Coordinated Location, Distribution, and Inventory Decisions in Supply Chain Network Design

The Proposed Sgp-based Solution Approach

Defining the goals of the objective functions

As we know, stochastic goal programming (SGP) needs an aspiration level for each objective. These aspiration levels are determined by DMs. In addition to the aspiration levels of the goals, we need max-min limits ( $u_g$ , $l_g$ ) for each goal. While the DMs decide the max-min limits, the linear programming results are starting points, and the intervals are covered by these results. Note that in non-linear programming (with a minimisation objective) the minimum limit of any non-linear objective may be calculated by the results of the other objectives. This situation may occur because the optimum value may be its local optimum.

Generally the DMs find estimates of the upper ( $u$ ) and lower ( $l$ ) values for each goal using payoff table (Table 1). Thus the feasibility of each stochastic goal is guaranteed.

Here, $Z_g(X)$ denotes the gth objective function, and $X^{(g)}$ is the optimal solution of the gth single objective problem. Solving the problem with $X^{(g) } (g=1,...,G)$ for each objective, a payoff matrix with entries $Z_{pg} =Z_g (X^{(p)}), g, p=1,...,G$ can be formulated as presented in Table 1. Here, $u_g= max (Z_{1g}, Z_{2g}, . . . , Z_{Gg})$ and $l_g=Z_{gg} , g =1,..., G$ .

Using the interactive paradigm can improve the flexibility and robustness of multi-objective decision making by:

Providing a learning process about the system, whereby the DMs can learn to recognise good solutions,
The DMs can control the search direction during the solution procedure and, as a result, the efficient solution achieves their preferences,
Various scenarios could be generated, based on a systematic procedure.

Solution methodology

To deal with multi-objectives and enable the DMs to evaluate a greater number of alternative solutions, three different approaches are implemented in this section.

Solution Approach 1. The weights of objective $Z_1$ and $Z_2$ are specified with $W_1$ and $W_2$ as follows:

$W_1: \text{Set of weights for the INV objective function (W_1, W_2, W_3,...}$ and

$W_2: \text{Set of weights for the TCOST objective function (1-W_1, 1-W_2, 1-W_3,...}$

Note that, based on the three presented objective functions and preferred DMs' service level (K), in this approach we generate several scenarios and the TDELT objective is not considered. (A more detailed explanation about the service level of the system is presented in Appendix) So problem 1 can be summarised as follows:

Generated Problem 1

$Z_{p_1} = Min W_1 Z_1 \cdot (PH \cdot Z2)$

Subject to:

$(4)-(9)$

TH allows one to sum the investment cost that occurs at the beginning of the planning horizon with the rate cost incurred by the entire network. In order to determine the weights, there are some good approaches in the literature, such as the analytical hierarchy process, the weighted least square method, and the entropy method. However, determination of the weights is not the focus of this study.

Solution Approach 2. In this approach the weights of the objectives ( Z1, Z2 ) and preferred DMs' service level are the same as in solution approach 1, but we consider Z3 (TDELT objective) as a new constraint.

Generated Problem 2

$Z_{p_2} = Min W_1 Z_1 \cdot (PH \cdot Z2)$

Subject to:

$Z_3 \leq Z_{33} + \gamma Z_{33}$

and

$(4)-(9)$

In the payoff table we calculate optimum (or local optimum) values for the three objective functions. In this approach, to compare each objective function against the others, we use the performance Index as a compensation rate. Since objectives Z₃ and Z₂ are very interactive, it is important for the DMs to evaluate the impact of increasing γ % in total delivery time (TDELT) on the system costs (INV and TCOST). To generate new scenarios we calculate the γ parameter based on the DMs preferences.

Solution Approach 3. In this approach, the objective function is TCOST and the other objectives (INV, TDELT) are added to the previous constraints (4)-(9). As in solution approach 2, it is important for DMs to consider Z₁ and Z₃ against the TCOST objective function.

Generated Problem 3:

$Z_{1} \leq Z_{11} + \eta Z_{11})$

$Z_3 \leq Z_{33} + \gamma Z_{33}$

and

$(4)-(9)$

To generate more scenarios we calculate η and γ parameters based on the DMs preferences.

Solution procedure

The interactive solution procedure of the proposed MOSNLP method for solving stochastic multi-objective DPD problems includes the following steps:

Step 1: Formulate the original stochastic MOSNLP model for the DPD problem.

Step 2: Obtain efficient extreme solutions (payoff values) used for constructing the right-hand side of the added constraints (first and third objective functions). If the DMs select one of them as a preferred solution, go to Step 10.

Step 3: Define upper and lower bounds of each objective functions from the payoff table.

Step 4: Formulate problems 1, 2 and 3.

Step 5: Ask the DMs if they want to modify the right-hand side of the newly-added constraints of problems 2 and 3.

Step 6: Introduce η and γ parameters to generate new scenarios - i.e., define a systematic rule for changing upper bound of Z₁ and Z₃

Step 7: Determine the values of the SC performance vector (W₁ , W₂, η, γ, K).

Step 8: Improve the generated scenarios with the performance vector determined in Step 7.

Step 9: Analyse outputs of generated scenarios and obtain non-dominated solutions. If the DMs select one of them as a preferred solution, go to Step 10; otherwise, go to Step 5.

Step 10: Stop.