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x2 300 No more than 300 drill hours are available for product 2, and 1 h per unit is required. x1 + x2 500 No more than 500 polishing hours are available for both products, with 1 hour per unit required in both cases. Solving the problem. The problem for the values given previously is as follows: Many solutions would satisfy all three constraints, but there is only one maximizing solution. The possible solutions are shown graphically in Fig. 11.5. The axes represent the number of product units; the remaining solid lines, the production characteristics. Any point in the shaded area will satisfy the constraints, but the maximizing solution will be at one of the four corners of this area point A, B, C, or D. This point is the exact one which can be found by drawing a line that represents the profit function 2x1 + 5x2 (the lower dotted line in Fig. 11.5). When this line is shifted upward, always parallel to its original position, it finally reaches the point where it touches the shaded solution space at only one point, B. This point, where x1 = 200 and x2 = 300, is the solution that gives the highest possible total contribution within the limited resource constraints. The solution gives the following results: Total contribution Lathe time used Drill time used Polishing time used $ 1900 (200 $2 + 300 $5) 200 h (200 1 h) 300 h (300 1 h) 500 h (200 1 h + 300 1 h)
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FIGURE 11.5
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There are 200 idle lathe hours, but these cannot be used under existing conditions without reducing the output of product 2. If more lathe time is used for product 1, then more polishing time must also be used; and the polishing time must be taken away from product 2, thus reducing product 2 output. Any reduction in product 2 output represents a loss of $5 per unit, whereas producing additional units of product 1 will add only $2 per unit. The net loss from such a substitution is therefore $3 per unit. On the other hand, the output of product 2 cannot be increased, since there is no more drill time available. The graphic solution described here is a rough approach to a fairly simple problem. More complex problems can be programmed into a computerized automated system.
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Manufacturing and construction firms may undertake one-time-only jobs, such as the construction of a building or the design and manufacture of a special piece of a large assembly system. In such projects, at any one time much of the company s time, resources, and prospects may be tied to a single large job. And whether the job will result in profit or loss for the company may depend on its being completed on time. If the completion date is to be met, the various segments of the job must be coordinated. Subunits must be scheduled, and they must be finished according to the schedule date. Project management techniques have been developed for planning the subunits that make up such a job. One of the most important of these methods is the project evaluation and review technique (PERT). Another one, very similar to PERT, is known as the critical path method (CPM).
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PERT is based on what is called a network plan. A flowchart the network is used to illustrate how the individual parts of a project (the activities and the starting and completion events) depend on one another and which task must be finished before others can be started. It is particularly concerned with dovetailing individual parts of a project into the schedule of the project as a whole. PERT analysis makes the following estimates from data about the activities and events that make up the project: Earliest expected time for completion of each activity, or task The latest allowable time for each completion event The critical path of the network that is, the path through a flowchart that has the least total amount of slack (Slack is the difference between the latest allowable time and the expected time of completion.) The probabilities that events will occur on a schedule
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