1. A well-known stock expert conducts a popular training session in will he will charge participants $150 to attend. The cost of the meeting room, speaker fee, equipment, and advertise amounted to $3000. Assume it costs $25 per person for the organizer to provide the course materials (15 marks).

a. Write the expression for total cost for the training session. (3 marks)

Profit $150 per participant Cost= $3000 Cost $25 per student cost material

Let (x)= number of participants needed to attend

C(x) = 3000 + 25(x)

b. Write the expression for total revenue for the training session. (3 marks)

TR(x) = 150(x)

c. How many persons would have to attend for the company to break even? (4 marks)

C(x) = TR(x)3000 + 25x = 150x 3000 =150x 25x 3000 = 125x 3000/125 = x 24 = x

= 24 students need to attend in order to break-even.

d. If the organizers think, realistically, that 20 people will attend: (i) write the expression for total revenue. (2 marks)

Let x= number of participants attendingTR(x)= Profit (x)TR(20) = 150 (20) = 3,000

(ii) what price should be charged per person for the organization to break even? (3 marks)

Let p = price to change per participant

C(x) = TR(x)

3000 + 25(20) = p(20)3500 = 20p3500/20 = p175 = p

= Price to be charged per participant should be $175 in order to break-even.

2. A grocer solves a problem of how much pastry to order every day. His profit depends on a demand that can be low, moderate, or high. Values of the profit per day (in $) for these situations and for small, medium or large order are shown in Table 2.2 (15 marks). Table 2.2 Payoff Table for Order PlanningDecision(order)State of nature

lowmoderateHigh

Small 120150150

Medium 90250350

Large 60350600

Determine the size of the order that the manager should order according to:(a) Maximax criterion(5 marks)

Decision(order)State of natureRowMaximum

lowmoderateHigh

Small 120150150150

Medium 90250350350

Large 60350600600

= With the highest value of 600 of the last column, the best decision is to do the large order.

(b) Maximin criterion(5 marks)

Decision(order)State of natureRow Minimum

lowmoderateHigh

Small 120150150120

Medium 9025035090

Large 6035060060

= With the highest value of 120 of the last column, the best decision is to do the small order.

(c) Equally likely criterion(5 marks)

Decision(order)State of natureRow Mean

lowmoderateHigh

Small 120150150(120+150+150)/3 = 140

Medium 90250350(90+250+350)/3 = 230

Large 60350600(60+350+600)/3 = 336.67

= With the highest mean of 336.67, the best decision is to do the large order.

3. A payoff table is given as (13 marks)

State of Nature

Decisions1s2s3

d120 1816

d224 2512

d317 1819

a.What decision should be made by the optimistic decision maker? (3 marks)

State of NatureRow Maximum

Decisions1s2s3

d120 181620

d224 251225

d317 181919

= With the highest value of 25 of the last column, decision ( d2) should be selected.

b.What decision should be made by the conservative decision maker? (3 marks)

State of NatureRow Minimum

Decisions1s2s3

d120 181616

d224 251212

d317 181917

= With the highest value of 17 of the last column, decision (d3) should be selected.

c.What decision should be made under minimax regret? (7 marks)

State of Nature

Decisions1s2s3

d124- 20 = 4 25- 18= 719-16= 3

d224- 24 = 0 25-25 = 019- 12=7

d324- 17 = 7 25- 18= 719-19 = 0

State of NatureRowMaximum

Decisions1s2s3

d14737

d20077

d37707

= As the values of all the decisions have same maximum value, all 3 decisions can be selected.

4. Meng and Kee run a small electrical appliance shop and they must order table fans for the coming season. Orders for the fans must be placed in quantities of twenty (20). The cost per bicycle is $40 if they order 20, $37 if they order 40, $35 if they order 60, and $34 if they order 80. The fans will be sold for $70 each. Any fans left over at the end of the season can be sold (for certain) at $30 each. Meng and Kee estimate that the demand for fans this season will be 10, 30, 50, or 70 fans with probabilities of 0.2, 0.4, 0.3, and 0.1 respectively. (12 marks)

a. Construct the payoff table. (8 marks)

State of Nature

Decision10305070

200600600600

400141013201320

600195020502100

800252026002680

Probability0.20.40.30.1

b. Find the Expected Monetary value (EMV) (4 marks)

State of NatureEMV

Decision10305070

200600600600480

4001410132013201092

6001950205021001605

8002520260026802056

Probability0.20.40.30.1

EMV(20) = 0(0.2) + 600(0.4) + 600(0.3) + 600(0.1) = 480EMV(40) = 0(0.2) + 1410(0.4) + 1320(0.3) + 1320(0.1) = 1092EMV(60) = 0(0.2) + 1950(0.4) + 2050(0.3) + 2100(0.1) = 1605EMV(80) = 0(0.2) + 2520(0.4) + 2600(0.3) + 2680(0.1) = 2056

= Decision to order 80 has the highest expected monetary value (EMV), hence it is the best decision to take.

5. ABC company faces a payoff table as given below (15 marks):

DecisionState of nature

S1S2S3

D1250750500

D2300-2501200

D3500500600

a. If the probabilities of S1, S2 and S3 are 0.2, 0.5, and 0.3, respectively, then what choice should be made under expected monetary value (EMV)? (5 marks)

State of NatureEMV

DecisionS1S2S3

D1250750500575

D2300-2501200295

D3500500600530

Probability0.20.50.3

EMV(D1) = 250(0.2) + 750(0.5) + 500(0.3) = 575EMV(D2) = 300(0.2) + (-250)(0.5) + 1200(0.3) = 295EMV(D3) = 500(0.2) + 500(0.5) + 600(0.3) = 530

= Decision 1 has the highest expected monetary value (EMV), hence it is the best decision to take.

b. Find the Expected value with perfect information (EVWPI). (6 marks)

EVWPI = 500(0.2) + 750(0.5) + 1200(0.3) = 835

c. A market research company is trying to persuade ABC company to hire them to do market research so that the management of ABC co. can make better informed decision. So, What is the maximum amount that ABC company is willing to pay for the market information (hint: find the expected value of perfect information (EVPI))? (4 marks)

EVPI= EvWPI Maximum EMV = 835 575 = 260 = The maximum amount that ABC Company is willing to pay for the market information is RM 260.

6. An investor faces a payoff matrix table as shown below (15 marks).

Probability0.70.3

a. What decision should maximize the expected payoff (EMV)?(3 marks)Decision( Purchase )States of NatureEMV

Good Economic ConditionsPoor Economic Condition

Apartment Building50,00030,00050,000(0.7) + 30,000(0.3)= 44,000

Office Building100,00-40,000100,000(0.7) + (-40,000)(0.3)= 58,000

Warehouse30,00010,00030,000(0.7) + 10,000(0.3) = 24,000

Probability0.70.3-

= The maximum EMV is $58,000, therefore the decision is to purchase the office building.b. Determine the expected value with perfect information (EVWPI) and expected value of perfect information (EVPI)(5 marks) EVWPI = 100,000(0.7) + 30, 000 (0.3) = 70,000 + 9,000 = 79,000

EVPI = EVWPI Maximum EMV = 79,000 58,000 = 21,000

c. Calculate the expected opportunity loss (EOL).(7 marks)

Decision( Purchase )States of NatureEOL

Good Economic ConditionsPoor Economic Condition

Apartment Building50,000050,000(0.7) + 0(0.3)= 35,000

Office Building070,0000(0.7) + 70,000(0.3)= 21,000

Warehouse70,00020,00070,000(0.7) + 20,000(0.3) = 55,000

Probability0.70.3-

= Since the decision to purchase the office building has the minimum value of EOL that is $21,000 ( =EVPI), it is thus the best alternative.

7. A factory produces bowl and plate. The product resource requirement and unit profit are given below: (15 marks)

Resource Requirements

ProductLabor (Hr/unit)Clay (pound/unit)Profit ($/unit)

Bowl1440

Plate2350

Given 40 labor hours per day and 120lb of clay per day, answer the following questions:

a.Define the decision variables and write out the objective function and resource constraints. (8 marks)

X = Amount of bowls producedY= Amount of plates produced

Objective function : Max 40x + 50y

Constraintss.t x + 2y 40 4x + 3y 120 x, y 0

b.Use a graph to show the feasible region. (3 marks)

4x + 3y 120

When x=0,

4(0) + 3y =120 3y = 120 y = 120 3 y = 40(0,40)

When y=0 4x +3(0)=120 4x=120 x=120 4 x=30

(30,0)

x + 2y 40

When x=0,

0 + 2y=40 2y=40 y=40 2 y=20(0,20)

When y=0,

x+2(0)=40 x=40

(40,0)

c.What are the values of X and Y at the optimal solution? What is the optimal value of the objective function? (4 marks)

4x + 3y =120 (1) x + 2y = 40 - (2)

From (2) : x= 40 2y (3) Subsitute (3) into (1)

4(40-2y) + 3y =120160 8y + 3y =120 160 - 5y =120 -5y = 120-160 -5y = -40 y = (-40) (-5) y = 8When y=8,

x+2y =40x+2(8) =40 x+16=40 x =40-16 x =24

Corner pointsMax 40x +50y

(0,20)40(0) + 50(20) = 1000

(24,8)40(24) + 50(8) = 1360

(30,0)40( 30) + 50(0) = 1200

=The optimal value of the objective function is (24,8)