Derechos exclusivos para moléculas previamente aprobadas

The problem introduced in Proposition 3.2 has exponentially many constraints and there- fore, we need to generate and add them progressively. Algorithm 2 describes the outline to generate and add the delayed constraints. In this section we compare the running times of the MILP solver for different models. Since the complexity of the models depends on the total number of scenarios, we represent the complexity of the different models with the size of scenario set for each model. For the F SCN D − u − stoch − det we only consider one scenario, the F SCN D − u − stoch − w has 3n _{scenarios, and therefore its complexity is}

proportional to O(|Ξ|) = O(3n) for a problem with n uncertain parameters (each param- eter is discretized into three low, medium, and high estimations). The complexity of the

F SCN D − u − stoch − w/o1 _{is proportional to O(3}2n_{) since we have a constraint for each}

scenario. Finally, by proposition 3.3, the complexity of the F SCN D − u − stoch − w/o2

is reduced and it is proportional to O(3n_).

We compare the running time of the Gurobi 7.0 MIP solver for all models with different number of uncertain parameters. In Figure 3.8 the horizontal axis shows the number of uncertain parameters. In this axis 0 means no uncertain parameters (deterministic model). Then, we add the demand uncertainty, then exchange rates, and finally the freight cost. This comparison is presented in figure 3.8:

Figure 3.8: Comparison of the running times for different models

As expected the MILP deterministic model is very fast for different problem sizes. The convexity of the plots suggest that the running times increase exponentially by increasing

model for problem sizes larger than 7 in less than 500 seconds. However, using Algorithm 2 we were able to solve these instances faster. Figure 3.9 compares the running times of the MILP solver with the cutting-plane algorithm 2 for  = 0.1%:

Figure 3.9: Comparison of the running times for MILP Solver and Cutting-plane Algorithm 2

The cutting-plane algorithm 2 was able to solve all instances. It seems the running time of the algorithm 2 is less sensitive to the problem size compared to that of MILP solver. The convexity of the curve for the running times of this algorithm suggests that algorithm 2 does not guarantee polynomial time solutions as we mentioned before. It is worth men- tioning that we consider a relatively small  for comparison purposes. Larger values of  can lead to different (sub-optimal) solutions than the exact optimal solution.

3.7. Concluding Remarks

We propose an optimization model for designing flexible SCNs operating under uncertainty through capacity planning. First, we consider an extension of current SCN models by co- ordinating several SCNs of common commodities of different products. In practice, we cannot coordinate the SCNs of all the parts in different products (vehicles) due to different designs, specifications, and requirements for different products. However, we can optimize the global sourcing strategies of the common parts to reduce initial (tooling) investments, and take advantage of economies of scale. Although in our case study we did not consider any suppliers capable of producing more than one commodity, the capacity of the suppliers can be shared among different commodities for different products in our models. Second, we focus on the correlation among uncertain parameters (demands, exchange rates, and freight costs). To do so, we employ distributionally robust optimization to coordinate the SCNs against the worst-case distribution with given marginal probabilities for demand, exchange rate, and freight cost uncertainties. Moreover, we investigate the price of corre- lation in the design parameters against the traditional methods in which the independence of the uncertain parameters is an assumption. To the best of our knowledge, our model is different from other approaches found in the literature and contributes in the following ways. (1) We include multiple sources of uncertainty in the model (demand, exchange rate, and freight cost). (2) Capacity planning (as opposed to holding inventory) is proposed to create flexibility in designing SCNs. In particular, holding inventory to create flexibility is not feasible in auto industry. (3) Distributionally Robust Optimization (DRO) is used to coordinate SCNs of different products and parts against the worst-case joint distribution

relation increases when we have less information about the uncertain parameters. The deterministic model is only reliable when we have low or medium exchange rates. The correlated model (with full margins) gives higher profit when exchange rates are high com- pared to the stochastic model (independent). The worst-case analysis of all models return similar results, but the correlated model has a slightly higher mean profit. The results indi- cate the value of the correlated model compared to other variations. Finally, we proposed a tractable algorithm to solve the correlated model in less amount of time than required by the commercial solvers.

For the future research, we suggest the study of risks involved with under/over estimat- ing the required tooling, specifically for parts with expensive tooling. Another interesting subject to investigate is the possibility of reserving capacity options. Finally, the study of the demand elasticity to the price in different markets can be another interesting subject.