课程名称︰ 管理科学模式 课程性质︰ 必带 课程教师︰ 蒋明晃 开课学院： 管理学院 开课系所︰ 工商管理学系 考试日期（年月日）︰ 2011/4/19 考试时限（分钟）： 60分钟 是否需发放奖励金： 是 （如未明确表示，则不予发放） 试题 : 1. Wilkinson Auto Dealership is contemplating selling one or more of the following types of automobiles - sedans, SUV's, and trucks. There is a fixed license fee per year for doing business in each line, $100,000, $250,000, and $50,000, respectively. The profit contribution exclusive to the fixed cost is $250.00 per unit for sedans, $700.00 per unit for SUV's, and $150.00 per unit for trucks. The company is planning the placement of orders with the manufacturer for next year. Dealer preparation takes 2hrs/sedan, 3.0 hrs/SUV, and 1.5 hrs/truck. They have 4400 hrs of preparation time next year. Sedans take 1 unit of space, SUV's take 1.5 units of space, and trucks take 1.1 units of space. 1200 units of space are available. How many sedans, SUV's, and trucks should be ordered in order to maximize total profit contribution less fixed costs incurred? Define the decision variables, constraints, and the objective function for this problem. (Allow the order quantities to be fractional). (26 points) 2. A cargo plane has two compartments for storing cargo, front and back. These compartments have capacity limits on both weight and space, as summarized below. ┌──────┬────────────┬──────────────┐ │Compartment │Weight capacity (Pounds)│Space capacity (Cubic feet) │ ├──────┼────────────┼──────────────┤ │Front │ 16,000 │ 5,000 │ ├──────┼────────────┼──────────────┤ │Back │ 24,000 │ 7,000 │ └──────┴────────────┴──────────────┘ They have five available cargos for an upcoming flight. The details are given below: ┌───┬────────┬───────────────┬─────┐ │Cargo │Weight (Pounds) │Volume of the cargo (cu. ft.) │Profit ($)│ ├───┼────────┼───────────────┼─────┤ │A │ 10,000 │ 2500 │ 1000 │ ├───┼────────┼───────────────┼─────┤ │B │ 12,000 │ 4200 │ 1560 │ ├───┼────────┼───────────────┼─────┤ │C │ 15,000 │ 4500 │ 1350 │ ├───┼────────┼───────────────┼─────┤ │D │ 8,000 │ 2240 │ 720 │ ├───┼────────┼───────────────┼─────┤ │E │ 11,000 │ 3520 │ 1210 │ └───┴────────┴───────────────┴─────┘ For maintaining balance, the total weight of the cargo loaded in the back must be more than the total cargo loaded in the front. Each cargo must be accepted in full and loaded entirely in front or back. The objective is to determine which must be accepted and where it should go in the plane, so as to maximize total profit. Formulate this as an integer linear program. (32 points) 3. Ken and Larry, Inc., supplies its ice cream parlors with three flavors of ice cream: chocolate, vanilla, and banana. Due to extremely hot weather and a high demand for its products, the company has run short of its supply of ingredients: milk, sugar, and cream. Hence, they will not be able to fill all the orders received from their retail outlets, the ice cream parlors. Due to these circumstances, the company has decided to choose the amount of each flavor to produce that will maximize total profit, given the constraints on supply of the basic ingredients. The chocolate, vanilla, and banana flavors generate, respectively, $1.00, $0.90, and $0.95 of profit per gallon sold. The company has only 200 gallons of milk, 150 pounds of sugar, and 60 gallons of cream left in its inventory. The linear programming formulation for this problem is shown below in algebraic form. Let C = gallons of chocolate ice cream produced, V = gallons of vanilla ice cream produced, B = gallons of banana ice cream produced. Maximize Profit = 1.00 C + 0.90 V + 0.95 B, subject to Milk: 0.45 C + 0.50 V + 0.40 B ≦ 200 gallons Sugar: 0.50 C + 0.40 V + 0.40 B ≦ 150 pounds Cream: 0.10 C + 0.15 V + 0.20 B ≦ 60 gallons and C≧0, V≧0, B≧0. This problem was solved using the Excel Solver. The spreadsheet (already solved) and the sensitivity report are shown below. [Note: The numbers in the sensitivity report for the milk constraint are missing on purpose, since you will be asked to fill in these numbers in part (f).] Resource Usage Per Unit of Each Activity │ Activity │ Resource Resource │ Chocolate Vanilla Banana │Totals Available ──────┼────────────────┼──────────── Milk │ 0.45 0.5 0.4 │ 180 ≦ 200 Sugar │ 0.5 0.4 0.4 │ 150 ≦ 150 Cream │ 0.1 0.15 0.2 │ 60 ≦ 60 ──────┼────────────────┼──────────── Unit Profit│ 1 0.9 0.95 │$341.25 Solution │ 0 300 75 │ Changing Cells ───────────────────────────────────── Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease ───────────────────────────────────── $B$8 Solution Chocolate 0 -0.0375 1 0.0375 1E+30 $C$8 Solution Vanilla 300 0 0.9 0.05 0.0125 $D$8 Solution Banana 75 0 0.95 0.021429 0.05 ───────────────────────────────────── Constraints ───────────────────────────────────── Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease ───────────────────────────────────── $E$4 Milk Totals $E$5 Sugar Totals 150 1.875 150 10 30 $E$6 Cream Totals 60 1 60 15 3.75 ───────────────────────────────────── For each of the following parts, answer the question as specifically and completely as is possible without solving the problem again on the Excel Solver. Note: Each part is independent (i.e., any change made to the model in one part does not apply to any other parts). (42 points) (a) What is the optimal solution and total profit? (b) Suppose the profit per gallon of banana changes to $1.00. Will the optimal solution change, and what can be said about the effect on total profit? (c) Suppose the profit per gallon of banana changes to $0.92. Will the optimal solution change, and what can be said about the effect on total profit? (d) Suppose the company discovers that 3 gallons of cream have gone sour and so must be thrown out. Will the optimal solution change, and what can be said about the effect on total profit? (e) Suppose the company has the opportunity to buy an additional 15 pounds of sugar at a total cost of $15. Should they? Explain. (f) Fill in all the sensitivity report information for the milk constraints given just the optimal solution for the problem. Explain how you were able to deduce each number. --

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