17. A manufacturer currently produces four products. Recent recessionary trends cause a decline in demand and the company is laying off workers and discontinuing its third shift.

The problem is that of rescheduling production during the first and second shifts for the remaining quarter of the Year. The production involves various processes and one limiting resource in production is the availability of machine hours for a particular process Z. For this process, the four products require 4, 5, 5, and 7 hours respectively.

The sales manager has forecast the expected sales for each of the four products in the last quarter of the year. The estimates are shown in the following table:

The production capacity in terms of process Z, hours available, is expressed by month and shift.

The labour cost of operating the process Z machines is Rs 100 per hour during the first shift and Rs 120 for the second shift. The other relevant cost is storage. It costs Rs 40 per month to store one unit of any of the four products. It may be noted that it will be necessary to store some units of the four products as there is not enough labour available during the December demand.
Assuming that the company wishes to produce as many products as the sales manager has forecast, formulate an LP model to determine a production schedule that will meet the demand at minimum cost.

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Month Forecast sales Product Product Product Product #1 #2 #3 #4 October 8,000 19,000 4,000 7,000 November 7,000 19,000 15,000 7,000 December 6,000 18,000 17,000 7,000