Microsoft Word – MGMT20005_S1_2017_Solutions
Question 1
a. Use a decision tree to recommend a decision for Giant.
[6 marks]
Recommended Decision: Purchase Component (d2)
b. Compute EVPI to determine whether Giant should attempt to obtain a better estimate of
demand.
[4 marks]
EVPI = $12,000,000
c. Suppose Giant will conduct the above market study. Use a decision tree to recommend a
decision for Giant.
[14 marks]
If Favourable
State of nature
s
i
Prior
probabilities
P(s
j
)
Conditional
probabilities
P(F | s
j
)
Joint
probabilities
P(F ∩ s
j
)
Posterior
probabilities
P(s
j
| F)
s1 0.4 0.1 0.04 0.13
s2 0.4 0.3 0.12 0.4
s3 0.2 0.7 0.14 0.47
P(F) = 0.3 1.00
1
2
-30
40
90
s1
0.4
s2
0.4
s3
0.2
3
10
30
50
s1
0.4
s2
0.4
s3
0.2
d1
d2
If Unfavourable
State of nature
s
i
Prior
probabilities
P(s
j
)
Conditional
probabilities
P(U | s
j
)
Joint
probabilities
P(U ∩ s
j
)
Posterior
probabilities
P(s
j
| U)
s1 0.4 0.9 0.36 0.51
s2 0.4 0.7 0.28 0.4
s3 0.2 0.3 0.06 0.09
P(U) = 0.7 1.00
Decision strategy:
If F then d1
If U then d2
d. Determine whether Giant should conduct the market study if it costs 3.5 million dollars.
[4 marks]
Giant should conduct the market study.
Question 2:
Iteration 1 2 3
Random Number 0.51 0.97 0.16
Units sold 800 1000 600
Revenue 17600 22000 13200
Variable cost 4800 6000 3600
Fixed cost 8000 8000 8000
Net Profit 4800 8000 1600
Question 3:
a. What are the optimal solutions (including the optimal values for decision variables and the
optimal objective function value)? [8 marks]
The optimal decision is x1=2.4, x2=3.2, and the optimal profit is 3*2.4+4*3.2=20.
b. Suppose the unit profit on x1 is increased from $3 to $7. Are the values for decision
variables in (a) still optimal? What is the optimal value of the objective function when this
unit profit is increased to $7? [8 marks]
Yes. The optimal objective function value 29.6.
Question 4:
a. Fill out the above tables. Note that the unfilled cells are marked by “xxx”. [3 marks]
The first cell 318.
The second cell is “not binding”.
The third cell is 0.
b. How much storage space is left unused? How many display racks are left unused?[2 marks]
Zero units of storage space are left. There are 36 display racks left unused.
c. By how much can the total profit on women’s umbrellas increase before the optimal
solution would change? [4 marks]
12
d. Interpret the shadow price for the storage space constraint. [2 marks]
The optimal profit increases by 2 per unit increase in the storage space, given the available
storage space is within the interval [96, 168].
e. Sky-Umbrella considers stocking a new arrival umbrella, Travel Umbrella. It uses 3 units of
storage space and 2 special display racks. Sky-Umbrella makes a profit of $7 per unit of
Travel Umbrella. What is your recommendation on whether to stock this new arrival
umbrella or not? [5 marks]
Sky-Umbrella should produce.
Question 5:
Let Pij = number of toy i to produce in month j
Sij = surplus (inventory) of toy i at end of month j
Min 12P11 + 12P 12 + 12P13 + 14P21 + 14P22 + 14P23 + 1S11 + 1S12 + 1S13 + 1.2S21 + 1.2S22 + 1.2S23
s.t. P11 − S11 = 195
S11 + P12 − S12 = 350
S12 + P13 − S13 = 600
S13 = 150
P21 − S21 = 395
S21 + P22 − S22 = 700
S22 + P23 − S 23 = 520
S23 = 60
5P11 + 4P21 ≤ 3500
5P12 + 4P22 ≤ 5000
5P13 + 4P23 ≤ 4800
2P11 + 3P21 ≤ 2100
2P12 + 3P22 ≤ 3000
2P13 + 3P23 ≤ 2500
Pij, Sij ≥ 0
Question 6:
Let xij be the units of products to be shipped from warehouse i to boutique j;
yi = 1 if a warehouse is constructed in suburb i, 0 otherwise
Minimize the total cost or z
= (2×11 + 4×12 + 6×13 + 8×14 + 7×15 + 4×21 + 5×22 + 2×23 + 4×24 + 4×25
+ 3×31 + 3×32 + 3×33 + 5×34 + 5×35 + 5×41 + 6×42 + 3×43 + 2×44 + 4×45)
+ (27,000y1 + 45,000y2 + 35,000y3 + 25,000y4)
Subject to x11 + x12 + x13 + x14 + x15 – 24,000y1 ≤ 0
x21 + x22 + x23 + x24 + x25 – 29,000y2 ≤ 0
x31 + x32 + x33 + x34 + x35 – 21,000y3 ≤ 0
x41 + x42 + x43 + x44 + x45 – 26,000y4 ≤ 0
x11 + x21 + x31 + x41 = 10,000
x12 + x22 + x32 + x42 = 12,000
x13 + x23 + x33 + x43 = 15,000
x14 + x24 + x34 + x44 = 8,000
x15 + x25 + x35 + x45 = 6,000
xij ≥ 0 and integer; yi = 0 or 1 for all i and j