Question

Mary is the Market Analyst of Nestle and she is asked to present an expansion plan for the company to the senior managers, which is either to construct a plant, open a distribution center, or do nothing. She was able to determine the expected payoff depending on the decision and type of market, which may be favorable, average, or unfavorable to the business. If the decision is to construct a plant, the company will gain $8,000 if the market turns out to be favorable, lose $1,000 if the company turns to be average, and lose $5,000 if the market turns out to be unfavorable. If the decision is to open a distribution center, the company will gain $4,000 if the market turns out to be favorable, gain $2,000 if the market turns out to be average, and lose $3,000 if the market turns out to be unfavorable. If the decision is to do nothing, the company will gain nothing, the company will gain nothing whether the market turns out to be favorable, average, or unfavorable.

Mary noted that the decision of each manager is affected by his or her personality. She described each manager as follows:

Edward is an optimistic person.
Agnes is very conservative.
Joel does not want to regret making the wrong decision.

What would be the decision of each manager based on the following criterion:

Laplace criterion
Hurwicz criterion (alpha level of 0.75)
Maximax criterion
Minimax regret (savage criterion)
Maximin criterion

Answer 1

Decision alternatives	States of nature
Favorable	Average	Unfavorable
Construct a plant	$8,000	-$1,000	-$5,000
Open a distribution centre	$4,000	$2,000	-$3,000
Do nothing	$0	$0	$0

Laplace criterion:

Under Laplace criterion, it is assumed that each of the state of nature has equal probability of occurrence. So, the market being favorable, average or unfavorable each have equal probability of occurrence. If total probability is 1, then probability of each state = 1/3

Now, expected value of each decision = Weighted average of payoffs in each state of nature with probabilities

So, expected value of ‘Construct a plant’ decision = 1/3 * $8,000 + 1/3 *(-$1,000) + 1/3*(-$5,000)

= 1/3 * ($8,000 - $1,000 - $5,000)

=$666.67

Expected value of ‘Open a distribution centre’ decision = 1/3*$4,000 + 1/3*$2,000 + 1/3*(-$3,000)

= 1/3* ($4000+$2000-$3000)

= $1,000

Expected value of ‘Do nothing’ decision = 1/3*$0 +1/3*$0 + 1/3*$0

=$0

So, the best decision would be ‘Open a distribution centre’ since it has maximum payoff.

Hurwicz criterion (alpha level of 0.75):

So, (1 – alpha) = 0.25

Weighted average =(alpha*maximum in that decision alternative) +( (1-alpha)*minimum in that decision alternative)

So, for e.g. weighted average for ‘construct a plant’ = (0.75*$8000)+(0.25*-$5000)

=$4,750

Decision alternatives	States of nature	Weighted average
Favourable	Average	Unfavourable
Construct a plant	$8,000	-$1,000	-$5,000	$4,750
Open a distribution centre	$4,000	$2,000	-$3,000	$2,250
Do nothing	$0	$0	$0	$0

The highest weighted average is $4,750, therefore, as per Hurwicz criterion, decision alternative ‘construct a plant’ should be chosen.

Maximax criterion:

Under maximum criterion, the maximum payoff associated with each decision alternative is considered. Out of these, the decision alternative with maximum payoff is chosen. So, the matrix will be as follows:

Decision alternatives	States of nature	Maximum
Favourable	Average	Unfavourable
Construct a plant	$8,000	-$1,000	-$5,000	$8,000
Open a distribution centre	$4,000	$2,000	-$3,000	$4,000
Do nothing	$0	$0	$0	$0

Since the maximum of maximum payoffs is $8,000, so ‘Construct a plant’ decision is chosen.

Minimax regret (savage criterion):

Under minimax regret approach, the decision alternative with minimum of maximum regrets is chosen. So, a regret table is constructed which shows regret associated with each decision across states of nature.

Regret = Best payoff in that state of nature – Payoff received

So, for favorable state of nature, best payoff is $8000. So, for ‘construct a plant’ regret is $8000-$8000 = $0, for ‘open a distribution centre’ regret = $8000-$4000 = $4000 and regret for ‘do nothing’ is $8000-$0 = $8000.

Decision alternatives	States of nature
Favourable	Average	Unfavourable
Construct a plant	$8,000	-$1,000	-$5,000
Open a distribution centre	$4,000	$2,000	-$3,000
Do nothing	$0	$0	$0

Regret table
Decision alternatives	States of nature	Maximum Regret
Favourable	Average	Unfavourable
Construct a plant	$0	$3,000	$5,000	$5,000
Open a distribution centre	$4,000	$0	$3,000	$4,000
Do nothing	$8,000	$2,000	$0	$8,000

The minimum of maximum regrets is $4,000.

So, ‘open a distribution centre’ alternative will be chosen.

Maximin criterion:

Under this criterion, the best of the minimum payoffs associated with each decision alternative is chosen.

Decision alternatives	States of nature	Minimum
Favourable	Average	Unfavourable
Construct a plant	$8,000	-$1,000	-$5,000	-$5,000
Open a distribution centre	$4,000	$2,000	-$3,000	-$3,000
Do nothing	$0	$0	$0	$0

The minimum payoffs for each decision alternative are considered in ‘Minimum’ column. Out of these, the best payoff is $0. So, the decision ‘Do nothing’ will be considered.