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cost is low fewer requisitions must be processed, and receiving operations are less frequent. This cost tradeoff is shown graphically in Fig. 11.4. The basic data for the inventory problem are the holding costs, the ordering costs, and the expected demand for raw materials and purchased parts. Other factors that may enter into purchasing decisions are the lead time (the length of time between purchase order and receipt of the materials or parts) and the availability of quantity discounts.
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Linear programming is a mathematical method for making the best possible allocation of limited resources (labor hours, machine hours, materials, etc.). It can help management decide how to use its production facilities most profitably. Suppose, for instance, that the firm produces more than one product, each with a different contribution rate. Management needs to know what combination of quantities produced, given the limitations of its facilities, will bring the highest profit. Three principal concepts must be considered in seeking a solution to the problem: The profit function The constraints of the problem The production characteristics
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11.4.11 The Profit Function
The profit function is a mathematical expression used to show how the profit will vary when different quantities of product 1 and product 2 are produced. It requires knowledge of the contribution that each product makes toward overhead and profit. If the symbols x1 and x2 stand for the number of each product to be produced, and if product 1 makes a profit contribution of $2 per unit and product 2 s profit contribution is $5 per unit, the profit function is 2x1 + 5x2. The problem is to find the values of x1 and x2 that will yield the highest total value (profit), given the constraints of the problem and the production characteristics.
11.4.12 The Constraints
Sometimes the profit-maximizing solution is obvious simply produce as many product units as possible with the resources available. However, the manufacturing process may have characteristics that prevent the use of resources in this way. The constraints of this problem are the limitations imposed by the scarcity of production resources. Suppose that a manufacturer uses lathes, drilling machines, and polishing machines in the production process. The lathes can be used for a maximum of 400 h a month, the drilling machine for 300 h, and polishing machine for 500 h. The machines cannot be used more than the hours indicated, but they need not be used to the limit of this capacity in order to maximize the profit function. The constraints are applicable only as long as the production capacity remains constant. The manufacturer could purchase new machines or add work shifts and expand the machine time available. But both of these actions will probably change the statement of the problem. The profit contribution of the product might be changed (by higher night-shift direct-labor costs, for example). Also, a capital budgeting analysis might be required. The linear programming problem is applicable for a time period during which constraints are fixed.
Production Characteristics
The term production characteristics refer to the machine times used in producing the product. Suppose that product 1 requires 1 h on a lathe, no time on a drilling machine, and 1 h on a polishing machine. Product 2 requires no lathe time, 1 h on a drilling machine, and 1 h on a polishing machine. As stated earlier, the lathe can be used for a maximum of 400 h a month, the drilling machine for 300 h, and the polishing machine for 500 h. The production characteristics would then be expressed as follows: x1 400 No more than 400 lathe hours are available for product 1, and 1 h is required for each unit. (If 2 h per unit were required, the expression would be 2x1 400.)
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