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In this paper we develop a framework for the semiconductor manufacturing process, including front-end (fab) operations and backend (assembly/test) operations. We then propose a Quadratic Programming (QP) formulation for the profit maximizing objective with flexible demand and price ranges (per product) and fab capacities. We demonstrate the model by applying it to a case study that is based on a real-world (distorted) dataset and show how the solution varies between a local optimization of a single fab (for minimum wafer cost) and a globally optimal solution for maximum profit of a network of fabs.

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 [

SCM is of paramount importance especially in the semiconductor industry, as highlighted by [

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.

In this section, we review some of the key research work to date on SCM and the semiconductor industry in particular. Jain [

Many approaches to building strategic plans to effectively operate these complex supply chains have been proposed. The most sophisticated rely on some form of mathematical optimization, e.g. linear programming [

Despite the extensive research on SCM, there seem to be lack of applied research pertaining to the SC network planning problem in the semiconductor industry. During 1993-1995, a team of researchers at SEMATECH developed a strategic decision support system to assist large semiconductor companies in managing their supply networks. The system included an optimization model that helped determine the configuration of the supply network and which products should be built in which fab. This work is summarized in [

Most of this literature review has been motivated by works that pertain to the semiconductor industry and its supply chain as this type of supply chain possesses unique characteristics in terms of the behavior of product ASP’s (average selling prices) over time or the breakdown of the supply chain by the components created during the process (i.e. wafers, then die). Yet, it is important to mention that there are many related works on similar problems for other industries. As an example, Kannegiesser et al. [

In summary, despite all the work that has been published, the modeling and solution approach to the semiconductor supply-chain network planning problem has not been explicitly addressed to date.

The contribution of the current paper emanates from the fact that it addresses pertinent aspects that were not addressed integrally in previous papers thus far. First, we consider a profit maximizing objective rather than a cost minimizing objective. Although many have addressed the profit maximizing objective before, some were noted in the literature section of this paper, it was not in conjunction with the next few differentiators.

Second, we assume that there is at least some flexibility in setting the demand for each product in sold units, such that there is a range rather than a point estimate, and the final value produced for each product is constrained within this range. Third, we extend the analysis to varying ASP’s (average selling prices) of any product by customer, such that different customers may be offered different ASP’s for the same product. This is embedded in order to reflect different prices ranges for differences in quantities sold. This covers, for example, the cases where customers are awarded discounts for purchasing higher volumes of a given product. Implied by this extension is that the mathematical formulation of the problem becomes quadratic rather than linear. Lastly, we also assume that there is some flexible capacity across the product mix such that each fab has a capacity range between a minimum and a maximum value rather than a fixed number.

The framework for the planning problem of this supply chain network in our paper is depicted in

Once production is completed at the fabs, the wafers are shipped for assembly, test and unit packaging, and then shipped as units, after order consolidation, to the various OEMs.

In order to illustrate the problem at hand, consider the following numerical example. We note here that in both the following numerical example and in the case study later on, the datasets are inspired by real fab networks. In our numerical example, there are two fabs, denoted as Fab 1 and Fab 2, are each with a 7000 weekly wafer start capacity. Demand per product and wafer cost per product per fab are given in

In solving this problem optimally for each of the fabs independently (i.e. to minimize that fab’s cost), while satisfying the demand, the resultant solutions are shown in

Demand | Fab1 | Fab2 | |
---|---|---|---|

Wafer Cost | |||

PROD 1 | 4000 | 1500 | 2500 |

PROD 2 | 5000 | 1000 | 1500 |

PROD 3 | 5000 | 2250 | 3000 |

Fab1 Optimal | ||
---|---|---|

Fab1 | Fab2 | |

Volume | ||

PROD 1 | 2000 | 2000 |

PROD 2 | 5000 | 0 |

PROD 3 | 0 | 5000 |

Cost | 8,000,000 | 20,000,000 |

Total Cost | 28,000,000 |

Fab2 Optimal | ||
---|---|---|

Fab1 | Fab2 | |

Volume | ||

PROD 1 | 2000 | 2000 |

PROD 2 | 0 | 5000 |

PROD 3 | 5000 | 0 |

Cost | 14,250,000 | 12,500,000 |

Total Cost | 26,750,000 |

Globally Optimal | ||
---|---|---|

Fab1 | Fab2 | |

Volume | ||

PROD 1 | 4000 | 0 |

PROD 2 | 0 | 5000 |

PROD 3 | 3000 | 2000 |

Cost | 12,750,000 | 13,500,000 |

Total Cost | 26,250,000 |

The following notation is utilized. Indices:

q Products:

q Fabs:

q ATs:

q Customers:

Inputs for stages 1 and 6:

q

q

q

q

q

Inputs for stages 2 and 3:

q

Note that MinUtilization is given per fab, where it is assumed that fabs have a minimum threshold for utilization due to taxation and other governmental obligations; and this is applicable to most international companies with presence in several countries. In case this is irrelevant it can simply be set to zero.

q

q

q

Inputs for stages 4 and 5:

q

q

q

q

q

Decision variables:

q

q

Next is a Quadratic Programing (QP) problem formulation to the problem at hand. QP means that the objective function is quadratic in the decision variables and is subject to linear constraints on these variables.

The objective function to be maximized in Equation (1) is a quadratic profit maximizing objective that incorporates revenue from sold units with a discount factor based on the volume of purchase (first term of the equation), fab production costs (second term), AT production costs (third term) and shipping costs from Fab to AT and from AT to customers (fourth and fifth terms respectively).

The constraints are reflected in Equations (2) to (6). Demand range for each product and each customer is set by Equation (2). Fab capacity range for each fab is accounted for in Equation (3). Note that to ensure that feasibility exists, the maximum capacity of the network (in units) has to exceed the minimum demand of the demand range (in units), for every product i (i.e.

The upper limits on AT capacity for each AT site are considered in Equation (4). The flow balancing constraints on wafers transformed into units (die) are represented by Equation (5) and lastly fab product qualifications are set by Equation (6) which ensures that if fab j is not qualified for product i (such that Q_{ij} = 0), then it would not produce any units for this product, but if it is qualified, then it would produce up to the maximum capacity of the fab.

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.

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

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

MIN Demand in Dies | |||||
---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C5 | |

PROD1 | 10,237,500 | 2,047,500 | 0 | 0 | 0 |

PROD2 | 0 | 10,725,000 | 21,092,500 | 0 | 6,077,500 |

PROD3 | 0 | 13,650,000 | 15,600,000 | 5,850,000 | 23,400,000 |

PROD4 | 35,100,000 | 0 | 0 | 0 | 23,400,000 |

PROD5 | 0 | 3,250,000 | 0 | 1,560,000 | 0 |

MAX Demand in Dies | |||||
---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C5 | |

PROD1 | 12,285,000 | 2,313,675 | 0 | 0 | 0 |

PROD2 | 0 | 14,657,500 | 31,460,000 | 0 | 8,937,500 |

PROD3 | 0 | 14,430,000 | 24,960,000 | 6,630,000 | 37,050,000 |

PROD4 | 53,300,000 | 0 | 0 | 0 | 36,400,000 |

PROD5 | 0 | 3,770,000 | 0 | 1,690,000 | 0 |

MAX Price Per Die | |||||
---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C5 | |

PROD1 | 68 | 71 | 0 | 0 | 0 |

PROD2 | 0 | 127 | 121 | 0 | 134 |

PROD3 | 0 | 98 | 106 | 112 | 99 |

PROD4 | 5.5 | 0 | 0 | 0 | 6.5 |

PROD5 | 0 | 630 | 0 | 650 | 0 |

MIN Price Die | |||||
---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C5 | |

PROD1 | 57 | 67 | 0 | 0 | 0 |

PROD2 | 0 | 97 | 91 | 0 | 104 |

PROD3 | 0 | 94 | 86 | 100 | 79 |

PROD4 | 4 | 0 | 0 | 0 | 5 |

PROD5 | 0 | 610 | 0 | 630 | 0 |

Capacity Fabs | Min Util | ||
---|---|---|---|

Week | Qtr. | ||

Fab1 | 10,000 | 130,000 | 60% |

Fab2 | 7800 | 101,400 | 65% |

Fab3 | 7500 | 97,500 | 70% |

PROD1 | PROD2 | PROD3 | PROD4 | PROD5 | |
---|---|---|---|---|---|

Fab1 | |||||

Fab2 | |||||

Fab3 |

Capacity AT | ||
---|---|---|

Week | Qtr. | |

AT1 | 8500 | 110,500 |

AT2 | 10,000 | 130,000 |

AT3 | 9000 | 117,000 |

Cost AT | ||
---|---|---|

Wafer | K Dies | |

AT1 | 110 | 120 |

AT2 | 100 | 90 |

AT3 | 115 | 115 |

Fab-to-AT wafer shipping Cost | |||
---|---|---|---|

AT1 | AT2 | AT3 | |

Fab1 | 30 | 30 | 30 |

Fab2 | 75 | 75 | 75 |

Fab3 | 50 | 55 | 50 |

AT-to-Customer Cost K die | |||||
---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C5 | |

AT1 | 53 | 51 | 60 | 58 | 55 |

AT2 | 53 | 52 | 56 | 54 | 54 |

AT3 | 58 | 56 | 56 | 52 | 51 |

DPW | |||
---|---|---|---|

PRODUCT | CTR | PRODUCT | DPW |

PROD1 | 1 | PROD1 | 630 |

PROD2 | 1 | PROD2 | 550 |

PROD3 | 1 | PROD3 | 600 |

PROD4 | 0.9 | PROD4 | 2000 |

PROD5 | 1.2 | PROD5 | 100 |

(1.25 seconds on an Intel Core i5 5200U CPU at 2.20 GHz.)

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

Fab specific Optimal | Globally Optimal | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

F1 | F2 | F3 | F1 | F2 | F3 | |||||||||

Fab Utilization | 1.00% | 1.00% | 1.00% | 100.00% | 100.00% | 100.00% | ||||||||

1.00% | 100.00% | |||||||||||||

Avg Wfr Cost | 4141.16 | 3842.97 | 4151.23 | 3863.30 | 4018.45 | 4107.00 | ||||||||

Fab Production Cost | 550,463,550 | 389,677,600 | 359,413,080 | 502,228,661 | 407,470,440 | 400,432,500 | ||||||||

1,299,554,230 | 1,310,131,601 | |||||||||||||

FabToAT Shipping Costs | 3,987,750 | 7,605,000 | 4,329,000 | 3,939,000 | 4,143,750 | 4,647,500 | ||||||||

15,921,750 | 12,730,250 | |||||||||||||

AT1 | AT2 | AT3 | AT1 | AT2 | AT3 | |||||||||

AT Utilization | 52.29% | 100.00% | 75.56% | 94.53% | 100.00% | 73.89% | ||||||||

78.18% | 89.47% | |||||||||||||

AT Die Cost | 13,637,520 | 17,582,760 | 16,056,300 | 11,490,050 | 13,000,000 | 9,941,750 | ||||||||

AT Wfr Cost | 10,703,550 | 13,000,000 | 10,764,000 | 4,138,194 | 11,686,538 | 7,848,750 | ||||||||

AT Production | 81,744,130 | 58,105,282 | ||||||||||||

ATTo Customer | 3,134,534 | 4,106,148 | 3,824,470 | 1,773,294 | 5,445,684 | 3,502,752 | ||||||||

Shipping Costs | 11,065,152 | 10,721,730 | ||||||||||||

Cost | 1,408,285,262 | 1,391,688,863 | ||||||||||||

Revenue | C1 | C2 | C3 | C4 | C5 | C1 | C2 | C3 | C4 | C5 | ||||

889,200,000 | 4,971,070,000 | 5,009,420,000 | 1,878,500,000 | 3,464,719,514 | 889,204,094 | 5,191,162,039 | 4,929,698,053 | 1,719,901,188 | 3,955,006,953 | |||||

16,212,909,514 | 16,684,972,326 | |||||||||||||

Profit | 14,804,624,253 | 15,293,283,464 | ||||||||||||

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

The results are depicted in

Fabs | ATs | Customers | Products | Instances | Profit Gain (avg) | Std. Dev. | Revenue Loss (avg) | Ratio |
---|---|---|---|---|---|---|---|---|

3 | 3 | 5 | 20 | 20 | 13,517,075 | 3,088,135 | 107,132 | 126.2 |

3 | 3 | 5 | 50 | 20 | 14,507,085 | 3,427,495 | 91,649 | 158.3 |

3 | 3 | 5 | 100 | 20 | 17,717,953 | 2,970,469 | 73,302 | 241.7 |

5 | 5 | 10 | 20 | 20 | 18,638,989 | 3,955,901 | 124,985 | 149.1 |

5 | 5 | 10 | 50 | 20 | 19,200,775 | 3,513,044 | 93,823 | 204.6 |

5 | 5 | 10 | 100 | 20 | 18,623,342 | 3,427,650 | 46,307 | 402.2 |

7 | 7 | 20 | 50 | 20 | 21,030,469 | 3,035,081 | 29,599 | 710.5 |

7 | 7 | 20 | 100 | 20 | 21,918,830 | 3,728,617 | 73,158 | 299.6 |

In this paper we develop a framework for the key components of the supply chain of the semiconductor manufacturing process, including front-end (fab) operations and backend (assembly/test) operations. Within this framework we consider the conversion from wafers to units. Then we propose a QP formulation for the profit maximizing objective function with flexible demand ranges per product and fab capacities. We demonstrate the model by applying it to a case study that is based on an industry dataset and show how the solution varies between a local optimization of a single fab (for minimum wafer cost) and a globally optimal solution for the network.

Our model extends on previous work in several respects; most notable is the consideration of varying ASP’s of any product by customer, to reflect segmentation in pricing for differences in quantities sold. Other extensions pertain to the flexibility in setting the demand for each product, and in setting capacity across the product mix such that each fab has a capacity range rather than a fixed number.

On top of these extensions, there are still opportunities for further work. One that immediately comes to mind is the consideration of a different objective function to compare with the proposed quadratic objective function. Other directions include the incorporation of stock points within the supply chain, as they are used in practice to mitigate changes in the demand, and the explicit consideration of sub-problems of the SC problem within the model framework, for example the capacity planning problem for each of the fabs and/or the AT sites.

Kalir, A.A. and Grosbard, D.I. (2017) Global Optimization of a Semiconductor IC Supply Chain Network. Journal of Service Science and Management, 10, 338-352. https://doi.org/10.4236/jssm.2017.103027