1. The number of pizzas ordered on Friday evenings between 5:30 and 6:30 at a pizza delivery location for the last 10 weeks is shown below. 58, 46, 55, 39, 43, 63, 54, 58, 62, 52 Use exponential smoothing with smoothing constants of .2 and .8 to forecast a value for week 11. Compare your forecasts using MSE. Which smoothing constant would you prefer? (Please do not just write the answer. Otherwise your will lose points.) 2. Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below. Fliptop Model Tiptop Model Available Plastic 3 4 36 Ink Assembly 5 4 52 Molding Time 2 2 30 The profit for Fliptop model is $1200 per lot, and the profit for Tiptop model is $1400 per lot. Please answer the following questions: a. What is the linear programming model for this problem? b. Find the optimal solution. (You can use graphic method. Please show your detailed steps the graph if you use graphic method. You can also use software package such as Excel Solver. Please provide the Excel file if you use Solver. Please do not just write the answer. Otherwise your will lose points.) 3. Consider the following linear program: Max 3x1 + 4x2 ($ Profit) s.t. x1 + 3x2 <=12 2x1 + x2 <= 8 x1 <= 3 x1, x2 >= 0 The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 20.000 Variable Value Reduced Cost X1 2.400 0.000 X2 3.200 0.000 Constraint Slack/Surplus Dual Price 1 0.000 1.000 2 0.000 1.000 3 0.600 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 1.333 3.000 8.000 X2 1.500 4.000 9.000 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 9.000 12.000 24.000 2 4.000 8.000 9.000 3 2.400 3.000 No Upper Limit a. What is the optimal solution including the optimal value of the objective function? b. Suppose the profit on x1 is increased to $8. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $8? c. If the unit profit on x2 was $9 instead of $4, would the optimal solution change? If yes, what is the optimal solution? 4. The ABC Club of sponsors driver education events that provide high- performance driving instruction on actual racetracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. XYZ company manufactures two types of roll bars. Model DRB is bolted to the car using exiting holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high-alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high-alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. XYZ's steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, XYZ estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $320 per unit for model DRW. Please answer the following questions: a. What are the optimal solution and the total profit contribution? (You can use graphic method. Please show your detailed steps the graph if you use graphic method. You can also use software package such as Excel Solver. Please provide the Excel file if you use Solver. Please do not just write the answer. Otherwise you will lose points.) b. Another supplier offered to provide XYZ company with an additional 500 of the steel alloy at $8 per pound. Should XYZ company purchase the additional pounds of the steel alloy? Explain. c. XYZ company is considering using overtime to increase the available assembly time. What would you advise XYZ to do regarding this option'? Explain. d. Because of increased competition, XYZ company is considering reducing the price of model DRB such that the new contribution to profit is $150 per unit. How would this change in price affect the optimal solution? Explain