Chapter 2

# Chapter 2 - Chapter 2 An Introduction to Linear Programming...

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Chapter 2 An Introduction to Linear Programming Case Problem 1: Workload Balancing 1. Production Rate (minutes per printer) Model Line 1 Line 2 Profit Contribution (\$) DI-910 3 4 42 DI-950 6 2 87 Capacity: 8 hours60 minutes/hour = 480 minutes per day Let D 1 = number of units of the DI-910 produced D 2 = number of units of the DI-950 produced Max 42 D 1 + 87 D 2 s.t. 3 D 1 + 6 D 2 480 Line 1 Capacity 4 D 1 + 2 D 2 480 Line 2 Capacity D 1 , D 2 0 Using The Management Scientist , the optimal solution is D 1 = 0, D 2 = 80. The value of the optimal solution is \$6960. Management would not implement this solution because no units of the DI-910 would be produced. 2. Adding the constraint D 1 D 2 and resolving the linear program results in the optimal solution D 1 = 53.333, D 2 = 53.333. The value of the optimal solution is \$6880. 3. Time spent on Line 1: 3(53.333) + 6(53.333) = 480 minutes Time spent on Line 2: 4(53.333) + 2(53.333) = 320 minutes Thus, the solution does not balance the total time spent on Line 1 and the total time spent on Line 2. This might be a concern to management if no other work assignments were available for the employees on Line 2. 4. Let T 1 = total time spent on Line 1 T 2 = total time spent on Line 2 Whatever the value of T 2 is, T 1 T 2 + 30 T 1 T 2 - 30 Thus, with T 1 = 3 D 1 + 6 D 2 and T 2 = 4 D 1 + 2 D 2 3 D 1 + 6 D 2 4 D 1 + 2 D 2 + 30 3 D 1 + 6 D 2 4 D 1 + 2 D 2 - 30 Hence, CP - 1

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Chapter 2 - 1 D 1 + 4 D 2 30 - 1 D 1 + 4 D 2 ≥ - 30 Rewriting the second constraint by multiplying both sides by -1, we obtain - 1 D 1 + 4 D 2 30 1 D 1 - 4 D 2 30 Adding these two constraints to the linear program formulated in part (2) and resolving using The Management Scientist , we obtain the optimal solution D 1 = 96.667, D 2 = 31.667. The value of the optimal solution is \$6815. Line 1 is scheduled for 480 minutes and Line 2 for 450 minutes. The effect of workload balancing is to reduce the total contribution to profit by \$6880 - \$6815 = \$65 per shift. 5. The optimal solution is D 1 = 106.667, D 2 = 26.667. The total profit contribution is 42(106.667) + 87(26.667) = \$6800 Comparing the solutions to part (4) and part (5), maximizing the number of printers produced (106.667 + 26.667 = 133.33) has increased the production by 133.33 - (96.667 + 31.667) = 5 printers but has reduced profit contribution by \$6815 - \$6800 = \$15. But, this solution results in perfect workload balancing because the total time spent on each line is 480 minutes. Case Problem 2: Production Strategy
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## Chapter 2 - Chapter 2 An Introduction to Linear Programming...

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