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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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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)

Modify the craps game simulator given in Example 7.10 so that it simulates a specified number of games and saves the outcome of each game in a data file. At the end of the simulation, read the data file to determine the percentage of wins and losses that the player has experienced. Test the program by simulating 1000 consecutive games. Based upon these results, estimate the odds of winning when playing craps. Extend the Visual Basic program written for Problem 8.39 (reordering a list of numbers four different ways) so that the original set of numbers is written to a random access data file, which is later updated by replacing the original set of numbers with the sorted numbers. Include all of the following features. (a) Enter a set of numbers from the keyboard and write it out to the data file. (b) Read the numbers from the data file, sort the numbers in one of the four ways indicated in Problem 8.39 and write the sorted numbers out to the data file. (c) Read the numbers (either sorted or unsorted) from the data file and display them on the screen. 85 10 51 49 34 4 71 45 19 50

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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