QUESTIONS FOR PRACTICE SMART Question 1 A chemical company is expanding its operations and a disused woollen mill is to be converted into a processing plant. Four companies have submitted designs for the equipment which will be installed in the mill and a choice has to be made between them. The manager of the chemical company has identified three attributes which he considers to be imp ortant in the decision: ‘cost’, ‘environmental impact’ and ‘reliability’. He has assessed how well each design performs on each attribute by allocating values on a scale from 0 (the worst design) to 100 (the best). These values are shown below, together with the costs which will be incurred if a design is chosen.
DESIGN A B C D
COST ($) 90,000 110,000 170,000 60,000
BENEFITS Environmental Environmental Impact 20 70 100 0
Reliability 100 0 90 50
(a) Eventually, the manager decides to alloca te ‘environmental impact’ a weight of 30 and ‘reliability’ a weight of 70. By plotting the benefits and costs of the designs on a graph, identify the designs which lie on the efficient frontier.
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(b) The manager also decides that if he was offered a hypothetical design which had the lowest reliability and the worst environmental impact he would be prepared to pay $120 000 to convert that design to one which had the best impact on the environment but which still had the lowest level of reliability. Which design should the manager choose?
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Solution (a) First calculate the Aggregate benefits
Attribute Environment impact
Given Weight 30
Reliability
70
= 30
Design A Design B NW x Score NW x Score Score 20 600 70 2100
= 70
100
Normalised Weights
Score
7000
0
7600
100
Aggregate Score
0
Design C NW x Score 100 3000
Score
90
6300
2100
9300
= 76
= 21
=93
Then draw efficient frontier
100
C
90
A
80 70 e r o c S e t a g e r g g A
60 50 40 30 20
D B
Score
Design D NW x Score 0 50
0 3500
3500
=35
Then draw efficient frontier
100
C
90
A
80 70 e r o c S e t a g e r g g A
60 50
D
40 30
B
20 10 0 0
20000
40000
60000
80000
100000 120000 140000 160000 180000
Costs ($)
Designs: A, C and D lie on the efficient frontier (b) The manager also chose to use Environmental Impact to break the tie Company prepared to Pay:
= N$4000 per benefit point ( NB: 30 is the normalized weight for Environmental impact)
D to A = A to C =
Therefore we chose design A
= N$732 per benefit point (It’s within the limit of $4000) = N$4706 per benefit point (It’s not within the limit of $4000)
Question 2 A local authority has to decide on the location of a new waste disposal facility and five sites are currently being considered; Inston Common, Jones Wood, Peterton, Red Beach and Treehome Valley. In order to help them to choose between the sites the managers involved in the decision arranged for a decision analyst to attend one of their meetings. He first got the managers to consider the factors which they thought were relevant to the decision and, after some debate, four factors were identified: i.
The visual impact of the site on the local scenery (for example, a site at Treehome Valley would be visible from a nearby beauty spot).
ii.
The ease with which waste could be transported to the site (for example, Red Beach is only two miles from the main town in the area and is close to a main highway while Inston Common is in a remote spot and its use would lead to a major increase in the volume of transport using the minor roads in the area).
iii.
The risk that the use of the site would lead to contamination of the local environment (e.g. because of leakages of chemicals into watercourses).
iv.
The cost of developing the site.
The decision analyst then asked the managers to assign scores to the sites to show how well they performed on each of the first three attributes. The scores they eventually agreed are shown below, together with the estimated cost of developing each site. Note that 0 represents the worst and 100 the best score on an attribute. In the case of risk, therefore, a score of 100 means that a site is the least risky.
Site
Benefits Visual Impact Ease of Transport
Risk
Cost ($ Millions)
Inston Common
100
0
60
35
Jones wood
20
70
100
25
Peterton
80
40
0
17
Red Beach
20
100
30
12
Treehome Valley
0
70
60
20
The decision analyst then asked the managers to imagine a site which had the worst visual impact, the most difficult transport requirements and the highest level of risk. He then asked them if they had a chance of switching from this site to one which had just one of the benefits at its best value, which would they choose? The managers agreed that they would move to a site offering the least risk of contamination. A move to a site with the best visual impact was considered to be 80% as preferable as this, while a move to one with the most convenient transport facilities was 70% as preferable.
(a.)
(b.)
(c.) (d.)
Can we conclude from the values which were assigned to the different sites for visual impact that, in terms of visual impact, the Inston Common site is five times preferable to Red Beach? If not what can we infer from these figures? An alternative way of allocating weights to the three benefit attributes would have involved asking the managers to allocate a score reflecting the importance of each attribute. For example, they might have judged that risk was five times more important and visual impact three times more important than ease of transport, so that weights of 5, 3, and 1 would have been attached to the attributes. What are the dangers of this approach? Assuming that mutual preference independence (no preference interactions) exists between the attributes; determine the value of aggregate benefits for each site. Although a weight of 80 was finally agreed for visual impact, this was only after much debate and some managers still felt that a weight of 65 should have been used while others thought that 95 would have been more appropriate. Perform sensitivity analysis on the weight assigned to visual impact to examine its effect on the aggregate benefits of the Inston Common and Jones wood and interpret your results.
SOLUTION Question 1
(a) It is the interval (or improvement) between the points in the scale which we compare. This is because the allocation of a zero to represent the worst alternative was arbitrary, and we therefore have what is known as an interval scale, which allows only intervals between points to be compared. Thus we can infer that the improvement in visual impact between Treehome Valley and Inston Common is perceived by the managers to be five times as preferred as the improvement in visual impact between Treehome Valley and Red Beach. The inference given in the question is not correct. (b) The problem with importance weights is that they may not take into account the range between the least- and most-preferred options on each attribute. If the options perform very similarly on a particular attribute, so that the range between worst and best is small, then this attribute is unlikely to be important in the decision, even though the decision maker may consider it to be an important attribute per se. In this case, the weight attached to the unimportant attribute should be zero because this attribute has no importance in discriminating between the different alternatives. (c) Weights
Original
Visual Impact
Ease of Transport
Risk
80
70
100
Normalised
Attribute
0.32
Weights
0.28
Inston Common
0.40
Jones Wood Peterton
Red Beach
(w)
Treehome Valley
Visual Impact 0.32
100
32
20
6.4
80
25.6
20
6.4
0
0
Ease
0
0
70
19.6
40
11.2
100
28
70
19.6
60
24
100
40
0
0
30
12
60
24
of 0.28
Transport Risk
0.40
Aggregate Benefits
56
66
36.8
46.4
(d) With 95 for Visual Impact
With 65 for Visual Impact
Visual
Ease
Visual
Ease
Impact
Transport
Impact
Transport
Original
95
70
100
65
70
100
Normalised
0.36
0.26
0.38
0.28
0.30
0.42
of Risk
of Risk
Visual Impact 95
Attribute
Weights Inston Common Jones Wood
Visual Impact
0.36
100
36
20
7.2
Ease of Transport 0.26
0
0
70
18.2
Risk
60
22.8
100
38
Aggregate Benefits
Visual Impact 65
0.38
58.8
63.4
43.6
Attribute
Weights (w) Inston Common Jones Wood
Visual Impact
0.28
100
28
20
5.6
Ease of Transport
0.30
0
0
70
21
Risk
0.42
60
25.2
100
42
Aggregate benefits
53.2
68.6
The sensitivity analysis shows that Jones Wood has the highest aggregate benefits whatever the weight is assigned to visual impact.
DECISION TREES Question 1 Even though independent gasoline stations have been having a difficult time, Susan Solomon has been thinking about starting her own independent gasoline station. Susan’s problem is to decide how large
her station should be. The annual returns will depend on both the size of her station and a number of marketing factors related to the oil industry and demand for gasoline. After a careful analysis, Susan developed the following decision table:
Good Market
Fair Market
Poor Market
(N$)
(N$)
(N$)
Size of the first Station Small
50,000
20,000
-10,000
Medium
80,000
30,000
-20,000
Large
100,000
30,000
-40,000
Very Large
300,000
25,000
-160,000
For example, if Susan constructs a small station and the market is good, she will realise a profit of N$ 50,000. (a.) (b.) (c.) (d.) (e.) (f.) (g.)
Develop a decision table for this decision What is the Maximax decision? What is the Maximin decision? What is the equally likely decision? What is the criterion of realism decision? Use an α value of 0.8. Develop an opportunity loss table What is the Minimax regret decision
Solution (a) Decision table
Alternatives
Maximum
Minimum
Average
Realism
Maximum
Choice
Choice
Choice
Choice
Choice( Based on opportunity loss)
Small
$50,000
-$10,000 Best $20,000
$38,000
$250,000
Medium
$80,000
-$20,000
$30,000
$60,000
$220,000
Large
$100,000
-$40,000
$30,000
$72,000
$200,000
Very large $300,000 Best -$160,000
$55,000 Best $208,000 Best
$150,000
Weighted Average = α(maximum in row) + (1 – α)(minimum in row)
(b) Maximax Decision: Very large station (c) Maximin Decision: small station (d) Equally likely Decision: Very large station (e) Realism Decision: Very Large station (f) Opportunity Loss Table Market Good
Fair
Poor
Row maximum
Station size Small
250,000
10,000
0
250,000
Medium
220,000
0
10,000
220,000
Large
200,000
0
30,000
200,000
Very Large
0
5,000
150,000
150,000
(e) Minimax Regret Decision: Very Large Station
Best
