Management Science term notes.docx

Expressing the problem mathematically the issue

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Expressing the Problem Mathematically The issue facing management is to make the following decision. Decision to be made: Number of watches to produce (if any). Since this number is not yet known, we introduce an algebraic variable Q to represent this quantity. Thus, Q = Number of watches to produce, where Q is referred to as a decision variable . Naturally, the value chosen for Q should not exceed the sales forecast for the number of watches that can be sold. Choosing a value of 0 for Q would correspond to deciding not to introduce the product, in which case none of the costs or revenues described in the preceding paragraph would be incurred. The objective is to choose the value of Q that maximizes the company’s profit from this new product. The management science approach is to formulate a mathematical model to represent this problem by developing an
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equation that expresses the profit in terms of the decision variable Q. To get there, it is necessary first to develop equations in terms of Q for the total cost and revenue generated by the watches. If Q = 0, no cost is incurred. However, if Q > 0, there is both a fixed cost and a variable cost. Therefore, the total cost would be Since each watch sold would generate a revenue of $2,000 for the company, the total revenue from selling Q watches would be Total revenue = $2,000 Q Consequently, the profit from producing and selling Q watches would be Analysis of the Problem This last equation shows that the attractiveness of the proposed new product depends greatly on the value of Q, that is, on the number of watches that can be produced and sold. A small page 9 value of Q means a loss (negative profit) for the company, whereas a sufficiently large value would generate a positive profit for the company. For example, look at the difference between Q = 2,000 and Q = 20,000.
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FIGURE 1.2 Break-even analysis for the Special Products Company shows that the cost line and revenue line intersect at Q = 10,000 watches, so this is the break-even point for the proposed new product. Profit = − $10 million + $1,000 (2,000) = −$8 million if Q = 2,000 Profit = − $10 million + $1,000 (20,000) = $10 million if Q = 20,00 Figure 1.2 plots both the company’s total cost and total revenue for the various values of Q. Note that the cost line and the revenue line intersect at Q = 10,000. For any value of Q < 10,000, cost exceeds revenue, so the gap between the two lines represents the loss to the company. For any Q > 10,000, revenue exceeds cost, so the gap between the two lines now shows positive profit. At Q = 10,000, the profit is 0. Since 10,000 units is the production and sales volume at which the company would break even on the proposed new product, this volume is referred to as the break-even point . This is the point that must be exceeded to make it worthwhile to introduce the product. Therefore, the crucial question is whether the sales forecast for how many watches can be sold is above or below the break-even point.
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  • Spring '19
  • Miller

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