Management Science Techniques for Consultants
Case Study 2: Factory Planning

Due October 10, 1996. You may work in groups of up to four people.

A factory makes six products (P1 to P6) on the following machines: four grinders, two vertical drills, and three horizontal drills. Each product yields a certain contribution to profit (defined as selling price minus cost of raw materials and other variable costs). These profits (in $/unit), together with the unit production times (in hours) are given in Table 1. A dash indicates that a product does not require a process.

  
Table 1: Production Data

In the coming six months (beginning in August), certain machines will be down for maintenance (for the entire month). These machines will be as follows:

The orders for each product in each month are given in Table 2. Unmet demand in any given month results in lost sales. It is possible to hold inventories of each product at at cost of $0.50 per unit per month. There are no inventories at present, and no inventories desired at the end of January.

  
Table 2: Orders

The factory works a six day week with two 8-hour shifts per day. Assume that there are 24 working days in every month.

A previous consultant created a linear program to optimize production without considering machine downtime due to maintenance. This spreadsheet is available at the web site http://mat.gsia.cmu.edu/mstc/. Your task is to update this model so as to consider machine downtime and answer the following questions:

1. Examine the current model (which, unfortunately comes with very little documentation). Define the variables and constraints and update the model to handle machine downtimes as given above.
2. When and what should the factory make in order to maximize the total profit? What is the effect of requiring integer production quantities of each product in every month? In all following questions, do not require integer production quantities.
3. Instead of stipulating when each machine is down for maintenance, it is desired to find the best month for each machine to be down. Each vertical and horizontal drill must be down for maintenance in one month of the six. Only two of the grinders (namely grinders 1 and 2) need be down in the next six months. In this question you will evaluate two alternative formulations for solving this problem.
   1. Define 0-1 integer variables to be 1 if grinder number i is down in month j, and 0 otherwise. Similarly define and for horizontal and vertical drills. Formulate and solve the integer program for maximizing profit subject to the maintenance and other constraints.
   2. Define the general integer variable to be the number of grinders down in month j, with and defined similarly. Formulate and solve the integer program for maximizing profit subject to the maintenance and other constraints.

   After solving both of these, compare the models and corresponding solutions. Some aspects to consider are problem size, computation time, number of branches enumerated, initial linear programming solution quality, and any other aspects you deem important.

4. Find the best month for each machine to be down when all the following additional restrictions must be met:
   • No more than one machine of each type can be maintained during a single month,
   • At most two machines can be maintained during the same month, and
   • When a grinder or horizontal drill is maintained, then no vertical drill can be maintained during the same month.

   Determine constraints for each of your two integer programming formulations, and solve. How do the two formulations compare for this problem? Consider ease of formulation, computation time, and any other relevant aspects.

Note: In all the questions, you can terminate a computer run if SOLVER requires more than ten minutes to solve the problem. Report all the information you can gather on such partial runs.

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