Module 1. Decision Making without Probabilities
A payoff table is a means of organizing and illustrating the payoffs from the different decisions,
given the various states of nature.
The Maximax criterion selects the alternative (decision) that maximizes the maximum payoff
over all alternatives. (an optimistic or aggressive criterion) (Maximum in Row, Max)
The Maximin criterion selects the alternative (decision) that maximizes the minimum payoff
over all alternatives. (a pessimistic or conservative criterion) (Minimum in Row, Max)
The equally likely (Laplace) criterion finds the decision alternative with the highest average
payoff. (Row Average, Max)
The criterion of realism (Hurwicz) criterion is a compromise between an optimistic and a
pessimistic decision. (weighted average) (α®1,optimistic) (α®0, pessimistic) (shortcoming)
(Hurwics Value, Max)
Opportunity loss (regret) is the difference between the optimal payoff and the actual payoff
received. Or, it is the amount lost by not picking the best alternative. (Maximum in Row, Min)
Since Chantal encountered severe traffic congestion 10 days out of 60 days, we can estimate a
10/60 = 0.167 probability that severe traffic congestion will prevail.
Module 2. Decision Making with Probability
The Expected Monetary Value (EMV) for an alternative is computed as the weighted average
of all possible payoffs for the alternative, where the weights are the probabilities of the different
states of nature. (long term)
States of Nature
Decision
EMV
Alternatives
Computed
Strong Demand Weak Demand
Residential Towers
29
-14
29 x 0.66 + (-14) x 0.34 = 14.38
Mid-Rise Condos
14
12
14 x 0.66 + 12 x 0.34 = 13.32
Bungalow
19
6
19 x 0.66 + 6 x 0.34 = 14.58 (MAX)
Probability
0.66
0.34
The Expected Opportunity Loss (EOL) for an alternative is the sum of all possible regrets of
the alternative, each weighted by the probability of the state of nature for that regret occurring.
States of Nature
Decision
Alternatives
Strong Demand Weak Demand
Residential Towers
0
28
Mid-Rise Condos
15
0
Bungalow
10
6
Maximum in Column
29
12
Decision
Alternatives
Apartment building
Office building
Warehouse
States of Nature
Strong Demand
Weak Demad
0
28
15
0
10
6
EOL
Computed
0 x 0.66 + 28 x 0.34 = 9.52
15 x 0.66 + 0 x 0.34 = 9.9
10 x 0.66 + 6 x 0.34 = 8.64 (MIN)
Probability
0.66
0.34
EOL and EMV decisions are the same (Maximum in Column)
EVwPI = 0.66 × 29 + 0.34 × 12 = 23.22
EVPI = 23.22 – 14.58 = 8.64 = MinEOL
It means that the investor can pay up to 8.64 for the information.
= Decision Node.
= Event Node
A decision tree presents the decision alternatives and states of nature in a sequential manner.
Probability Revisions Given a Favourable Report
State of
Prior
Conditional Probability
Joint
Posterior Probability
Nature
Probability
𝑃(Fav.|State of Nature)
Probability
𝑃(State of Nature|Fav.)
Strong
0.66
x 0.9
= 0.594
0.9202
Weak
0.34
x 0.15
= 0.051
0.0791
𝑃(Favourable) = 0.645
P=1
Probability Revisions Given an Unfavourable Report
State of
Prior
Conditional Probability
Joint
Posterior Probability
Nature
Probability
𝑃(Unfav.|State of Nature)
Probability
𝑃(State of Nature|Unfav.)
Strong
0.66
x 0.1
= 0.066
0.1859
Weak
0.34
x 0.85
= 0.289
0.8141
𝑃(Unfavourable) = 0.355
In case of favourable report,
EMV(Residential Towers) = 25.6
EMV( Mid-Rise Condos) = 13.84
EMV(Bungalow Complex) = 17.97
P=1
In case of unfavourable report,
EMV(Residential Towers) = -6.01
EMV(Mid-Rise Condos) = 12.37
EMV(Bungalow Complex) = 8.42
EMV(hire) = 20.9 (chon cai nay)
EMV(do not hire) = 14.58
Hire. If report is favourable, then choose residential tower. If report is unfavourable, then choose
Mid-rise condos
Can pay up to EVSI = 6.32
The efficiency of the sample information is the ratio of the expected value of sample
information to the expected value of the perfect information.
Efficiency = EVSI/EVPI = 6.32/8.62 = 0.7332 = 73.1%
The analyst’s report is viewed by the investor to be 73.1% as efficient as perfect information.
If $50,000 is hiring fee,
EVwSI = 20,900,000 – 50,000 = 20,850,000
EVSI = EVwSI – EvwoSI = 20.85 – 14.58 =
LINEAR PROGRAMMING
§ Decision variables are mathematical symbols representing levels of activity.
§ The objective function is a linear relationship reflecting the objective of an operation.
§ A constraint is a linear relationship representing a restriction on decision making.
Example:
There are two decision variables representing the number of grams of ingredient 1 (that can be
denoted by 𝑥) and the number of grams of ingredient 2 (denoted by 𝑦) that must go into the drug.
The company’s objective is to minimize cost. If ingredient 1 costs $80 per gram, and ingredient 2
costs $50 per gram, then total cost can be expressed as Cost = $80𝑥 + $50𝑦.
The objective function is expressed as :
Minimize C = $80𝑥 + $50𝑦
A gram of ingredient 1 contributes 3 units of antibiotic Alpha, and a gram of ingredient 2
contributes 1 unit of antibiotic Alpha. Including 𝑥 grams of ingredient 1 and 𝑦 grams of
ingredient 2 contributes 3𝑥 + 1𝑦 units of antibiotic Alpha. As the minimum requirement of
antibiotic Alpha is 6 units, each drug should contain 6 units of antibiotic Alpha or more.
We get:
3𝑥 + 𝑦 ≥ 6 (antibiotic Alpha constraint)
The constraint for antibiotic Beta is formulated the same way. Including 𝑥 grams of ingredient 1
and 𝑦 grams of ingredient 2 contributes 1𝑥 + 1𝑦 units of antibiotic Beta. As At least 4 units of
antibiotic Beta are required, each drug should contain 4 units of antibiotic 2 or more. We get
𝑥 + 𝑦 ≥ 4 (antibiotic Beta constraint)
Finally, including 𝑥 grams of ingredient 1 and 𝑦 grams of ingredient 2 contributes 2𝑥 + 6𝑦 units
of antibiotic Gamma. At least 12 units of antibiotic Gamma are required, each drug should
contain 12 units of antibiotic Gamma or more. Then, we get 2𝑥 + 6𝑦 ≥ 12. Simplifying the
inequality by dividing both sides by 2, we get
𝑥 + 3𝑦 ≥ 6 (antibiotic Gamma constraint)
The final restrictions are referred to as nonnegativity constraints:
𝑥 ≥ 0, 𝑦 ≥ 0 (nonnegativity constraints)
Therefore, the complete LP model for the Panacea Drug Company’s case can be mathematically
stated as follows:
Minimize Cost = 80𝑥 + 50𝑦
subject to
3𝑥 + 𝑦 ≥ 6 (antibiotic Alpha constraint)
𝑥 + 𝑦 ≥ 4 (antibiotic Beta constraint)
𝑥 + 3𝑦 ≥ 6 (antibiotic Gamma constraint)
𝑥 ≥ 0, 𝑦 ≥ 0 (nonnegativity constraints)
where 𝑥 – number of grams of ingredient 1 and 𝑦 – number of grams of ingredient 2 that must go
into the drug.
The nonnegativity constraints imply that we are working only in the first (or positive) quadrant
of a graph. To represent a constraint graphically, we replace the inequality (≥) with an equality
sign (=) and obtain a linear equation.
3x + y = 6
x 0 2
If x = 0 => y = 6 (𝑦-intersect)
If y = 0 => x = 2 (𝑥-intersect)
y 6 0
x+y=4
x 0 4
If x = 0 => y = 4 (𝑦-intersect)
If y = 0 => x = 4 (𝑥-intersect)
y 4 0
x + 3y = 6
If x = 0 => y = 2 (𝑦-intersect)
x 0 6
If y = 0 => x = 6 (𝑥-intersect)
y 2 0
Find the point of intersection of the lines 3𝑥 + 𝑦 = 6 and x + 𝑦 = 4.
ð x = 1, y =3 => We get the point of intersection (1 , 3)
Find the point of intersection of the lines 𝑥 + 𝑦 = 4 and x + 3𝑦 = 6.
ð x = 3, y =1 => We get the point of intersection (3 , 1)
For this problem, the cost analysis at each of the four corner points is as follows:
Point A (𝑥 = 0 , 𝑦 = 6) Cost = 80(0) + 50(6) = 300 or Cost = $300
Point B (𝑥 = 1 , 𝑦 = 3) Cost = 80(1) + 50(3) = 270 or Cost = $270 (MIN)
Point C (𝑥 = 3 , 𝑦 = 1) Cost = 80(3) + 50(1) = 290 or Cost = $290
Point D (𝑥 = 6 , 𝑦 = 0) Cost = 80(6) + 50(0) = 480 or Cost = $480
The point B is the optimal.
Answer: To meet minimum antibiotics requirements and get the minimum cost of $230 the
Panacea Drug Company should put 1 gram of ingredient 1 and 3 grams of ingredient 2 in the
drug.
There is no feasible region where all the problem’s constraints
overlap. The constraints are conflicting, and this is the case of
infeasibility.
Answer: Infeasibility
Note that the shaded area has a missing (upper)
boundary. For example, when point 𝐴 (18 , 𝑡) moves up
(𝑡 → ∞), then 𝑃 = 300(18) + 500𝑡 = 5400 + 500𝑡
becomes infinitely large. Thus, this is a case of
unboundedness.
Answer: Unboundedness
3𝑥 + 5𝑦 = 0.01(300𝑥 + 500𝑦) = 0.01𝑃
It means that the left part of the constraint 3𝑥 + 5𝑦 ≤ 30 is
the one hundredths of the objective function P . The isoprofit
line and the boundary line are parallel as their slopes are
equal. Thus, all points on 3𝑥 + 5𝑦 = 30 between two corner
points corresponds to the same profit value; all such points
areoptimal. This is an example of alternate optimal solutions.
Answer: Alternate Optimal Solutions
Note that all points of the yellow shaded area satisfy
the constraints 2𝑥 + 𝑦 ≤ 8 and 𝑦 ≤ 4, and the
constraint 3𝑥 + 5 𝑦 ≤ 30 does not affect the feasible
region. Thus, we can remove the constraint 3𝑥 + 5𝑦 ≤
30 from LP model.
Answer: Redundancy