SilComputer needs to meet the demand of its largest corporate and educational cu

Question

SilComputer needs to meet the demand of its largest corporate and educational customers for notebook computers over the next four quarters (before its current model becomes obsolete). SilComputer currently has 5,000 notebook computers in inventory. Expected demand over the next four quarters for its notebook is 7,000; 15,000; 10,000; and 8,000. SilComputer has sufficient capacity and material to produce up to 10,000 computers in each quarter at a cost of $2000 per notebook. By using overtime, up to an additional 2,500 computers can be produced at a cost of $2200 each. Computers produced in a quarter can be used either to meet that quarter's demand, or be held in inventory for use later. Each computer in inventory is charged $100 for each quarter to reflect carrying costs. How should SilComputer meet its demand for notebooks at minimum cost? Provide a specific model, and then a generic model. Make sure you define all the variables, parameters clearly for the generic model.

Explanation / Answer

Answer:

Decision Variables.

The decision variables represent (unknown) decisions to be made. This is in contrast to problem data, which are values that are either given or can be simply calculated from what is given. For this problem, the decision variables are the number of notebooks to produce and the number of desktops to produce.

We will represent these unknown values by x1 and x2 respectively. To make the numbers more manageable, we will let x1 be the number of 1000 notebooks produced (so x1 = 5 means a decision to produce 5000 notebooks) and x2 be the number of 1000 desktops.

Note that a value like the quarterly pro t is not (in this model) a decision variable: it is an outcome of decisions x1 and x2.

Objective. Every linear program has an objective. This ob jective is to be either minimized or maximized. This ob jective has to be linear in the decision variables, which means it must be the sum of constants times decision variables. 3x1 10x2 is a linear function. x1x2 is not a linear function. In this case, our objective is to maximize the function 750x1 + 1000x2 (what units is this in?).

Constraints. Every linear program also has constraints limiting feasible decisions. Here we have four types of constraints: Processing Chips, Memory Sets, Assembly, and Nonnegativity.

In order to satisfy the limit on the number of chips available, it is necessary that x1 + x2 10. If this were not the case (say x1 = x2 = 6), the decisions would not be implementable (12,000 chips would be required, though we only have 10,000). Linear programming cannot handle arbitrary restrictions: once again, the restrictions have to be linear. This means that a linear function of the decision variables must be related to a constant, where related can mean less than or equal to, greater than or equal to, or equal to.

So 3x1 2x2 10 is a linear constraint, as is x1 + x3 = 6. x1x2 10 is not a linear constraint, nor is x1 + 3x2 < 3.

Our constraint for Processing Chips x1 + x2 10 is a linear constraint. The constraint for memory chip sets is x1 + 2x2 15, a linear constraint. Our constraint on assembly can be written 4x1 + 3x2 25, again a linear constraint. Finally, we do not want to consider decisions like x1 = 5, where production is negative. We add the linear constraints x1 0, x2 0 to enforce nonnegativity of production.

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