課程名稱︰管理科學模式 課程性質︰必帶 課程教師︰蔣明晃 開課學院：管理學院 開課系所︰工商管理學系科技管理組 考試日期（年月日）︰2010.03.26 考試時限（分鐘）：50 min. 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.(20 points) Midwest Money Manger (MMM), an investment firm, has $4 million to invest. They have four choices, namely stocks, bonds, money-markets and goverment securities. The respective projected yields are: 15%, 9%, 6%, and 4.5%. The respective risk indices are: 0.2, 011, 0.07, and 0.01. They can also put their money in a vault (safe deposit vault), earning 0% and having a risk index of 0. It is assumed that the risk index of a portfolio is equal to the weighted average value of the individual index, using the proportion of investment as weights. MMM wants to limit its investment in stocks and bonds to a maximum of 50% of the total investment. Investment in money markets should always be less than or equal to investment in government securities. MMM wants to earn at least $80,000 in the next year and minimize the risk of its portfolio. Formulate this as a linear program. Specify the decision variables, constraints, and the objective function. 2.(24 points) A certain restaurant operates 7 days a week. Waiters are hired to work 6 hours per day. The union contract specifies that each must work 5 consecutive days and then have 2 consecutive days off. All waiters receive the same weekly salary. Staffing requirements are shown in the Table 1. Assume that this cycle of requirements repeats indefinitely, and ignore the fact that the number of waiters hired must be an integer. The manager wishes to find an employment schedule that satisfies these requirements at a minimum cost. Formulate this problem as a linear program. Specify the decision variables, constraints, and the objective function. ┌──────┬───┬────┬───┬────┬───┬────┬───┐ │Day │Monday│Tuesday │Wed. │Thursday│Friday│Saturday│Sunday│ ├──────┼───┼────┼───┼────┼───┼────┼───┤ │Minimum │ 150 │ 200 │ 400 │ 300 │ 700 │ 800 │ 300 │ │number of │ │ │ │ │ │ │ │ │waiter hours│ │ │ │ │ │ │ │ │requirements│ │ │ │ │ │ │ │ └──────┴───┴────┴───┴────┴───┴────┴───┘ Red Brand Canners case: http://140.112.110.2/[email protected]@.21d1e881 3.(56 points) For the following question assume 3 grades of tomatoes, as in the alternate questions in the original Red Brand Canners case. The LP formulation and solution are shown as follows: AC=lbs.A in canned; BC=lbs.B in canned; CC=lbs.C in canned; AJ=lbs.A in juice; BJ=lbs.B in juice; CJ=lbs.C in juice; AP=lbs.A in paste; BP=lbs.B in paste; CP=lbs.C in paste; Max 8.22(AC+BC+CC)+6.6(AJ+BJ+CJ)+7.4(AP+BP+CP) s.t. AC+BC+CC≦14,400,000 (canned) AJ+BJ+CJ≦1,000,000 (juice) AP+BP+CP≦2,000,000 (paste) AC+AJ+AP≦600,000 (A limits) BC+BJ+BP≦1,600,000 (B limits) CC+CJ+CP≦800,000 (C limits) AC-2BC-5CC≧0 (canned quality) 3AJ-3CJ≧0 (juice quality) All variables nonnegative The Excel solution is shown in Figure 1. ┌─────────────────────────────────────┐ │ Red Brands Canners Solution │ ├───────┬────┬──────┬──────┬─────┬────┤ │Variable Name │Final │Reduced Cost│Objective │Allowable │Allowable │ │Value │ │Coefficient │Decrease │Increase│ ├───────┼────┼──────┼──────┼─────┼────┤ │AC │ 600000│ 0│ 8.22│ 2.43│Infinity│ ├───────┼────┼──────┼──────┼─────┼────┤ │BC │ 300000│ 0│ 8.22│ 0.972│ 1E+30│ ├───────┼────┼──────┼──────┼─────┼────┤ │CC │ 0│ 2.43│ 8.22│ 1E+30│ 2.43│ ├───────┼────┼──────┼──────┼─────┼────┤ │AJ │ 0│ 0│ 6.6│ 1E+30│ 2.43│ ├───────┼────┼──────┼──────┼─────┼────┤ │BJ │ 100000│ 0│ 6.6│ 4.86│ 0.8│ ├───────┼────┼──────┼──────┼─────┼────┤ │CJ │ 0│ 2.43│ 6.6│ 1E+30│ 2.43│ ├───────┼────┼──────┼──────┼─────┼────┤ │AP │ 0│ 2.43│ 7.4│ 1E+30│ 2.43│ ├───────┼────┼──────┼──────┼─────┼────┤ │BP │ 1200000│ 0│ 7.4│ 0.8│ 2.43│ ├───────┼────┼──────┼──────┼─────┼────┤ │CP │ 800000│ 0│ 7.4│ 2.43│ 1E+30│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint │Shadow │Final Value │Constraint │Allowable │Allowable │ │Price │ │R. H. Side │Decrease │Increase│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 1 │ 0│ 13500000│ 14400000│ 900000│ 1E+30│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 2 │ 0│ 900000│ 1000000│ 100000│ 1E+30│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 3 │ .8│ 0│ 2000000│ 900000│ 100000│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 4 │ 9.03│ 0│ 600000│ 600000│ 200000│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 5 │ 6.6│ 0│ 1600000│ 100000│ 900000│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 6 │ 6.6│ 0│ 800000│ 100000│ 900000│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 7 │ -8.1│ 0│ 0│ 200000.0│ 600000│ ├───────┼────┼──────┼──────┼─────┼────┤ │Constraint 8 │ -8.1│ 0│ 0│ 0│ 1800000│ └───────┴────┴──────┴──────┴─────┴────┘ Figure 1 ￣￣￣￣ a. What is the net profit obtained after netting out the cost of the crop? cost = $180,000. b. Myers claims the net profit from his policy of producing 2,000,000 lb paste and 1,000,000 lb juice is $89,600. Is this correct? If not, what is his net profit? c. Use the above sensitivity output to determine whether the additional purchase of up to 80,000 pounds of grade A tomatoes should be undertaken. Can you tell how much should be purchased? d. Suppose that the Market Research Department feels they could increase the demand for paste by 3000 cases by means of an advertising campaign. How much should Red Brand be willing to pay for such a campaign? e. Suppose that the price of canned whole tomatoes decreased by 16 cents per case. Does your computer output tell you whether the optimal production plan will change? f. Suppose that the Market Research Department suggests that if the average quality of paste is below 4 the product will not be acceptable to customers. Would an additional computer run be necessary to determine the optimal production plan if this Constraint were added to the model? g. Suppose that an additional lot of grade C tomatoes is available. The lot is 200,000 lb. How much would RBC be willing to pay for this lot of grade C tomatoes? --

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