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ECO00030M DEPARTMENT OF ECONOMICS AND RELATED STUDIES
创建日期：2024-04-09 14:33:14
所属分类：经济Economics
标签： ECO00030M
DEPARTMENT OF ECONOMICS AND RELATED STUDIES

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ECO00030M


Candidates should attempt to answer ALL questions of Section A and ONE
question from Section B.

The use of hand-held, battery-operated, electronic calculators will be permitted subject to
the regulations governing their use which are specified by the University.

Turn Over
ECO00030M Management Decision Analysis

Section A (85 marks)

Candidates should attempt to answer BOTH questions in this section.

1. [45 Marks] Consider the following LP problem:

Maximise 321 510 xxx 
Subject to 2 1x + 2x + 83 x
1x + 2 2x - 3x 12
5 1x + 2 103 x
321 ,, xxx 0
a) Use the simplex method to solve the maximization problem. [25 marks]
b) Is the optimal solution you obtained above unique? If unique, why? If not
unique, then what is (are) the alternative solution(s)? [5 marks]
c) Which constraint(s) is (are) non-binding? [4 marks]
d) Determine the range of optimality for 2C , i.e., the coefficient of 2x . [5 marks]
e) Find the dual price for the first, second and third constraints from the final
simplex tableau you get, respectively. Interpret these dual prices? [6 marks]

a) In tableau form: (note that M is very large positive number)
Maximise 1321321 000510 Masssxxx 
Subject to 2 1x + 2x + 3x - 1s + 1a = 8
1x + 2 2x - 3x + 2s = 12
5 1x + 2 3x + 3s = 10
0,,,,,, 1321321 asssxxx
First tableau:
X1 X2 X3 S1 S2 S3 A1
Basis CB 10 5 1 0 0 0 -M
A1 -M 2 1 1 -1 0 0 1 8
S2 0 1 2 -1 0 1 0 0 12
S3 0 5 0 2 0 0 1 0 10
Zj -2M -M -M M 0 0 -M -8M
Cj-Zj 10+2M 5+M 1+M -M 0 0 0
10+2M is the most positive number in Cj-Zj row, X1 is the entering variable. Compare the ratios in
this column, we find S3 is the leaving variable. 5 is the pivot element.

Second tableau:
X1 X2 X3 S1 S2 S3 A1
Basis CB 10 5 1 0 0 0 -M
A1 -M 0 1 1/5 -1 0 -2/5 1 4
S2 0 0 2 -7/5 0 1 -1/5 0 10
X1 10 1 0 2/5 0 0 1/5 0 2
Zj 10 -M -M+4 M 0 2M/5+2 -M 20-
4M
Cj-Zj 0 5+M M-3 -M 0 -2M/5-2 -M-5
Here, X2 is the entering variable, A1 is the leaving variable. 1 is the pivot element.
Third tableau:
X1 X2 X3 S1 S2 S3 A1
Basis CB 10 5 1 0 0 0 -M
X2 5 0 1 1/5 -1 0 -2/5 1 4
S2 0 0 0 -9/5 2 1 3/5 -2 2
X1 10 1 0 2/5 0 0 1/5 0 2
Zj 10 5 5 -5 0 0 5 40
Cj-Zj 0 0 -4 5 0 0 -M-5
Here, S1 is the entering variable, S2 is the leaving variable. 2 is the pivot element.

Fourth tableau:
X1 X2 X3 S1 S2 S3 A1
Basis CB 10 5 1 0 0 0 -M
X2 5 0 1 -7/10 0 1/2 -1/10 0 5
S1 0 0 0 -9/10 1 1/2 3/10 -1 1
X1 10 1 0 2/5 0 0 1/5 0 2
Zj 10 5 1/2 0 5/2 3/2 0 45
Cj-Zj 0 0 1/2 0 -5/2 -3/2 -M
Here, X3 is the entering variable, X1 is the leaving variable. 2/5 is the pivot element.

Final tableau:
X1 X2 X3 S1 S2 S3 A1
Basis CB 10 5 1 0 0 0 -M
X2 5 7/4 1 0 0 1/2 1/4 0 8.5
S1 0 9/4 0 0 1 1/2 3/4 -1 5.5
X3 1 5/2 0 1 0 0 1/2 0 5
Zj 45/4 5 1 0 5/2 7/4 0 47.5
Cj-Zj -5/4 0 0 0 -5/2 -7/4 -M

Optimal solution: X1=0, X2=8.5, X3=5, S1=5.5, S2=S3=A1=0, and the value of objective function is
47.5.


b) Here, a non-basic variable X1 has non-zero value in the Cj-Zj row, so there is no alternative solution.

c) The first constraint is nonbinding (S2=S3=0) whereas the second and third constraints are binding.

d) Range of optimality of C2:

X1 X2 X3 S1 S2 S3 A1
Basis CB 10 C2 1 0 0 0 -M
X2 C2 7/4 1 0 0 1/2 1/4 0 8.5
S1 0 9/4 0 0 1 1/2 3/4 -1 5.5
X3 1 5/2 0 1 0 0 1/2 0 5
Zj (7C2+10)/4 C2 1 0 C2/2 (2+C2)/4 0 8.5C2+5
Cj-Zj (30-7C2)/4 0 0 0 -C2/2 -
(2+C2)/4
-M

The optimal solution remains optimal for C2>=30/7.

e) For “less than or equal to” constraint, the dual price is found as the Zj value of the corresponding slack
variable for the constraint in the final tableau. For “greater than or equal to” constraint, the dual price is
the negative of the Zj value of the corresponding surplus variable for the constraint in the final tableau.
Hence, it is 0 for the first constraint, but for the second and third constraints, the dual prices are 5/2 and
7/4, respectively.

A dual price for a constraint is the increase in the objective function value resulting from a one unit
increase in its right-hand side value. Here, if the RHS value of the 2nd and 3rd constraints are increased by
one unit, respectively, the value of the objective function is increased by 5/2 and 7/4, respectively. But for
the 1st constraint, since it is non-binding, a unit increase in the RHS would not help at all, and hence the
dual price is 0.
2. [20 marks] Solve the following two decision making problems.

(1) A decision maker is faced with four decision alternatives and four states of nature,
and has the following profit payoff table.

State of nature
Decision alternative S1 S2 S3 S4
D1 14 19 16 5
D2 11 10 25 6
D3 10 9 12 16
D4 20 15 11 12

Suppose the decision maker knows nothing about the probabilities of the four states of
nature, what is the recommended decision using the minimax regret approach? [5 marks]

(2) A committee in charge of promoting a College Basketball League tournament is
trying to determine how best to advertise the event during the three months prior to the
tournament. The committee obtained the following information about the four advertising
media that they are considering using.

Category Audience reached
per advertisement
Cost per
advertisement
Maximum number
of advertisement
TV 250,000 £6,000 10
Radio 50,000 £400 8
Newspaper 200,000 £5000 5
Internet 300,000 £2,000 15

The committee established the following goals for the campaign.

Priority level 1 goal
Goal 1: Reach at least 2 million people.

Priority level 2 goal
Goal 2: The number of TV advertisements should be at least 20% of the total number of
advertisements.

Priority level 3 goal
Goal 3: The number of radio advertisements should not exceed 20% of the total number
of advertisements.

Priority level 4 goal
Goal 4: The total audience reached by Internet advertisements should be at least 1 million.

Priority level 5 goal
Goal 5: Limit the total expenditure for advertising to £35,000.

Formulate a goal programming model for this problem (but do not solve it). [15 marks]
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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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