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Operations Processing and Linear Programming

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1. VeriFast Semiconductor makes memory chips for digital electronics manufacturers, and it has expensive equipment in its plant that has to be used for multiple product lines. Photolithography is one of the critical steps in wafer fabrication facilities, and lithography unit R2D2 in their plant is their newest and most expensive to date.

The photolithography process is very similar to the use of film cameras; a light source is directed through a (very expensive) lens that has the integrated circuit diagram printed on it, at a silicon surface where we wish to carve the image of the circuit. In the VeriFast process, the silicon wafer surface is coated with special light sensitive "resist" materials, and the light exposure essentially hardens the exposed coating. The next step is to dip the wafers in a chemical bath that dissolves away the unexposed regions, so that a permanent circuit is etched on the wafer surface.
A single wafer may have up to a thousand identical circuits printed on its surface, and eventually, the identical and individual dies are cut out of the wafer to be packaged as the electronic chips were are all familiar with. Of course all this printing happens at the micron level so that we, as customers, get to buy our bite-sized digital cameras in the market.

The VeriFast factory is currently set-up to process two variants of semiconductor wafers: Wafer A and Wafer B. The two products A and B flow through the plant in boxes of 30 and 50 wafers respectively. The only two process steps of interest to us for this problem are the lithography and the chemical bath steps. R2D2 experiences a setup of 20 minutes to switchover to a box of Wafer A, and a setup of 30 minutes to switchover to a box of Wafer B. (The setup times occur only when switching product types.) Because the switchover times are significant, 5 boxes of Wafer A are exposed before switching over to a sequence of 5 boxes of Wafer B. This cycle is repeated over throughout the day. Once the setup is complete, the machine processes wafers, independent of type, at the rate of 2 wafers per minute.

From lithography, the boxes are transferred to the chemical bath or "etching" at the SpA
machine. In the etching step, a box of wafers of either Wafer A or Wafer B takes 15 minutes to setup as the chemicals in the SpA need to be changed or refilled for each box. However, Wafer A needs to be dipped in the SpA for 2 minutes longer than Wafer B, with the recommended processing time for a box of Wafer B currently at 3 minutes. There is no work-in-process inventory between R2D2 and SpA machines.

Luke, the area manager has been informed that product mix is likely to change in the coming
months, and he has been asked to keep an eye out for something called a bottleneck as the
product mix changes in the next few months.

a.) Draw the process flow diagram, and determine the product mix for the current
system of production.
Wafer A : Wafer B = _____:_____

b.) The manager wants to know the percentage of the time that R2D2 (at the
lithography step) spends on each of the wafers under the current product mix. What is the
percentage of the time that R2D2 spends on Wafer A?

_________________ %

c.) Calculate R2D2's capacity (in terms of processing rate) for the lithography step.

Lithography capacity: wafers/hour

d.) Which machine among R2D2 and SpA is the bottleneck? What is the capacity
of the system?

Bottleneck: .

Bottleneck capacity: wafers/hour.

Demand of Wafer B is expected to decrease for the following month. Moreover, the new CEO
Darth, is trying to implement "lean manufacturing." Thus, the batch size has been reduced to 25
wafers for both types A and B wafers.

e.) What is the capacity of the SpA machine under this rule?

Etching step capacity: wafers/hour

f.) What is the bottleneck under this new batch size rule? What is the capacity of the system?

Bottleneck: .

Bottleneck capacity: wafers/hour.

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Solution Summary

Word file contains process flow diagram and calculations of the percentage of the time that R2D2 spends on Wafer A,Bottleneck capacity and capacity of unit.

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The Thompson Furniture Company produces inexpensive tables and chairs using a production process common for both products. Each table requires 4 hours in the carpentry department and 2 hours in the painting and varnishing department. Each chair requires 3 hours in carpentry and 1 hour in painting and varnishing. During the current production period, 240 hours are scheduled in carpentry and 100 hours are scheduled in painting and varnishing. Also, each product must undergo an inspection and possible rework process. A table requires an average of 1/2 hour for this process, each chair requires 3/5 hour on average. For the current production period, 36 hours have been scheduled for the inspection/rework process. Each table requires 32 linear feet of lumber in the manufacturing process while 10 linear feet are needed to complete each chair. There are 1,248 linear feet of lumber currently available for production of these products.
Each table contributes $23 towards profit while each chair contributes $13. The current schedule reflects a maximum demand for tables at 40 without a similar constraint for chairs.

a. Formulate a linear programming model that can be used to determine the production schedule for the two products for the current period.

b. Additional lumber is available to Thompson at $0.38 per linear foot in minimum quantity orders of 500 linear feet each. An order for the next production period has already been placed under a continuing vendor contract to arrive at the end of the current production period. That order will require all but 300 linear feet of available lumber storage space. A maximum of 8 hours of additional time is available for the current period in each of the carpentry and painting/varnishing departments using part-time employees at an hourly rate of $10.50. A maximum of 4 hours of additional time in inspection/rework is available in the current period at an hourly rate of $9.50. All production hours previously scheduled are fixed by employment contract. All unused resources, including lumber not stored, is considered wasted. Determine and justify which additional resources, if any, would result in a better solution for Thompson.
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