Maths - Linear Programming

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I'm having trouble formulating this as a linear program. If anyone could help at all it would be much appreciated. Thanks

A manufacturer has contracted to produce 2,000 units of a particular product over the next eight months. Deliveries are scheduled as follows:

Month Units
January 100
February 200
March 300
April 400
May 100
June 100
July 500
August 300
Total 2,000

The manufacturer has estimated that it costs her $1 to store one unit of product for one month. She has a warehouse capacity of 300 units. The manufacturer can produce any number of units in a given month, since the units can be produced mostly with part-time labour, which can be easily obtained. However, there are costs of training new personnel and costs associated with laying off personnel who have been hired. The manufacturer has estimated that it costs approximately 75 cents per unit to increase the production level from one month to the next ( e.g., if productionin January is 200 and is increased to 300 in February, the cost is $75 for training the additional people required to produce at the 300-unit level ).

Similarly, it costs 50 cents per unit to reduce production from one month to the next.

At the end of eight months, all employees will be laid off, with the corresponding production-reduction costs. Assume the production level before January is zero.

  • Maths - Linear Programming -

    What i have done so far is:

    Inventory in Storage (Ii) = Xi + Yi-1 - Orders <= 300 for i = {1,....,8}
    Xi is that months storage
    Yi-1 is last months storage

    In the first period, storage is:
    X1 - 100 <= 300 -> X1<=400
    Cost of storage is $1(X1 - 100) {so cost of storage is the amount of storage}
    Total Cost in first period is (X1 - 100) + 0.75X1 = 1.75X1 - 100

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