4. (18pts) (Multiperiod/Sensitivity Analysis) Swirlpool Manufacturing produces a variety of household appliances at a single manufacturing facility. The expected demand for one of these appliances during the next four months is shown in the following table along with the expected production costs and the expected capacity for producing these items: Month Demand 1 420 Production Cost ($) Production Capacity 49.00 550 400 2 580 45.00 46.00 500 3 310 540 47.00 550 It costs $1.50 per month for each unit of this appliance carried in inventory for one month. This cost figure is based on the assumption that inventory is measured at the end of each month, after production and meeting the monthly demand. It is also possible to store appliances in an external facility, at a unit cost of $2.50 per two months. Namely, the contract with the external storage firm mandates that we store each unit for exactly two months. This is accounted for by measuring the inventory we put in external storage at the end of each month. For example, if we put an appliance in external storage at the external facility at the end of month 1, it becomes available again in month 3. Currently, Swirlpool has 120 units in inventory on hand for this product and no units in external storage. To maintain a level workforce, the company wants to produce at least 350 units per month. Swirlpool wants to determine how many units of this appliance to manufacture during each of the next four months, and how much to put in the respective storages, to meet the expected demand at the lowest possible total cost. A B C D E F G H 1 2 Month 1 2 3 3 Demand 420 580 310 4 540 Initial inventory 120 4 Production Cost ($) 49.00 45.00 46.00 47.00 5 Production Capacity 550 400 500 550 Holding costs 6 7 Minimum level 350 350 350 350 1.5 Standard (1 month) 2.5 External (2 months) 8 Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$9 Amount produced month 1 $D$9 Amount produced month 2 $E$9 Amount produced month 3 $F$9 Amount produced month 4 $C$13 Standard storage month 1 $D$13 Standard storage month 2 $E$13 Standard storage month 3 $F$13 Standard storage month 4 $C$14 External storage month 1 $D$14 External storage month 2 $E$14 External storage month 3 $F$14 External storage month 4 480 0 49 1E+30 5.5 400 -5.5 45 5.5 1E+30 350 0.5 46 1E+30 0.5 500 0 47 0.5 48 180 0 1.5 1E+30 5.5 0 6.5 1.5 1E+30 6.5 40 0 1.5 48 0.5 0 48.5 1.5 1E+30 48.5 0 6 2.5 1E+30 6 0 6 2.5 1E+30 6 0 48 2.5 1E+30 48 0 49.5 2.5 1E+30 49.5 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $C$16 Available to meet demand month 1 $D$16 Available to meet demand month 2 $E$16 Available to meet demand month 3 $F$16 Available to meet demand month 4 420 49 420 70 130 580 50.5 580 70 130 310 45.5 310 40 150 540 47 540 50 150 Swirlpool has implement a linear programming model in Excel and solved it to optimality. The resulting model is depicted on the previous page. The associated sensitivity report can be found above. a) (3pts) Let variable x; be the production in month i, let variable y; be the standard storage at the end of month i, and let variable z; be the external storage at the end of month i, for i=1, 2, 3, 4. Write down the inventory balance constraint for month 3 in terms of these variables. Answer the following questions independently, based on the sensitivity report. Do not implement the spreadsheet model. Justify your answers succinctly (in one or two sentences). For each question, "Cannot determine from the sensitivity report" is a possible answer (but also that answer needs to be justified). b) (3pts) Is the optimal solution unique? Are the shadow prices unique? c) (3pts) Suppose that the demand of month 3 is actually 350 units. What would be the corresponding optimal objective value? d) (3pts) Currently no external storage is utilized. How low must the cost for external storage (of one unit per two months) be so that we would start using it? e) (3pts) Suppose we would increase the production capacity of month 1 by 50 units. Would this change the optimal solution? What would be the total cost? f) (3pts) Suppose the standard storage cost for month 1 (only) increases to $6 per unit per month. Would this change the optimal solution? What would be the total cost?