# If a decision variable is not positive in the optimal

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10. If a decision variable is not positive in the optimal solution, its reduced cost is a. what its objective function value would need to be before it could become positive. b. the amount its objective function value would need to improve before it could become positive. c. zero. d. its dual price. Quantitative Problems – (25 marks each) 11. The Prince George RCMP schedules officers for 8 hour shifts. The shifts start at 8 am, noon, 4 pm, 8 pm, midnight and 4 am. An officer starting one of these shifts works for the next 8 hours. During normal weekday operations, the number of officers needed varies on the time of day. The staff guidelines require the following minimum number of officers on duty: Minimum Number Needed Shift 1: 8 am to noon 5 Shift 2: Noon to 4 pm 6 Shift 3: 4 pm to 8 pm 10 Shift 4: 8 pm to midnight 7 Shift 5: midnight to 4 am 4 Shift 6: 4 am to 8 am 6 What is the LP model to determine the minimum number of officers required to report for each shift? Use x i for the decision variables. 12. Use the following Management Scientist output to answer the questions. a. Give the solution to the problem. b. Which constraints are binding? c. What would happen if the coefficient of x 1 increased by 3? d. What would happen if the right-hand side of constraint 1 increased by 10? LINEAR PROGRAMMING PROBLEM MAX 31X1+35X2+32X3 S.T. 1) 3X1+5X2+2X3>90 2) 6X1+7X2+8X3<150 3) 5X1+3X2+3X3<120 Comm 251 F04 Page 3 of 6 Term Exam 1

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OPTIMAL SOLUTION Objective Function Value = 763.333 Variable Value Reduced Costs ---------- ------- ------------------ X1 13.333 0.000 X2 10.000 0.000 X3 0.000 10.889 Constraint Slack/Surplus Dual Prices -------------- ---------------- ------------------ 1 0.000 -0.778 2 0.000 5.556 3 23.333 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit ------------ ------------------- ---------------- --------------- X1 30.000 31.000 No Upper Limit X2 No Lower Limit 35.000 36.167 X3 No Lower Limit 32.000 42.889
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