operational supply chain planning. The objectives include cost, customer service levels (fill rates), and flexibility (volume or delivery). They tested their mathematical programming model on an example system consisting of three raw materials, two finished products, five vendors, three plants, four distribution centers, and five customer zones.

Nozick and Turnquist (2001) present an optimization model which is closely related to ours: minimize cost and maximize service coverage. They use a linear function to approximate the safety stock inventory cost function, which is then embedded in a fixedcharge facility location model. Using a linear function to approximate a nonlinear function (in fact it is a square root function of the total demand variance, see Shen,

Coullard, and Daskin, 2003) allows them to easily solve the location-inventory problem, but the error induced will be significant unless the number of DCs is very large.

3.

Formulation of the model

In this section we formulate a multi-objective model that can be used to explore the tradeoff between the location, transportation and inventory costs on the one hand and the level of service provided to the customers on the other hand. We explore this tradeoff in the context of a multi-echelon distribution system composed of a single plant, multiple distribution centers (DCs) and multiple retailers. The location/inventory model at the heart of this tradeoff finds the optimal number and locations of the DCs and the assignment of retailers to the DCs to minimize the sum of the (1) fixed distribution center (DC) location costs, (2) transportation costs from the plant to the DCs, (3) working inventory costs at the DCs, (3) safety stock costs at the DCs, and (5) transportation costs from the DCs to the retailers. Service is measured by the percentage of demands that can be served within an exogenously specified coverage distance. We begin by summarizing the formulation of the location/inventory model (Shen, 2000; Daskin, Coullard and Shen, 2002; Shen , Coullard and Daskin, 2003). We define the following notation: Inputs and sets: I J f

j

set of retailers set of candidate distribution centers fixed cost of locating at candidate DC j

µi

mean daily demand at retailer i

σ i2

variance of the daily demand at retailer i days per year

χ

Lj

lead time (in days) to deliver from the plant to candidate site j

d ij h Fj gj

distance from candidate site j to retailer i holding cost per item per day (at the DC) fixed cost of placing an order to the plant from candidate DC site j

aj

cost per unit of a shipment from the plant to candidate site j

β

weight on transportation costs weight on inventory costs critical value of a standard Normal random variable, Z, such that

fixed cost of a shipment from the plant to candidate site j

θ

zα

P (Z > z α ) = α

β and θ are used to allow us to vary the relative importance of transportation and

inventory costs relative to the fixed costs.

We also define the following decision

variables:

Xj

=

1 if we locate at candidate site j  0 if not

Y ij

=

1 if demands at retailer i are assigned to a DC at candidate site j  0 if not

With this notation, we can formulate an integrated location/inventory model as follows: Minimize

Subject to

∑ f j X j + ∑ ∑ dˆ ij Y ij + ∑ K j ∑ µ i Y ij + ∑ Θ ∑ σˆ ij2 Y ij

j∈J

j ∈J

i∈ I

j ∈J

i∈ I

(1)

∀i ∈ I

(2)

∀i ∈ I ; ∀j ∈ J

(3)

X j ∈ {0,1}

∀j ∈ J

(4)

Y ij ∈ {0,1}

∀i ∈ I ; ∀j ∈ J

(5)

∑ Y ij = 1

j∈J

Y ij ≤ X

where

j∈J i∈ I

j

(

βχ µ i d ij + a j

dˆ ij

=

Kj

=

Θ

=

θ h zα

σˆ ij2

=

L j σ i2

2θ hχ

)

F j+β g j

The objective function (1) is composed of four terms. The first term represents the fixed cost of locating DCs. The second term captures the cost of shipping goods from the DCs to the retailreers as well as the variable costs of shipping from the plant to the DCs. The third term represents the working inventory costs and includes the fixed cost of ordering from the DC to the plant as well as the fixed shipment costs from the plant to the DCs.

Finally, the fourth term captures the safety stock carrying costs.

The first

constraint (2) ensures that each demand node is assigned to exactly one DC. The second constraint (3) stipulates that demands can only be assigned to open DCs.

Finally,

constraints (4) and (5) are integrality constraints. Note that were it not for the final two terms of the objective function, the model would be structurally identical to an uncapacitated fixed charge location model. The working inventory carrying costs represent the optimal EOQ (economic order quantity) costs while the safety stock costs capture the cost of holding sufficient inventory to ensure that the probability of stocking out during a lead time is less than or equal to α . In essence we are approximating a (Q,r) inventory model with type I service (Hopp and Spearman, 1996; Nahmias, 1997). We make one additional simplifying assumption that the ratio of the variance of the demand per unit time to the mean demand per unit time is the same for every retailer. In other words, we assume that σ i2 µ i = γ for every demand node i. While this may seem like an overly restrictive assumption, we note that if the retailer demands are

Poisson – and the Poisson distribution can be approximated very well by the Normal distribution if the mean is sufficiently large – then this assumption is exact with γ = 1 . With this additional assumption, the objective function can be rewritten as

Minimize

∑ f j X j + ∑ ∑ dˆ ij Y ij + ∑ Kˆ j ∑ µ i Y ij

j∈J

j ∈J i ∈ I

j ∈J

i∈ I

(6)

where Kˆ j = K j + Θ L j γ .

This model is still a non-linear integer programming model, but Daskin, Coullard and Shen (2002) show that the subproblems that result from relaxing constraint (2) can be solved in O ( I log I ) time for each such problem and there are O ( J ) such subproblems that must be solved at each Lagrangian iteration. When coupled with intelligent heuristics for finding good upper bounds and when the Lagrangian problem is embedded in branch and bound, problems with hundreds of retailers and hundreds of candidate DCs can be typically be solved in minutes on today’s computers. While the model above captures important facility location, transportation and inventory costs, some retailers may be served very well, in the sense that they are located very close to the distribution centers to which they are assigned, while other retailers may be served very poorly by this criterion. The maximal covering location problem (Church and ReVelle, 1974) maximizes the number of demands that can be covered by a fixed number of facilities. Demands at node i are covered if demand node i is assigned to a facility that is within D c of node i, where D c is the coverage distance. Instead of maximizing the number of covered demands, for fixed total demand, we can minimize the number of uncovered demands. In a manner similar to Daskin (1995), we can then

formulate a problem that simultaneously minimizes the fixed costs of the facilities and a weighted sum of the uncovered demands as follows:

∑ f j X j + W ∑ ∑ d~ ij Y ij

Minimize

j∈J

(7)

j ∈J i ∈ I

Subject to (2)-(5) where W is the weight on the uncovered demands and if d ij > D c ~  βχ µ d ij = 0 i if not 

Objectives (6) and (7) can be combined as follows: Minimize

h ∑ f j X j + ∑ ∑ d ij Y ij + ∑ Kˆ j ∑ µ i Y ij

j∈J

j ∈J i ∈ I

j ∈J

i∈ I

(8)

h where d ij = dˆ ij + W d~ ij . This model is structurally identical to (6). The only difference is

that we penalize all assignments of demand nodes to DCs that are more than D c away from the distribution center. By varying the weight W on uncovered demands, we can trace out an approximation to the set of non-inferior solutions to the tradeoff between location/inventory costs (6) and uncovered demands. Very small values of W correspond to minimizing the total location/inventory cost; very large values of W are equivalent to minimizing the total location/inventory cost subject to the constraint that all demands are covered.

4.

Generating the tradeoff between cost and service

We now illustrate how to determine appropriate values for W, the weight to be placed on uncovered demands. Suppose we already have two different non-inferior solutions with location/inventory costs C 1 and C 2 respectively and uncovered demands U 1 and U 2 respectively as shown in Figure 1. The weighting method (Cohon, 1978)

attempts to find solutions to the south and west of the line connecting any two noninferior solutions when plotted in objective space. If we equate the objective function values at each of the two solutions we obtain C1 + W U 1 = C 2 + W U 2 which, when solved for W, yields W=

C 2 − C1 U 1 −U 2

(9)

By selecting two adjacent non-inferior solutions, we can use the weight determined in (9) in objective function (8) to determine whether or not there is a solution to the south and west of the line connecting the two solutions; i.e., in region A. The weighting method will not find non-inferior solutions that might exist to the north and east of the line; i.e., in region B.

Location/Inventory Cost

(U 1, C1) B A

(U 2 , C 2) Uncovered Demand

Figure 1 – Determining the weight, W

Determining the first two points to use in this process is relatively easy. To determine the rightmost point, the one that minimizes the location/inventory cost, we simply minimize (6) subject to (2)-(5). For real-world systems there is unlikely to be a solution with equal (minimal) cost and smaller uncovered demand. For artificial data sets or for cases in which the analyst believes that such solutions might exist, objective function (6) can be minimized and the total location/inventory cost recorded. Then, objective function (8) can be minimized subject to (2)-(5) with a small weight on the uncovered demand (e.g., W=0.5). The total cost (excluding any penalties for uncovered demands) from this model can be compared to the total cost recorded for objective function (6) alone. If the two objective function values are the same, then the solution to (8) provides the optimal rightmost solution in the objective space shown in Figure 1. If the costs are not the same (i.e., the cost from (8) exceeds that found using (6) alone), the weight W should be halved in objective function (8) and the process repeated until the two costs are identical.

In the computational results outlined below, we did not

implement this refinement since we were dealing with real data for which the likelihood of there being alternate optimal solution with smaller levels of uncovered demand is very small. To find the leftmost point of Figure 1, the solution that covers all demand at minimum cost, consider any demand node m that is not covered in the solution that minimizes cost without regard for coverage.

The contribution of this node to the

(

)

objective function due to the second term of (8) alone is β χ µ m d mj (m ) + a j (m ) + W where j (m ) is the index of the facility to which demand node m is assigned and W is the weight

that will be assigned to uncovered demands. One way for node m to be covered is to

locate a facility at the node at a fixed cost of f m . If the fixed cost of locating a facility at node m is less than the cost of assigning node m to the facility at j (m ) to which the node was assigned in the pure minimum cost solution, the node will surely be covered. In other W>

words, fm

βχ µ m

if

(

f m < β χ µ m d mj (m ) + a j (m ) + W

− d mj (m ) − a j (m ) ,

)

or,

solving

for

W,

if

then node m will be covered. Thus, to find the solution that  − d mj (m ) − a j (m ) , m∈U    βχ µ m 

covers all demand at minimal cost, we can set W > max 

fm

where U

is the set of demand nodes that are not covered in the solution to the pure minimum cost problem. Somewhat arbitrarily, we adopted an even more conservative approach setting   f m − W = 4 • max   d ( ) mj m m∈U    βχ µ m

when finding the leftmost point on the tradeoff curve.

Once the two extreme solutions are known – the solution that minimizes the location/inventory cost and the solution that covers all demand at minimal cost – the procedure outlined above can be used iteratively until the solution to (8) for every pair of adjacent solutions in the solution space fails to identify a new point on the tradeoff curve. Note that if there are n points on the final tradeoff curve that is identified, we must solve 2n-1 problems of the form of (8) subject to (2)-(5). Each of these problems is identical in

structure to the model outlined in Daskin, Coullard and Shen (2002) and is solved using the algorithm outlined in that paper. In solving problems other than those needed to get the extreme points of the tradeoff curve, however, we can provide an initial upper bound to the Lagrangian algorithm. In finding solutions that may lie between two solutions with coordinates

in

objective

space

of

(U 1, C1)

and

(U 2 , C 2) ,

we

know

that

C 1 + W U 1 = C 2 + W U 2 is an upper bound on the objective function. In most of the results

reported in the next section we used this bound though one subsection discusses the efficacy of the bound. Qualitatively, this bound will affect the step size used at each iteration of the Lagrangian procedure. More importantly, it may allow us to prune certain nodes in the branch and bound tree earlier than we might be able to do in the absence of this bound.

5.

Computational Results

5.1

Input conditions

Five real-world datasets were used to test the model outlined above.

Basic

characteristics of the datasets as well as key input parameters for the runs are shown in Table 1. The datasets ranged in size from 49 nodes to 263 nodes. Sortcap is a 49-node dataset representing the 48 capitals of the contiguous United States as well as Washington, D.C.

The nominal demand ( µ i ) at each node is equal to the 1990

population of the state (or city in the case of Washington, D.C.). The fixed cost is equal to the 1990 median home value in the city. City1990 is the Sortcap dataset expanded to 88 nodes to include the 50 largest cities in the continental U.S. according to the 1990 census.

Nominal demands in this dataset equal the 1990 city population.

150city

represents the 150 largest cities in the continental U.S. according to the 1990 census. Again, the nominal demand is equal to the city’s population. All fixed costs, however, are set equal to 100,000 in this dataset. Two different coverage distances were tested for this dataset. USA2000 263 Big Cities represents the 263 largest cities in the continental U.S. according to the 2000 census. Demands correspond to the city population and all

fixed costs again equal 100,000. Finally, Europe150 includes the 150 largest cities in Europe. As before, demands equal the city population and all fixed costs equal 100,000.

Total Distance Coverage Lead Datafile Nodes Demand metric Distance Time Sortcap 49 247,051,601 miles 300 1 City1990 88 44,840,571 miles 300 1 150city 150 58,196,530 miles 300 1 150city 150 58,196,530 miles 150 1 USA2000 263 Big Cities 263 78,396,814 miles 300 5 Europe150 150 77,968,385 kilometers 150 5

Holding cost Fixed order cost at a Transportation Days per per unit per DC Cost Weight year year 1 10 0.00001 1 1 10 0.00001 1 1 10 0.00001 1 1 10 0.00001 1 1 10 0.0000005 250 25 100 0.000001 365

Table 1 – Summary of test data

In addition to summary data about the five datasets, Table 1 includes information on those parameters of the model that differed between datasets. Thus, for four of the five test cases using U.S. datasets, the coverage distance was set to 300 miles; for the fifth the distance was 150 miles. For the European dataset, the coverage distance was 150 kilometers. All distances were computed as great circle distances. Table 2 gives the values of the model parameters that are common to all five datasets. Table 3 summarizes the Lagrangian parameter settings for the computational results. Model Parameters (common to all datasets) 1 Variance to mean ratio 1.96 Z-alpha 0.01 Inventory cost weight 1 Dollars per demand mile in local delivery

Table 2 – Model parameters common to all runs

5.2

Model results

Table 4 contains a summary of the basic results for the five datasets. The time required to solve for the entire tradeoff curve generally increases as the number of nodes increases. For the two datasets with under 100 nodes, the total solution time was under 5 minutes. For the larger data sets with 150 nodes and 263 nodes, the solution time ranged from 12 minutes to over 5 hours. (All computation times are on a Dell Latitutde C640 laptop computer running at 2.0 GHz. The program was coded in Delphi.) The number of Lagrangian iterations and the number of branch and bound nodes required to solve for the entire tradeoff curve exhibit similar patterns. The computation time increases as the coverage distance decreases since additional solutions are typically found. Key Lagrangian Parameters Substitution Option at End of Lagrangian Critical Percentage Difference Number of iterations Number of iterations at root node Minimum Alpha Value Allowed Number of failures before changing Alpha # of failures before changing Alpha at root node Crowder damping term Restart failure count on improved solution Assignment search on improved solution Root node forcing CONSTANT in initial Lagrange multipliers SLOPE in initial Lagrange multipliers AVERAGE in initial Lagrange multipliers DEMAND in initial Lagrange multipliers FIXED in initial Lagrange multipliers Do Lagrangian Bounding

Exchange Algorithm 0 400 1200 1E-08 12 36 0.3 TRUE TRUE TRUE 0 0 10 0 10 TRUE

Table 3 – Lagrangian parameters common to all runs

Datafile Sortcap City1990 150city 150city USA2000 263 Big Cities Europe150

Nodes 49 88 150 150 263 150

Solution Lagrangian time iterations 57.25 16,688 277.22 44,717 723.87 50,312 3,345.19 222,964 8,147.92 194,561 18,381.16 1,063,861

Branch and bound No. Min no. nodes solutions sites 35 12 5 133 20 3 143 17 2 777 38 2 541 24 3 4,619 52 4

Max no. sites 14 15 13 38 15 52

Figure 3 – Results for 150city with coverage distance of 150 miles

In addition to information about the computation times, Table 4 also provides information on the number of solutions found for each dataset. For the 4 datasets based on US data and a coverage distance of 300 miles, the number of solutions ranged from 12 to 24; when the coverage distance decreased to 150 miles, 38 solutions were identified. For the European data with a coverage distance of 150 kilometers or about 90 miles, 52 solutions were identified. The minimum number of sites – the number associated with the minimum cost solution – ranged from 2 to 5 while the maximum number – the number of sites needed to cover all demands within the coverage distance – ranged from 13 to 52. As expected, the maximum number of sites increases as the coverage distance decreases since more facilities are needed to cover all demands. Figure 2 shows typical results that we obtained. The figure shows the tradeoff curve for the largest dataset representing the 263 largest cities in the United States and a coverage distance of 300 miles.

The step function is the location/inventory model

objective function while the bar graph, with a scale on the right hand side of the figure, is the number of facilities located. The minimum cost solution is at the far right of the figure; 3 facilities are located at a total annual cost of approximately $6.4 million leaving over 38 million demands (or nearly 49% of the total demand) uncovered. However, for

less than a 6% increase in annual cost, the uncovered demand can be reduced 40% to less than 30% of the total demand. This entails locating one additional facility. On the other hand, covering all demands necessitates a 160% increase in cost and the siting of 15 facilities. Figure 3 shows the minimum cost solution while figure 4 shows a solution that reduces the uncovered demand by 40% using only one additional facility. In the minimal cost solution, distribution centers are located in Glendale, CA, Garland, TX, and Harrisburg, PA; in the shown in figure 4, facilities are located in Bakersfield, CA, Dallas, TX, Indianapolis, IN, and Elizabeth, NJ. Essentially, this solution adds a facility in the Midwest in Indianapolis. Figure 5 shows the tradeoff curve that resulted from using the 150city dataset with a coverage distance of 150 miles. Again, significant reductions in the uncovered demand can be achieved at relatively little increase in cost while covering all demands requires a very large cost increase. With the smaller coverage distance, the increase in cost associated with covering all demands is even greater (over 700% more than the minimal cost).

Figure 2 – Results for USA2000 263 Big Cities with coverage distance of 300 miles

Figure 3 – Minimal cost solution for 263 node problem

Figure 4 – A solution that reduces uncovered demand by 40% while increasing cost by less than 6%

Figure 5 – Results for 150city with coverage distance of 150 miles

5.3

Effects of upper bounding and different optimality gaps

In this section, we evaluate the impacts of changing the convergence criterion and using or not using the bounding rule outlined in section 4 above, which states that, the upper bound on any solution between two solutions C1 + W U 1 = C 2 + W U 2 .

(U 1, C1)

and

(U 2 , C 2)

is

The effects are evaluated in terms of three measures of the

solution difficulty – the solution time, the number of branch and bound nodes evaluated, the number of Lagrangian iterations needed to identify the tradeoff curve – and one measure of the solution quality, the number of solutions found. The solution time, number of branch and bound nodes, and the number of Lagrangian iterations all go down as the convergence criterion is relaxed from optimality (0.0%) to 0.5% and finally to 1.0%. The three measures all increase by about 50% if bounding is not employed. Bounding has no effect on the final tradeoff curve that is identified if the optimality gap is 0; however, for non-zero optimality gaps, the number of solutions found on the tradeoff curve depends on the optimality gap and on whether or not bounding is used. For nonzero optimality gaps, the tradeoff curve that is found is an approximation of the actual curve. At least in the limited testing reported below, the number of solutions found for these approximate curves is less than the number identified in the optimal curve. Despite the fact that the tradeoff curve identified with an optimality gap of 0 is the “optimal” curve, it does not necessarily include all non-inferior solutions, as there may be solutions in the duality gap region. This is region B of figure 1. In the experiments outlined below, one such solution was identified. When the optimality gap was set to 1% and bounding was not used, one additional non-inferior point was identified. It lies between the second and third least costly solutions shown in Figure 6, which depicts the

optimal tradeoff curve for the 150city dataset with a coverage distance of 300 miles. Thus, while allowing a non-zero optimality gap will result in a tradeoff curve that is not guaranteed to be optimal, doing so may allow the analyst to find solutions that lie in the duality gap region of Figure 1 which would not otherwise be found using the weighting method.

Optimality Gap 0.00% 0.50% 1.00%


Solution times Bounding No Bounding 723.87 1,097.76 227.69 341.89 139.54 217.80 Iterations Bounding 50,312 15,274 10,138

No Bounding 76,215 24,772 15,971


B&B Nodes Bounding 143 33 21

No Bounding 265 60 33

No. Solutions Found Optimality Gap Bounding No Bounding 0.00% 17 17 0.50% 13 12 1.00% 11 15

Table 5 – Effects of different optimality gaps and bounding using 150city dataset with a coverage distance of 150 miles

Figure 6 – “Optimal” tradeoff curve for the 150city dataset with a coverage distance of 150 miles

6.

Summary and Conclusions

In this paper, we have extended an integrated location/inventory model to incorporate a measure of customer service quality. Key costs represented by the model include fixed distribution center location costs, working and safety stock costs at the distribution centers, fixed and variable shipment costs from the plant to the DCs, and transportation costs from the DCs to the customers. Customer service is measured by the fraction of all customer demands that are within a specified distance or service standard of the DC to which they are assigned. Through tests of the model on real world datasets ranging in size up to 263 demand nodes, we showed that significant improvements in the customer service can often be achieved at relatively little cost. While this often entails locating additional distribution centers, the cost of the incremental facilities is largely offset by reduced outbound transportation costs. As indicated in the discussion above, many firms maintain a number of different service levels (e.g., 2 hour service, 8 hour service and 24 hour service). We are exploring ways of extending the model to explore the tradeoff between cost and service using this expanded definition of service. We are also considering means of generating the entire tradeoff curve and not just the supported points that can be found using the weighting method outlined above.

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