BUS606: Operations and Supply Chain Management

The case study presented here, with an example from the food industry, illustrates the algorithm proposed in Section 4, as well as the applicability and effectiveness of the model. This food industry company is the leading producer of two main categories of Iranian food and drink (rice and tea). The basic distribution data are presented in the next sub-section.

Setup

A case study inspired by a food producer in Iran is presented to demonstrate the validity and practicality of the model and solution method. The company owns one production site and six potential DC/warehouse sites in the different customer zones (Figure 2). There are three types of products and twenty main retailers.

Lingo 8.0 optimisation software is used as the problem solver. All scenarios are solved on a Pentium 4 (Core 2 Duo) with 1GB RAM and 4 GHz CPU.

Because of confidentiality, the input data are randomly generated. However, the generation process is done so that it will be close to the real data available in the company. Without loss of generality and just to simplify generation of the stochastic parameters, we apply the pattern of a systematically normal distribution for our numerical test.

The required throughput capacity of any warehouse for product ' is as follows: s,= 2, s2= 5, s3= 4. Tables 2 and 3 list some of the other basic distribution data.

Performance analysis

The interactive solution procedure using the proposed SGP method for the case study is as follows:

First, formulate the original stochastic multi-objective DPD problem according to equations (1)-(9). The goal of the model is to select the optimum numbers, locations, and capacity levels of warehouses to deliver the products to the retailers at the least cost, while satisfying the desired service level of the retailers. The proposed model is distinguished from the other models in this field in the modelling approach. Because of the somewhat uncertain nature of retailers' demand and DMs' aspiration levels for the goals, a stochastic modelling approach is used. Additionally, a novel and generic SGP-based solution approach is proposed to determine the preferred compromise solution.

Second, obtain efficient extreme solutions for each of the objective functions. These extreme solutions of the case study are presented in Table 4.

It is assumed that the DMs do not choose any of the efficient extreme solutions as the preferred compromise solution, and proceed to the next step.

Considering the efficient extreme solutions given in Table 4, the lower and upper bounds of the objectives can be determined. In our case, the corresponding minimum and maximum values of the efficient extreme solutions are determined as the lower and upper bounds respectively, as presented in Table 5.

After calculating the upper and lower bounds of each objective function, the next step is formulation of problems 1, 2 and 3. A summary of the results for the various scenarios is given in Tables 6, 9 and 11.

As stated previously, the relative weights for the first and second objective functions in problem 1 can be determined by DMs using various methods. For the presented case study, DMs determine three weights for the INV and TCOST objectives as follows: (0.7, 0.3), (0.5, 0.5) and (0.3, 0.7). For this problem, no constraint on delivery time is included and TH=1000 (planning horizon) hereafter. By fixing the values of W₁ and W₂, the solution given in Table 6 is obtained. In this table, for three values of each objective function and three levels for the customer service performance index (K), nine scenarios have been generated.

In Table 6, the warehouse load ratio percentage (WRL) column shows the efficiency of the opened warehouses. The average WRL in approach 1 is 0.9865, and since Z_p1 is a non-linear objective function, the range of the CPU time for solving this problem is very wide, from 6 to 180 seconds.

Note that in scenarios 5 and 6, although the customer service performance (90%, 75%) is lower than in the 4th scenario (97.5%), the objective function is higher. Therefore these scenarios are inferior and must be removed from the scenario list. Figure 3 shows the results of equal weights for scenarios 1, 2 and 3, comparing them with non-equal weights for scenarios 7, 8 and 9 in approach 1. A comparison of the first and third scenarios in Table 7 shows that total cost is increased slightly from 6,988,496 to 7,111,742 (1.7%) when CSPI is increased from 75% to 97.5% (23%). This situation is the same for scenarios 7, 8 and 9 in approach one and for the other scenarios in the second and the third approaches (Tables 9 and 11). As can be seen, the effect of customer service level decreasing on cost improvement is negligible. This may support management's preference to select K=97.5% because a large increase in CSPI results in a small cost penalty. Selecting the first or the seventh scenario in this approach is based on DMs' preferred objective weights.

To solve problem 2, first the γ parameter must be calculated based on the DMs' preferences for the right-hand side of the new constraint (TDELT). Table 8 shows three preferred values for the delivery time performance index ( γ ).

Based on three values for W₁, W₂ and γ, eighteen scenarios have been generated. The results of these scenarios are presented in Table 9. In approach 2, the WRL average (0.9644) is lower than approach 1 (0.9865); and by considering the TDELT objective in approach 2, this effect was predictable. Considering the sixth column in Table 8, it can be determined that since Z_p2 is a non-linear objective, the range of the CPU time to solve this problem is very wide, from 32 to 1,693 seconds. Comparing the CPU times in Table 6 and 9 shows that these times for problem 2 are significantly larger than those for problem 1. Unfortunately, LINGO optimisation software could not solve the 16th scenario in 180 minutes. The results presented in Table 9 are illustrated graphically in Figures 4 and 5.

Table 10 shows the preferred values for η and γ in problem 3. For this problem, six scenarios are examined. The performance vectors and the other results are presented in Table 11, and illustrated graphically in Figure 6. It is interesting to note that in approach 3 the WRL average is 0.9878, and it is higher than the other approaches.

In summary, we make the following observations from our case analysis:

• Ten cases out of 33 scenarios are dominated by the other ones.
• The solution results indicate that the proposed model is not very sensitive to CSPI, so the preferred value for this parameter is 97.5%.
  

It can be concluded that the proposed SGP solution using approach 3 may provide different and even more preferable results when compared with approaches 1 and 2