In our proposed example for stage #1, we use three cost functions and one QL in each echelon. In total, there are nine cost functions and three QLs in the whole mathematical model. The COQ functions of each echelon are presented in Table 5-2. The total COQ functions are quadratic functions, and they are greatly affected by the QL. When the QL is small, the COQ is high and as the QL increases the COQ decrease until a certain point when it starts to increase again. In addition, the QL directly affects the number of supplied components to each echelon (i.e., Mi, Mj
& Mk) and the output components from each echelon (Oi,j, Pj,k, & Qk,c).
Table 5-2: PA, F, and OC cost functions
Supplier Manufacturer Retailer
fPA(xi) = 0.085e6.41x fPA(y) = 0.18e6.01y fPA(uk) = 0.27e5.69u
fF(xi) = 8x-2 – 5x – 2 fF(y) = 11y-2 – 8y – 3 fF(uk) = 21u-2 – 7u – 14
fOC(xi) = 5x -2 – 5x fOC(yj) = 5y-2 – 5y fOC(uk) = 5u-2 – 5u fCOQ(xi) = fPA(xi) + fF(xi) + fOC(xi) fCOQ(yj) = fPA(yj) + fF(yj) + fOC(yj) fCOQ(uk) = fPA(uk) + fF(uk) + fOC(uk)
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The retailer is considered to work at certain QL, for which it spends a certain COQ, which depends on the optimization result. We set the production cost/unit (αi, αj, and αk) at the supplier,
manufacturer and retailer to be equal to $10/unit. The total customer demand (Dc) is 4500
components. We assume that the fOC(xi), fOC(yj) and fOC(uk) functions are identical among the SC entities, however, after the optimization, the OC will be different in each echelon according to the optimum QL. We assume that there is one facility at each echelon in the model to be compatible with the developed SD model and therefore I, J, and K = 1. We used Excel Solver (GRG Nonlinear) to solve the mathematical model. Although the the nonlinearity exists in the objective function and the constraints, Solver, could provide the same solution for different runs. It gives the best solution for the proposed model (Stage #2). According to Ramudhin et al. (2008), the model can be linearized and solved as a linear model using different methodologies, which are available in the literature. Based on the mathematical model, the solution minimizes the COQ functions at each echelon simultaneously. Table 5-3 presents the best (minimum) obtained solution for the following parameters: total PA, F and OC costs at the supplier, manufacturer and retailer, as well as, total PA, F and OC costs for the whole SC, which is the aggregate of PA, F, and OC at the supplier, manufacturer and retailer. In addition, the table provides the total COQ at each facility and the equivalent COQ for the whole SC. Moreover, the QL and material flow into and out of each facility are provided in the table. The objective value of the model is equal to $850,264.8. In general, the results show that to obtain the best minimum value for the model; the supplier spends the lowest portion of COQ, then manufacturer the medium, and the highest portion of COQ is allocated at the retailer. The retailer spends on PA costs more than what the supplier and manufacturer spend together. The manufacturer and retailer will work at a lower F costs, while the supplier incurs in the highest F costs in the model.
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Table 5-3: Optimum results for each scenario in the centralized SC
Supplier (i) Manufacturer (j) Retailer (k) Total SC Customer Demand (unit) PAi, PAj & PAk ($/unit) 15.8 28.8 47.5 92.2 4500 Fi, Fj & Fk ($/unit) 6.0 5.7 5.1 16.8 OCi, OCj & OCk ($/unit) 3.5 2.8 1.5 7.8
COQi, COQj & COQk($/unit) 25.3 37.3 54.2 116.8
Mi, Mj & Mi(unit) 7209 4870 4959 ---
xi, yj & uk 0.81 0.84 0.91 ---
Oi,j, Pj,k & Qk,u (unit) 5870 4959 4500 ---
Objective Value ($) 850,264.8
5.4.2. SD model
To run the SD model and to obtain more logical and accurate results, we have constrained our SD model by imposing a set of assumptions on it. These assumptions may, however, limit the external usability of the SD model. The model assumptions are as follows:
1 All supplied parts from the supplier are accepted by the manufacturer. 2 All supplied parts from the manufacturer are accepted by the retailer. 3 All supplied parts from the retailer are accepted by the customers.
4 The initial values of the potential customers are the same for each echelon and redefined. 5 The number of new customers at each echelon depends on the QL of each echelon. 6 The SC products are 100% inspected at each echelon.
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After considering the inputs from the mathematical model, which are based on the optimization results, the SD model was run at higher PA values than the PA obtained in the Table 5-3 (i.e., supplier = $15.8/unit, manufacturer = $28.8/unit and retailer = $47.5/unit). The results show that at the supplier, the number of new customers can be increased from around 20 to 44 customer/month (higher by 120%). This happens by increasing the PA from $15.8/unit to about $28.2/unit, after which that the number of new customers decreases as shown in Figure 5-6 (run no. 5). In addition, it is found that the number of manufacturer’s new customers can be increased to reach around 54 new customers per month (higher by 32%) by increasing the PA from $28.8/unit to about $35.5/unit.
Figure 5-6: The SD model results for the supplier
As shown in Figure 5-7, run no. 5 resulted in less new customers than run no. 4. Finally, in Figure 5-8, run no. 5 was computed with around 92 new customers, which is less than the
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number of new customers in run no. 4. Therefore the retailer’s new customers can be increased from an average of 76 to 114 new customers per month (higher by around 50%) by increasing the PA from $47.5/unit to about $54.6/unit. Figure 5-9 shows the overall effect of the five runs of the supplier, manufacturer, and retailer on the expected sales. It is obvious that the expected SC sales decrease as the number of new customers decreases in the SC, which occurred in the fifth run.
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Figure 5-8: The SD model results for the retailer
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5.4.3. Mathematical model
After adding the constraint to the supplier only, the constraint becomes PAs = SDs. For the manufacturer and retailer, the added constraints are PAm = SDm and PAr = SDr, respectively. For the whole SC, the constraint is PAs + PAm + PAr = SDSC
Where:
PAs, PAm, and PAr are the Prevention and Appraisal costs for the supplier, manufacturer, and
retailer, respectively. SDs,SDm, SDr,and SDSC are the SD values, which resulted in the highest
number of new customers for the supplier, manufacturer, retailer, and SC, respectively. Excel Solver (GRG Nonlinear) is also used to solve the mathematical model in this subsection. For the first constraint when PAs = SDs = $28.2/unit, the results are shown in Figure 5-10, in which the QL distribution for the supplier, manufacturer and retailer are 0.90, 0.85 and 0.91, respectively. For the supplier constraint, the results show that the retailer has to work at the same (highest) QL and he should also spend the highest amount of COQ on PA costs for the products. Accordingly, the value of new objective function increased to $875,098.3.
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Figure 5-11 shows the results after applied the PA constraint at the manufacturer (PAm = SDm = $35.5/unit), where the QL for the supplier, manufacturer and retailer are 0.81, 0.88 and 0.91 respectively. The retailer still has to work at the highest QL and it should also spend the highest amount of COQ on PA costs for the products. The overall increase in the objective function is lower than what was observed when the PA constraint was applied at the supplier. The increase is only around 2% with the final value of the objective value of $854,621.5. When we subject the objective function to the PA constraint at the retailer (i.e., PAr = SDr = $54.6/unit), the results show that the QL for the retailer should be 0.93 to satisfy the constraint (see Figure 5-12). The overall increase in the objective value as seen in Figure 5-12 is even less than the increase of the objective function at the supplier or manufacturer cases.
Figure 5-11: PA constraint is applied to the manufacturer
This implies that increasing the number of new customers at the retailer (downstream of the SC) requires lower investment in the COQ. In Figure 5-13, we subject the model to a general SC PA cost constraint. The constraint is the summation of the lowest PA costs of SD, i.e. PAs + PAm
+ PAr = SDSC = ($28.2/unit + $35.5/unit + $54.6/unit = $118.25 /unit). This is done to distribute
the PA costs freely among the SC entities. According to the results in Figure 5-13, we can notice that the optimization suggests to allocate a high portion of PA costs at the supplier ($62.9/unit),
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then the second-highest portion of PA costs at the manufacturer ($36.5/unit) and the smallest portion of PA costs is allocated at the retailer.
Figure 5-12: PA constraint is applied to the retailer
The optimization results recommend that to allocate the PA costs in a different way than they suggested by SD i.e. by 33% (less), 3% (high), and 15% (high) for the supplier, manufacturer, and retailer, respectively. We conclude that using one methodology is inadequate for our model and integrating the two methodologies (hybrid SC DSS) is beneficial and recommended to improve the decision variables.
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