YORK UNIVERSITY
FACULTY OF ARTS
DEPARTMENT OF MATHEMATICS AND STATISTICS
MATH 4570 6.0, YEAR 2006-07
APPLIED OPTIMIZATION (TEST #2 )
NOVEMBER 28, 2006
Instructor : Igor Poliakov
Name (print) ______________________________________________
(Family) (Given)
Student I.D. ______________________________________________
Instructions:
1.
This is a closed-book Test. Calculators are permitted.
2.
Show all significant steps. No marks will be given for the answer alone.
3.
Solve the problems in the spaces provided. If you need more space, use the back of the page (indicate this fact on the original page).
4.
YOU ARE ALLOWED TO USE AID SHEET.
5.
USE PEN TO WRITE THE FINAL ANSWER.
6.
DRAW A BOX AROUND YOUR FINAL ANSWERS.
Test # 2 contains 8 questions on 9 pages. Read the instructions carefully and sign below:
__________________
(Signature)
QUESTION 1 2 3 4 5 6 7 8
MARKS 9 9 6 6 2 6 6 6
SCORE
Total
50
Page 1 of 9

1. [9] Sodaco is considering producing a new product: Chocovan soda. Sodaco estimates that the annual demand for Chocovan, D (in thousand of cases), has the following mass function: P(D = 30) = 0.3, P(D = 50) = 0.4, P(D = 80) = 0.3. Each case of Chocovan sells for $5 and incurs a variable cost of $3. It costs $800’000 to build a plant to produce Chocovan. Assume that if $1 is received every year
(forever), this is equivalent to receiving $10 at the present time. Considering the reward for each action and state of the world to be in terms of net present value, use decision criterions (max min, max max, min max) to determine whether
Sodaco should build the plant.
The reward matrix (in hundreds of thousands of dollars)is as follows:
Annual Demand
30,000 50,000 80,000
----------------------------------------
Build 6-8 = -2 10-8 = 2 16-8 = 8
----------------------------------------
Not Build 0 0 0
----------------------------------------
To find the Maximin decision look at
Minimum Reward
---------------------
Build -2
---------------------
Don't Build 0*
---------------------
Thus Maximin decision is Don't Build.
To find the Maximax decision look at
Maximum Reward
---------------------
Build 8*
---------------------
Don't Build 0
---------------------
Thus Maximax decision is to Build.
The regret matrix is as follows
Demand
30,000 50,000 80,000 Maximum Regret
----------------------------------------------
Build 2 0 0 2*
----------------------------------------------
Don't Build 0 2 8 8
----------------------------------------------
Thus Minimax Regret decision is to Build.
Expected Reward from Build = .30(-2) + .4(2) + .3(8) = $260,000
Expected Reward from Don't Build = $0
Thus expected reward is maximized by Build decision.
Page 2 of 8

2. [ 9] Suppose that Pizza King and Noble Greek stop advertising but must determine the price they will charge for each pizza sold. Pizza King believes that Noble Greek’s price is a random variable “D” having the following mass function: P(D = $6) = 0.25, P(D = $8)
= 0.5, P(D = $10) = 0.25. If Pizza King charges a price “p1” and Noble Greek charges a price “p2”, Pizza King will sell (100+25*(p2 – p1)) pizzas. It costs Pizza King $4 to make a pizza. Pizza King is considering charging $5, $6, $7, $8, or $9 for a pizza. Use the decision criterions ( max min, max max, min max) to determine the price that Pizza
King should charge.
The reward matrix is computed to be
Noble Greek Price
Pizza King Price $6 $8 $10
----------------------------------------------
$5 $125 $175 $225
----------------------------------------------
$6 $200 $300 $400
----------------------------------------------
$7 $225 $375 $525
----------------------------------------------
$8 $200 $400 $600
----------------------------------------------
$9 $125 $375 $625
----------------------------------------------
Pizza King Charges Minimum Reward
* ----------------------------------------------
$5 $125
----------------------------------------------
$6 $200
----------------------------------------------
$7 $225*
----------------------------------------------
$8 $200
----------------------------------------------
$9 $125
----------------------------------------------
Thus Maximin action is to charge $7.
To determine Maximax action look at
Page 3A of 9

Pizza King Charges Maximum Reward
----------------------------------------------
$5 $225
----------------------------------------------
$6 $400
----------------------------------------------
$7 $525
----------------------------------------------
$8 $600
----------------------------------------------
$9 $625*
----------------------------------------------
Thus Maximax action is to charge $9.
The regret matrix is
Pizza King Price Noble Greek Price
----------------------------------------------
$6 $8 $10 Maximum Regret
----------------------------------------------
$5 $100 $225 $400 $400
----------------------------------------------
$6 $25 $100 $225 $225
----------------------------------------------
$7 $0 $25 $100 $100
----------------------------------------------
$8 $25 $0 $25 $25*
----------------------------------------------
$9 $100 $25 $0 $100
----------------------------------------------
Thus Minimax regret action is to charge $8.
Expected reward for $5 = $175
for $6 = $300
for $7 = $375
for $8 = $400
for $9 = $375
Thus Pizza King maximizes their expected reward by charging $8.
Page 3B of 9

3. [6] We now have $5,000 in assets and are given a choice between investment 1 and investment 2. With investment 1, 80% of the time we increase our asset position by
$295,000, and 20% of the time we increase our asset position by $95,000. With investment 2, 50% of time we increase our asset position by $595,000, and 50% of the time we increase our asset position by $5,000. Our utility function for final asset position
“x” is u(x). We are given the following values for u(x): u(0) = 0, u(640,000) = 0.8, u(810,000)= .90, u(0)=0, u(90,000)= .30, u(1,000,000)= 1, u(490,000)= .7. a) Are we risk-averse, risk-seeking, or risk-neutral? Explain b) Will we prefer investment 1 or investment 2?
Solution: a) From the graph we see that u(x) is concave, so we are risk averse. b) After reading utilities off the graph we find
Expected Utility for Investment 1 = .8u(300,000) + .2u(100,000)
= .8(.55) + .2(.32) = .50
Expected Utility for investment 2 = .5u(600,000) + .5u(10,000) = .5(.77) + .5(.1) = .44, so choose investment 1.
Page 4 of 8

4. [6] My current income is $40,000. I believe that I owe $8,000 in taxes. For $500, I can hire a CPA to review my tax return; there is a 20% chance that she will save me
$4,000 in taxes. My utility function for (disposable income) = (current income) –
(taxes) – (payment to accountant) is given by sq. root (x), where (x) is disposable
Hiring CPA has expected utility of .2u(35,500) + .8u(31,500) =
.2(35,500)
1/2
+ .8(31,500)
1/2
= 179.7 while Not Hiring CPA has expected utility of
(32,000)
1/2
= 178.9. Thus the CPA should be hired. income. Should I hire the CPA?
Page 5 of 9

5.[2] Find optimal strategies for each player and the value of the two-person zerosum game in the given table.
20 1
12 10
2
4
24 8 -2
. Since max (row min) = min (col max) = 4, we find that the value of the game to the row player is 4. Row choosing row 2 and column choosing column 3 is a saddle point.
Page 6 of 8

6.[6] A total of 90,000 customers frequent the Ruby and the swamp supermarkets. To induce customers to enter, each store gives away a free item. Each week, the giveaway item is announced in the Monday newspaper. Of course, neither store knows which item the other store will chose to give away this week. Ruby’s is considering giving away a carton of soda or a half gallon of milk. Swamp’s is considering giving away a pound of butter or a half gallon of orange juice. For each possible choice of items, the number of customers who will shop at Ruby’s during the current week is shown in table below.
Each store wants to maximize its expected number of customers during the current week.
Determine an optimal strategy for each store and the value of the game. Interpret the value of the game.
Swamp Chooses Swamp Chooses
Ruby Chooses Butter
Soda 40,000
Orange Juice
50,000
Milk
SOLUTION:
60,000 30,000
. We assume that all customers who do not go to Ruby go to Swamp. Thus we may solve this as if it were a zero-sum game. Let x
1
= probability that Ruby chooses soda and y
1
= probability that Swamp chooses butter. Then Ruby chooses x
1
to maximize min(40,000x
1
+ 60,000(1 - x
1
), 50,000x
1
+ 30000(1 - x
1
))
= min(60,000 - 20,000x
1
, 30,000 + 20,000x
1
)
The maximum occurs for x
1
= 3/4. Value of game = 45,000 to Ruby.
Swamp chooses y
1
to minimize max(40,000y
1
+ 50,000(1 - y
1
), 60,000y
1
+ 30,000(1 - y
1
))
= max(50,000 - 10,000y
1
, 30,000 + 30,000y
1
)
This minimum occurs for y
1
= 1/2. Thus Ruby should choose soda 3/4 of time and milk 1/4 of the time. Swamp should choose butter half the time and orange juice half the time. Value of the game is 45,000 customers to Ruby. Thus by choosing their optimal strategy, Ruby can ensure that an average of 45,000 customers per week will come to their store.
Page 7 of 8

7.[6] Two competing firms are deciding whether to locate a new store at point A, B, or
C. There are 52 prospective customers for the two stores. Twenty customers live in village A, 20 customers live in village B, and 12 customers live in village C. Each customer will shop at the nearer store. If a customer is equidistant from both stores, then assume there is a ½ chance that he or she will shop at either store. Each firm wants to maximize the expected number of customers that will shop at its store. Where should each firm locate its store? (AB = BC = 10 miles, points A, B, C belong to the same straight line).
SOLUTION:
. This is a constant-sum game with the following (expected) reward matrix (Firm 2 receives
52 - (expected reward to Firm 1)).
Locate at A Locate at B Locate at C Row Min
Locate at A 26 20 30 20
------------------------------------------------
Locate at B 32 26 40 26
------------------------------------------------
Locate at C 22 12 26 12
------------------------------------------------
Column Max 32 26 40
Thus both firms locating at B (and sharing the customers) is a saddle point.
Page 8 of 9

7.
[6] Mo and Bo each have a quarter and a penny. Simultaneously, they each display a coin. If the coins match, then Mo wins both coins; if they don’t match, then Bo wins both coins. Determine optimal strategies for this game.
SOLUTION:
Reward matrix is as follows:
Put out quarter +25 -25
Bo
Put out quarter Put out penny
Mo
Put out penny -1 +1
Let x i
= probability that Mo plays row i and y i
= probability that
Bo plays column i. Against each column Mo obtains
Against Column 1: 25x
1
-(1-x
1
) = 26x
1
-1
Against Column 2: -25x
1
+(1-x
1
)= 1-26x
1
.
Mo should choose x
1
to max min{26x
1
-1,1-26x
1
}. This occurs when
26x
1
- 1 = 1 - 26x
1
. This yields x
1
= 1/26.
Against each row Bo loses
Against Row 1: 25y
1
- 25(1-y
1
) = 50y
1
- 25
Against Row 2: -y
1
+ (1-y
1
) = 1 - 2y
1
.
Bo should choose y
1
to min max {50y
1
-25, 1 - 2y
1
}. This occurs when
50y
1
-25=1-2y
1
or y
1
= 1/2. Value of game is 0 to each player.
THE END
Total marks are 50
Page 9 of 9