Global Optimization of a Semiconductor – Engineering Assignment Help

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1. Introduction

Supply chain management (SCM) is defined as a set of approaches used to efficiently integrate suppliers, manufacturers, warehouses, and stores so that products are produced and distributed at the right quantities, to the right locations, and at the right time, in order to minimize overall costs while meeting service level requirements [1]. SCM also pertains to the set of actions and decisions that attempt to synchronize demand and supply with in-process inventories, in order to ensure on-time delivery of product commitments to customers, and optimize the overall manufacturing operations from start to end. SCM is of paramount importance especially in the semiconductor industry, as highlighted by [2]. The criticality of SCM for this industry stems from the fact that semiconductor wafer fabrication facilities represent large capital investments (usually in the range of Billions of dollars), and the assembly and test facilities are quite expensive as well (with some of the individual testers costing a few Millions of dollars). The products produced by these facilities are of high value, both in the form of wafers and in the form of integrated circuit (IC) chips. In case of processor chips, each unit can be of the order of a few hundred dollars. With these high capital investments and cost of products, it is critical for semiconductor manufacturers to maintain high utilization of the equipment with minimal inventory. Supply chain management can help in achieving these goals and provide large savings for the semiconductor industry. Only a handful of papers were published to date that offer a mathematical formulation for the semiconductor supply-chain network planning problem. By network planning, we refer to the strategic long term planning of satisfying forecasted demand and addressing the question of what products to manufacture and where. In this paper, we formulate this problem as a quadratic programming model and provide insight to the benefits of such an optimization to the supply-chain of a semiconductor company. Before going into the details of the model, we first review the existing literature. Then we develop the notation and model formulation of the problem in Section 3, followed by a case study via a numerical example in Section 4. Conclusions and suggestions for further work are depicted in Section 5.

2, Case Study.

In this section, we demonstrate the proposed model discussed in the previous section by applying it to a case study inspired by a real semiconductor setting.
We evaluate two cases, one in which the optimization solution is subject to having a specific fab running fully utilized (and at minimal average wafer cost), and the other in which the global optimization is attained.

2.1. Input Data.
The dataset for the case study is depicted in Tables 4-8. The demand range, as expressed by minimum and maximum by product by customer (in units) is in Table 4. Similarly, in Table 5 the ASP per product and customer is given. Table 6 contains information about fab capacities, minimum utilizations, and qualifications. Table 7 depicts the AT capacity and production costs (in wafers and K units) and Table 8 has the shipping costs from Fab to AT and from AT to the customer. Lastly, the CTR and DPW (die/unit per wafer) per product are given in Table 9. Note that in this case study, Products 1 to 3 are the baseline with a CTR of 1.0, and products 4, 5 require less/more capacity with a CTR of 0.9 and 1.2 respectively.

2.2. Results for the Case Study.

Next, we demonstrate the usage of the model. The formulation in Section 3.3 was populated with the dataset from the previous section and executed using ILOG CPLEX Studio IDE Version 12.6.1.0. Execution time per instance is very fast

(1.25 seconds on an Intel Core i5 5200U CPU at 2.20 GHz.) Table 10 contains the details of the optimal solution in two cases: Case (a): Fab-specific optimal solution The first case that is evaluated is the case where the optimization is solved subject to the requirement that a specific fab (in this case study, Fab2) would achieve minimum wafer cost and chooses each subsequent variable following a greedy algorithm for the specific stage of the supply chain be fully utilized (i.e., 100% utilization). Case (b): Globally optimal solution The second case that is evaluated is the case of global optimization without any additional constraints such as in case (a).

As can be depicted by the left-hand-side in Table 10, by enforcing a minimum wafer cost at 100% utilization on Fab2 as in case (a), the resultant total fab production costs are higher and the revenues drop. Consequently, the profit is lower. On the other hand, when allowing the model to find the global optimum, depicted by the right-hand-side in the table, it identifies a solution at which none of the fabs is fully utilized but the total profits are significantly higher (by 3.4%). 

4.3. Sensitivity Analysis

The results of the case study demonstrate the importance of searching for the global optimal solution for the network. In this section, we extend upon the case study that was presented and vary the inputs, to reflect different network sizes. Specifically, we executed the model for several values as follows:

• Number of fabs: 3, 5, and 7.
• Number of ATs: 3, 5, and 7.
• Number of customers: 5, 10, and 20.
• Number of products: 20, 50, and 100.

A note on the computation time before we proceed is appropriate. An expected non-linear increase has been observed in the computation time as the problem size grows, but even with the large scale problems of 100 products, the solutions were obtained in approximately 400 seconds, a reasonable time by all means. For the smaller scale problems, computation time decreased drastically, with about 15 seconds for the 20 product scenarios and 60 seconds for the 50 product scenarios. However, for each QIP solution we compared an equivalent rounded QP solution, where the decision variables were considered to be continues in the solution process and the final solution was the rounded down to the nearest integer. Across 24 different instances, 3 for each scenario depicted in Table 11 we found a maximum gap of 3.55e-09 ?tween the optimal solution of the QIP and the rounded QP solution. The cause for this negligible gap is due to the large volumes of the integer units in the decision variables. Having observed such minimal differences between the rounded QP and QIP solutions, we chose to solve the instances as rounded QPs since the run time for such a rounded QP solutions remained approximately 3.4 seconds for even the large models. We selected a subset of all possible combinations as this would suffice to provide insight. 20 instances were generated for each combination that was selected. For each instance two algorithms were tested. A greedy algorithm that first maximizes the revenue (considering only feasibility of the demands at the fabs and ATs) and then minimizes the supply chain costs. This algorithm was compared with the results from the proposed algorithm discussed in section 3.3, which maximizes profit.

The results are depicted in Table 11. The Profit Gain column shows the average increase in profit across the 20 instances generated when using the proposed
profit maximizing algorithm instead of a greedy, revenue maximizing, algorithm while the Revenue Loss column shows decrease in revenue. The Ratio column provides the quotient of the Profit Gain/ Revenue Loss. A common practice in the industry today is to first focus on generating maximum sales (revenue) and let the companies’ supply chain attempt to minimize the subsequent costs of fulfillment. However, the results of Table 11 indicate that a “holistic” approach which considers both the revenues and the costs may considerably increase the profit with almost no impact on the overall revenue, generating between a $126 – $700 of profit for each dollar of revenue loss. Furthermore, the results indicate that the benefit of such an approach increases with the size and complexity of the network.

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