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）︰2010.04.16
考試時限（分鐘）：60 min.
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(15 points) Market research corporation of Toledo Inc. (MRCT) has to finalize
recommendation for the 2006 advertising campaign for Miracle Motors Inc. (MM),
a major hydrogen-powered automobile manufacturer. MRCT has developed 5 distinct
campaigns for MM Inc. MM Inc. has 7 distinct customer segments, and it wants
all of its customers to be "hit" by at least one of its campaigns. It wants to
minimize the total cost of reaching its customer base through these campaigns.
┌────┬────┬────┬────┬────┬────┬────┐
│Campaign│Customer│Customer│Customer│Customer│Customer│Customer│
│ # │ #1 │ #2 │ #3 │ #4 │ #5 │ #6 │
├────┼────┼────┼────┼────┼────┼────┤
│ 1 │ 1 │ 0 │ 0 │ 1 │ 0 │ 0 │
├────┼────┼────┼────┼────┼────┼────┤
│ 2 │ 1 │ 1 │ 0 │ 0 │ 0 │ 0 │
├────┼────┼────┼────┼────┼────┼────┤
│ 3 │ 0 │ 0 │ 1 │ 0 │ 1 │ 1 │
├────┼────┼────┼────┼────┼────┼────┤
│ 4 │ 0 │ 0 │ 1 │ 0 │ 0 │ 1 │
├────┼────┼────┼────┼────┼────┼────┤
│ 5 │ 1 │ 0 │ 0 │ 0 │ 1 │ 0 │
└────┴────┴────┴────┴────┴────┴────┘
┌────┬──────┐
│Customer│Cost of the │
│ #7 │ campaign │
├────┼──────┤
│ 0 │ 12000 │
├────┼──────┤
│ 0 │ 15000 │
├────┼──────┤
│ 1 │ 30000 │
├────┼──────┤
│ 1 │ 20000 │
├────┼──────┤
│ 0 │ 10000 │
└────┴──────┘
2.(15 points) 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.)
3.(70 points) The party Nut Company has on hand 550 pounds of peanuts, 150
pounds of cashews, 90 pounds of brazil nuts, and 70 pounds of hazelnuts. It
packages and sells four varieties of mixed nuts in standards 8-ounce
(half-pound) cans. The mix requirements and net wholesale prices are shown in
Figure 1. The solution of Excel is shown in Figure 2. The firm can sell all
that it can produce at these prices. What mixes of products should it produce?
┌───────┬─────────────┬───────┐
│ Mix │ Contents │Price per can │
├───────┼─────────────┼───────┤
│1 (peanuts) │Peanuts only │ $0.26 │
├───────┼─────────────┼───────┤
│2 (party mix) │No more than 50% peanuts; │ 0.40 │
│ │At least 15% cashews; │ │
│ │At least 10% brazil nuts │ │
├───────┼─────────────┼───────┤
│3 (cashews) │Cashews only │ 0.51 │
├───────┼─────────────┼───────┤
│4 (luxury mix)│At least 30% cashews; │ 0.52 │
│ │At least 20% brazil nuts; │ │
│ │At least 30% hazelnuts │ │
└───────┴─────────────┴───────┘
Figure 1
￣￣￣￣
The LP formulation and solution are shown as follows:
P = pounds of peanuts used in mix i
i
C = pounds of cashews used in mix i
i
B = pounds of brazil nuts used in mix i
i
H = pounds of hazelnuts used in mix i
i
Max 52 P +80P +80C +80B +80H +102C +104P +104C +104B +104H
1 2 2 2 2 3 4 4 4 4
subject to
P +P +P ≦550
1 2 4
C +C +C ≦150
2 3 4
B +B ≦90
2 4
H +H ≦70
2 4
0.5P -0.5C -0.5B -0.5H ≦0
2 2 2 2
-0.15P +0.85C -0.15B -0.15H ≧0
2 2 2 2
-0.1P -0.1C +0.9B -0.1H ≧0
2 2 2 2
-0.3P +0.7C -0.3B -0.3H ≧0
4 4 4 4
-0.2P -0.2C +0.8B -0.2H ≧0
4 4 4 4
-0.3P -0.3C -0.3B +0.7H ≧0
4 4 4 4
P ,P ,P ,C ,C ,C ,B ,B ,H ,H ≧0
1 2 4 2 3 4 2 4 2 4
┌─────────────────────────────────────┐
│ The Party Nut problem solution │
├───────┬────┬──────┬──────┬─────┬────┤
│Variable Name │Final │Reduced Cost│Objective │Allowable │Allowable
│ │Value │ │Coefficient │Decrease │Increase│
├───────┼────┼──────┼──────┼─────┼────┤
│P1 │ 380│ 0│ 52│ 12│ 6│
├───────┼────┼──────┼──────┼─────┼────┤
│P2 │123.3333│ │ 80│ 6│ 9│
├───────┼────┼──────┼──────┼─────┼────┤
│C2 │ 80│ 0│ 80│ 6│ 24│
├───────┼────┼──────┼──────┼─────┼────┤
│B2 │ 43.3333│ 0│ 80│ 56│ 36│
├───────┼────┼──────┼──────┼─────┼────┤
│H2 │ 0│ -24│ 80│ 1E+30│ 24│
├───────┼────┼──────┼──────┼─────┼────┤
│C3 │ 0│ │ 102│ 1E+30│ 6│
├───────┼────┼──────┼──────┼─────┼────┤
│P4 │ 46.6667│ 0│ 104│ 36│ 1E+30│
├───────┼────┼──────┼──────┼─────┼────┤
│C4 │ 70│ 0│ 104│ 24│ 56│
├───────┼────┼──────┼──────┼─────┼────┤
│B4 │ 46.6667│ 0│ 104│ 36│ 56│
├───────┼────┼──────┼──────┼─────┼────┤
│H4 │ 70│ 0│ 104│ 24│ 1E+30│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint │Shadow │Final Value │Constraint │Allowable │Allowable
│ │Price │ │R. H. Side │Decrease │Increase│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 1 │ │ 550│ 550│ 380│ 1E+30│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 2 │ │ 150│ 150│ 61.4286│ 93.3333│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 3 │ 108│ 00│ 90│ 23.3333│143.3333│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 4 │ 132│ 70│ 70│ 70│ 56│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 5 │ 56│ 0│ 0│ 61.6667│ 93.3333│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 6 │ 0│ 43│ 0│ 1E+30│ 43│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 7 │ │ 18.6667│ 0│ 1E+30│ 18.6667│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 8 │ -56│ 0│ 0│ 70│ 46.6667│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 9 │ -56│ 0│ 0│ 46.6667│ 23.3333│
├───────┼────┼──────┼──────┼─────┼────┤
│Constraint 10 │ -80│ 0│ 0│ 56│ 28│
└───────┴────┴──────┴──────┴─────┴────┘
Figure 2
a. Explain the calculation that justifies 102 as the coefficient of C in the
3
objective function value.
b. How many cans of mix 2 are produced in the optimal solution?
c. Is the shadow price on the second constraint ≧0 or ≦0 ? Explain why.
d. What is the value of the shadow price on the seventh constraint? How do you
know?
e. What is the meaning of the fact that the surplus variable on constraint 6 is
positive? Explain.
f. What is the effect on the optimal solution and the objective function value
if the sales price of mix 1 (peanuts only) increases to $0.27 per can?
g. What is the numerical value of the shadow price on constraint 1? (Hint: What
would you do with another pound of peanuts if you had them?)
h. Why is the allowable increase for the right hand side value of constraint 1
1E+30?
i. Provide an explanation as to why it is possible to have an optimal solution
even though there is a positive surplus on constraint 6.
j. What are the numerical values for the reduced cost for Peanuts in the party
Mix and Cashews in the cashew Mix?
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