MATHEMATIC PROGRAMMING MODEL AND PARAMETER ESTIMATION

Location analysis models have been studied for a long period of time. According to Daskin (1995), there are four common types of problem formulations: coverage problems, median problems, P-center problems, and fixed charged facility location problems (later referred to as fixed charge models).

For the models and the solution algorithms of location analysis, Mirchandani and Francis (1990) as well as Dresner (1995) also serve as an excellent source.

Based on the background described in the previous section, this study chose the fixed-charge problem to model the decision problem. In the first subsection, the mathematical programming model is presented. In particular, the definitions of the parameters have been modified so as to apply the classical model originally for two-staged problems to deal with the case with four transportation links as illustrated in Figure 2. In the second subsection, the approach and result for estimating the parameters in the mathematical programming model is described.

Mathematical programming model

The fixed-charge model is usually used to determine the number and location of the facilities by taking both transportation cost and facility cost into consideration. The transportation cost is assumed to linear, and there is no economy of scale. On the other hand, facility cost can be site-dependent, but is a fixed constant. The mixed integer programming model and the typical definitions of the parameters and variables are as follows:

• i: index for potential facility sites (U is set of all potential sites.)

• j: index for customers (V is set of all customers.)

• fi: fixed cost of setting up a facility at potential site i

• hj: amount of demand at customer j

• cij: variable cost of serving customer j from a facility i

• Xi: binary decision variable representing that potential site i is selected

• Yij: variable representing the portion of demand at customer j served by a facility i

The objective function (1) is to minimize the total cost by adding the facility cost determined by the facility decision (Xj) and the transportation cost determined by the assignment decision (Yij). Constraint (2) requires that the demand of each customer is fully served. In Constraint (3), the relation between the

Total Facility Cost

Process Volume Fixed Facility

Cost, fi

Variable Facility Cost, gi

Slope

site selection decision and the assignment decision is ensured by forcing the assignment variable to be zero if the associated facility is not open. In Constraint (4), the site selection variable is specified as a binary variable. Finally, Constraint (5) describes that the assignment variable is a non-negative real number.

For most cases, the facility cost is assumed to be a constant (fi), and is independent of the amount of demand served, though the capacity limit can be imposed by adding one more constraint. However, for KKL, the produced liquor is usually stored by block stacking, and several cost components about the facility are in fact highly related to the process volume of a DC. In Figure 3, the possible relation between the total facility cost and the process volume of a DC for KKL is shown. It is assumed that there are two cost categories about setting and operating a DC. First, the administration cost of a DC (IT systems such as ERP, WMS, or TMS as well as the communication equipment and service) is basically independent of the process volume. A fixed cost is incurred as long as a DC is set up. Second, owing to the characteristics of KKL, the cost related to land purchase and construction expenditure as well as the direct operating cost (the machinery such as forklifts and air conditioners, the wage for the on-site workers, and the utility expenses for electricity and water etc.) can be proportional the process volume. As shown in Figure 3, the fixed component of the facility is the intercept of the vertical axis (fi), and the variable component of the facility cost is the slope of the straight line (gi). fi is simply the fixed charge in the objective function (1). On the other hand, based on the technique provided in Ghiani et al. (2004), gi can be incorporated into the variable cost, cij, in the objective function (1) as described in the following paragraphs.

Figure 3 – Fixed and variable components of the facility cost at a DC

The variable cost in the objective function (1) is traditionally used (e.g., Daskin, 1995) to represent the transportation cost between the facility and the demand nodes, i.e., Link 4 in Figure 2. Therefore, the fixed-charge model is usually applicable only to the case of a two-echelon system. However, based on several assumptions, this study incorporated the transportation cost of all four links into the variable term in the objective function. For each potential facility site i, the inbound transportation cost (Link 3) can be merged with the outbound transportation cost (Link 4), if site i is always accessed through the sea port with the least inbound transportation cost. This arrangement is in fact the reality for the practice of KKL. Besides, it doest not affect the optimality of the overall MIP model if the freight rate is linear (i.e., no scale of economy) for the link between the sea port in Kinmen and the sea port in

Taiwan, and there is no capacity limit on the link. Similarly, once the pairing of the sea port in Taiwan and the potential facility site is pre-determined, the cost for the sea transportation (Link 2) is determined for each potential facility site i. Finally, the transportation cost between the plant of KKL and the Liaolo Harbor in Kinmen (Link 1) is actually a constant as a whole. This cost is thus can be attached to the assignment decision (Yij) and becomes part of the variable cost for a potential facility site i.

The procedure of combining the transportation cost of all four links into the variable cost (cij) in the objective function (1) can be summarized as the following. Especially, this study assumed that the transportation cost for all links are linear with respect to distance and the cost can be derived by multiplying the distance by the given per-kilometer freight rate. Suppose the plant of KKL is indexed by k, the Liaolo Harbor in Kinmen is indexed by l, and the three sea ports in Taiwan are indexed by m.

The transportation costs of all four links originally should be computed as (6). However, since there is only one plant (k=1) and one seaport (l=1) in Kinmen, and the potential facility site i has been paired with the seaport in Taiwan m, the relation (6) is reduced to the relation (7). Therefore, the transportation cost can further be merged with the variable facility cost to generate the variable term (cij) in the objective function (1) as shown in the relation (8). Then, cij should not be interpreted as the variable (transportation) cost for the case of a two-echelon system in a classical way; instead, it refers to the variable cost related to the assignment (flow or process volume) decision (Yij) and can be thought as the cost for shipping one unit of products from the plant in Kinmen via the DC i (though a specific sea port) to the customer j.

4

Model settings and parameter estimation

The estimation of the parameters in the model (1) to (8) as well as the set of the potential facility sites and the information about the customer set are determined based on several intensive meetings and communications with the staff of KKL. However, in order to protect the sensitive commercial data of KKL, the numbers in this subsection are intentionally adjusted to some extent. Nonetheless, the essence of this case study is still preserved, and the results presented in the next section are informative from the application point of view.

The set of the potential facility sites (U) consists of 25 industry parks around Taiwan. In general, one site is selected for each county or city (county-level or above), and one extra is included for those geographically large.

As for the set of the customers (V), 30 demand nodes are considered including the major wholesalers of KKL and its branch companies. In terms of the amount of demand for each demand node (hj), an overall volume of 20 million liters per year are allocated to each demand node according to its historical sales data. In particular, some of the demand nodes are mega-wholesalers to KKL, whose allocated volume can be greater than 1 million liters.

There are two categories for the facility cost. For the fixed part (fi), it is assumed that setting up a DC involves a cost of 3 million TW dollars per year. Regarding the variable facility cost (gi), it is broken into the following sub-categories:

1. Land purchase: It is estimated that 0.0005m² is required in average for handling one liter of liquor based on block stacking storage. In addition, the land price is based on the information provided by the Department of Land Administration, Minister of the Interior, Taiwan (DLA-MOI, 2007) for each of the industry park considered as a potential site.

2. Construction expenditure: It is estimated as 0.29 TW dollars per liter per year. This is in fact independent of the potential sites. Thus, it is not affected by the decisions and a constant in term of the total cost in the objective function (1).

3. Directing operating cost: It is estimated as 0.35 TW dollars per liter per year. It is also not affected by the decisions and a constant in term of the total cost in the objective function (1).

There are four transportation links as shown in Figure 2. For land transportation, the delivery of all three links (Link 1, Link 3 and Link 4) is assumed to be done by trucks, either privately owned or outsourced. On the other hand, the sea transportation relies on the scheduled-service carriers.

1. Sea transportation: There are three routes between Liaolo Harbor in Kinmen and the sea ports near Keelung, Taichung, and Kaohsiung in Taiwan. The cost per liter (ri

2di

2) ranges from 2.0 to 2.5 TW dollars.

2. Land transportation: The base rate (ri3

and rij4

) is set as 0.035 TW dollars per liter per km.

However, it is adjusted by a factor of a=10% to reflect the possible rate difference between DC inbound and outbound shipments. For the DC inbound transportation (Link 3), the contract trucking companies are likely to reduce the rate as the replenishment from the plant in Kinmen to an DC is usually scheduled in advance, and the shipment size is large. Thus, the value of 0.035(1-a) is used. On the other hand, for the DC outbound transportation (Link 4), the charge tends to be higher, as the shipment initiated by a customer’s order is relatively uncertain, and the shipment size is small. Thus, the value of 0.035(1+a) is used.