1. Consider the following linear program: MIN 6x1 + 9x2 ($ cost) s.t. x1 + 2x2  8 10x1 + 7.5x2  30 x2  2 x1, x2  0 The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 27.000 Variable Value Reduced Cost X1 1.500 0.000 X2 2.000 0.000 Constraint Slack/Surplus Dual Price 1 2.500 0.000 2 0.000 0.600 3 0.000 4.500 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 0.000 6.000 12.000 X2 4.500 9.000 No Upper Limit RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 5.500 8.000 No Upper Limit 2 15.000 30.000 55.000 3 0.000 2.000 4.000 a. What is the optimal solution including the optimal value of the objective function? b. Suppose the unit cost of x1 is decreased to $4. Is the above solution still optimal? What is the value of the objective function when this unit cost is decreased to $4? c. How much can the unit cost of x2 be decreased without concern for the optimal solution changing? d. If simultaneously the cost of x1 was raised to $7.5 and the cost of x2 was reduced to $6, would the current solution still remain optimal? e. If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal solution? 2. BP Cola must decide how much money to allocate for new soda and traditional soda advertising over the coming year. The advertising budget is $10,000,000. Because BP wants to push its new sodas, at least one-half of the advertising budget is to be devoted to new soda advertising. However, at least $2,000,000 is to be spent on its traditional sodas. BP estimates that each dollar spent on traditional sodas will translate into 100 cans sold, whereas, because of the harder sell needed for new products, each dollar spent on new sodas will translate into 50 cans sold. To attract new customers BP has lowered its profit margin on new sodas to 2 cents per can as compared to 4 cents per can for traditional sodas. How should BP allocate its advertising budget if it wants to maximize its profits while selling at least 750 million cans? Develop a linear model for this problem 3. Joseph King has ambitions to be mayor of Williston, North Dakota. Joe has determined the breakdown of the steps to the nomination and has estimated normal and crash costs and times for the campaign as follows (times are in weeks). Normal Crash Immediate Activity Time Cost Time Cost Predecessors A. Solicit Volunteers 6 $5,000 4 $ 9,000 --- B. Initial "Free" Exposure 3 $4,000 3 $ 4,000 --- C. Raise Money 9 $4,000 6 $10,000 A D. Organize Schedule 4 $1,000 2 $ 2,000 A E. Hire Advertising Firm 2 $1,500 1 $ 2,000 B F. Arrange TV Interview 3 $4,000 1 $ 8,000 B G. Advertising Campaign 5 $7,000 4 $12,000 C,E H. Personal Campaigning 7 $8,000 5 $20,000 D,F Joe King is not a wealthy man and would like to organize a 16-week campaign at minimum cost. a. Draw a network diagram b. Write and solve a linear program to accomplish this task. c. Solve the model and find the solution 4. Daily demand for packages of five videotapes at a warehouse store is found to be normally distributed with mean 50 and standard deviation 5. When the store orders more tapes, the ordering cost is $42 and the orders take 4 days to arrive. Each pack of tapes costs $7.20 and there is a 24% annual holding cost for inventory. Assume the store is open 360 days a year. a. What is the EOQ? b. If the store wants the probability of stocking out to be no more than 5%, and demand each day is independent of the day before, what reorder point should be set? c. How much of your reorder point in part b) is safety stock? 5. Bank Drugs sells Jami Michelle lipstick. The Jami Michelle Company offers a 6% discount on orders of at least 500 tubes, a 10% discount on orders of at least 1,000 tubes, a 12% discount on orders of at least 1,800 tubes and a 15% discount on orders at least 2,500 tubes. Bank sells an average of 40 tubes of Jami Michelle lipstick weekly. The normal price paid by Bank drugs is $1 per tube. If it costs Bank $30 to place an order, and Bank's annual holding cost rate is 27%, determine the optimal order policy for Bank Drugs.