Proposed approaches

This research proposes a linear programming (LP) formulation approach to solve this problem. This approach could be very efficient and does not alter the cell process or add equipment. The results were benchmarked against two heuristics used in practice. The two heuristics will be discussed in the next section followed by the proposed LP approach.

1. Random heuristic

The simplest method is to match TFT and CF using a random approach. This approach randomly chooses a pair of cassettes and a pair of plates for cassette and plate matching. The advantages of random matching are that it is quick and easy to perform. A possible disadvantage is that not considering glass yield information might lead to LCD scrap and yield losses.

2. Best-first heuristic

This approach uses sorting techniques to improve the post-mapping yield as follows.

Step 1: Sort the N TFT cassettes in queue in descending order by yield rate.

Step 2: Sort the twenty TFT plates in each TFT cassette in descending order by yield rate.

Step 3: Based on the sequence from step 1, perform the “best” cassette-matching sequentially. “Best” indicates the highest yield. For example, the first TFT cassette in queue (after sorting) has the highest priority to choose the best matching CF cassette from those N CF cassettes in queue. Step 4 discusses how the cassette-matching yield is calculated. The second TFT cassette in queue then chooses its best matching CF cassette from those remaining N − 1 CF cassettes. This procedure continues until the last TFT cassette in

the queue is matched with the last CF cassette in queue.

Step 4: When the ith TFT cassette (after sorting) and the jth CF cassette are selected, the proposed procedure performs the “best” plate matching sequentially. It is similar to the above “best” cassette matching procedure.

Based on the sequence from step 2, the first TFT plate in TFT cassette i has the highest priority to choose the “best” matching CF plate from those 20 CF plates in CF cassette j. When a TFT plate and a CF plate are chosen, their post-mapping yield is a direct compound as shown in Figure 3.4. The second TFT plate then chooses its “best” matching CF plate from those remaining 19 plates. This matching procedure continues until the last TFT plate is matched with the last CF plate.

Best-first search can potentially improve the post-mapping yield better than random heuristic, but cannot assure the optimal solution. The best yield-matching for any one TFT glass (or cassette) may not be consistent with a maximum-value when all TFT glasses (or cassettes) are considered. The best-first heuristic can be implemented on a program.

3. Linear programming formulation

Linear programming involves restrictions or constraints for determining optimal solutions to problems. The proposed LP formulation first solves the plate-matching problem for all of the possible cassette matches. The result then becomes the input to the cassette-matching problem. Notation is defined before the linear programming formulation as follows:

=

N the total number of cassettes in queue.

=

r the plate quantities of cassette.

ij =

φ the optimal matching yield of the ith TFT cassette and the jth CF cassette.

This value is the result from the plate-matching LP solution.

ikjl =

f the mapping function represents the matching yield for the kth plate of the ith TFT cassette and the lth plate of the jth CF cassette.

=1

xikjl when the kth plate from the ith TFT cassette is matched with the lth plate from the jth CF cassette. Otherwise, x_ikjl =0. This is the decision variable of the plate-matching LP formulation.

=1

yij when the ith TFT cassette is matched with the jth CF cassette. Otherwise, . This is the decision variable of the cassette-matching LP formulation.

=0 yij

Then, the plate-matching problem can be formulated as equations (3.1) – (3.4).

Maximize

∑∑

Equation (3.1) is the objective function to maximize the yield when the ith TFT cassette and the jth CF cassette are chosen. Equation (3.2) assures that each CF plate has exactly one matching TFT plate. Equation (3.3) assures that each TFT plate has exactly one matching CF plate. Equation (3.4) is the {0, 1} constraints for the decision variables.

The proposed LP approach will solve the plate-matching LP formulation N × N times for all of the possible cassette-matching instances. Although this formulation is a combinatorial problem and for each pair matched cassettes there are

different matches, it has the special structure of a typical assignment problem that can be solved efficiently using a special algorithm, the Hungarian method. In the Hungarian method a one-to-one match is required. Readers are referred to Taha [25]

and Winston [11] for a detailed discussion of the assignment problem and Hungarian method.

The proposed methodology then uses theφ_ij from the plate-matching solution results as the input to model the optimal cassette matching problem as shown in equations (3.5) – (3.8).

Equation (3.5) is the objective function that maximizes the yield through cassette matching. Equation (3.6) assures that each CF cassette is matched to exactly one TFT cassette. Equation (3.7) assures that each TFT cassette has exactly one matching CF cassette. Equation (3.8) is the {0, 1} constraint for the decision variables. The cassette matching formulation also has the special assignment problem structure and can be solved efficiently using the Hungarian method.