Question 1 (15 marks)

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					                   STAT379 OPERATIONS RESEARCH II

                                   Revision Exercises

Question 1

A hardware store sells wire for 80 cents per metre which it purchases from a supplier for
45 cents per metre who charges $15 to process an order. Holding costs for the hardware
store amount to 13 cents per dollar of inventory per year. Demand for the wire averages
30000 metres per year while lead time demand follows a normal distribution with mean
600 metres and standard deviation 50 metres. If the hardware store runs out of stock no
backorders are taken and any shortages are estimated to cost 20 cents per metre short.

a) Write down the parameters of the problem.

b) Using no more than two iterations determine the economic order quantity, the reorder
   level and the expected shortage.

c) Calculate the total relevant annual expected cost for implementing the inventory

Question 2
Consider a zero-sum game that has the following payoff matrix.

                                             Player B
                                        B1           B2
              Player A        A1        3             5
                              A2        4             3

(Note: the payoffs described above are the payoff to Player A.)

How should player A play the game in order to maximise his payoff?
HINT: In order to answer this question you will need to use both graphical and algebraic

Question 3
Consider the following transition matrix that describes the probability of a breakdown (or
no breakdown) of a machine from one week to another.

                        Week 2     Good working           Broken down
         Week1                        order
         Good working order            0.8                      0.2
         Broken down                   0.3                      0.7

   (i)      What are the conditional probabilities for a two week transition?

   (ii)     Given that initially there is an equal chance of the machine being broken
            down or being in working order, determine the probability of the machine
            being in working order in week 3.

Question 4

A company has orders for at least 50 bicycles and can produce them in three different
factories: one at Asquith, one at Berowra and one at Cowan. In each factory there is a
start-up cost which is only incurred if at least one item is produced in that factory.

For the factories in Asquith, Berowra and Cowan respectively the start-up costs are $200,
$180 and $150. The production costs for each bicycle are $40 in the Asquith factory, $45
in the Berowra factory and $47 in the Cowan factory.

The labour time taken to produce a bicycle is 4.3 hours in the Asquith factory, 5.1 hours
in the Berowra factory and 6.4 hours in the Cowan factory. The company can pay for up
to 260 hours of labour.

The bicycle frames are made of steel tubing of which there is a total of 75 metres
available. For each bicycle the length of steel tubing used is 1.4 metres in the Asquith
factory, 1.6 metres in the Berowra factory and 1.7 metres in the Cowan factory.

If more than 20 bicycles are made in the Asquith factory then less than 25 bicycles must
be made in the Cowan factory.

The bicycles must NOT all be produced in the one factory. The total number of bicycles
made in the Berowra factory and the Cowan factory combined must be at least 15.

Formulate the problem as one in integer programming to satisfy all the constraints at
minimum cost. Clearly define all your variables.

Question 5
Kim buys flowers at the market for $5 per bunch and sells them by the roadside on the
weekend for $7 per bunch. Kim knows from experience that the demand for flowers on
the weekend is equally likely to be for three, four or five bunches. Kim takes the unsold
flowers home at the end of the weekend to decorate the house.

a) Draw up a payoff matrix showing the possible profits Kim could make for the
   undominated actions.

b) Determine the action that Kim should take using the maximin criterion and state the
   resultant profit from that action. Show your working using the table in a).

c) Draw up the opportunity loss table that results from your answer to a).

d) Using the minimax criterion on the opportunity loss determine the action Kim should
   take and determine the worst possible result for Kim if that action is taken.

e) If Kim decides to use the expected value criterion to make a decision what will be the
   resulting action and the expected payoff.

Question 6
There is a project which will at best yield $100 and at worst will yield a loss of $50.

a) If Jan is indifferent between obtaining a yield of $10 and a lottery in which she has a
   probability of 0.4 of having $100 and a probability 0.6 of losing $50, what is her
   utility for the $10 yield?

b) Bill is considering the same project and he is indifferent between lotteries L1 and L2.
   Lottery L1 gives a payoff equal to $30 with certainty. Lottery L2 gives a payoff equal
   to $100 with probability 0.6 and loss of $50 with probability 0.4.

   i)      Calculate Bills' utility for $30.

   ii)     What is the certainty equivalent for L2?

   iii)    Calculate the risk premium for L2

   iv)     Use your answer to iii) to determine Bills' attitude to risk.

Question 7

An investor who is risk neutral wants to maximise her profit on the purchase and resale of
an apartment. She can buy one in Brisbane for $120,000, one in Sydney for $250,000 or
one in Melbourne for $190,000.

If the current housing boom continues she will be able to sell the Brisbane unit for
$190,000, the Sydney unit for $340,000 and the Melbourne unit for $250,000.

If there is a downturn in the property market she will only be able to sell the Brisbane
unit for $120,000, the Sydney unit for $220,000 and the Melbourne unit for $200,000.

The housing experts predict that there is a 70% chance of a downturn.

a) Determine what action should the investor take and what is the expected profit for
   this action.

b) What is the maximum amount the investor should be prepared to pay for information
   on the investment climate?

Question 8

A company has monthly performance requirements. In the next month the company
requires the agents to make 240 contacts with existing customers and 120 contacts with
new customers.

It allocates two hours of agents time to contact an existing customer and three hours to
contact a new customer. There are only 640 hours of contact time available.

There is no penalty for over-achieving a goal.
   a) Both goals are equally important. How many existing customers and how many
       new customers should be contacted?
   b) It is now decided that contact with existing customers are twice as important as
       contact with new customers. How many existing customers and how many new
       customers should be contacted?

Question 9
A company makes desks and chairs. Each chair takes 1.5 hours to join and each desk
takes 5.0 hours to join. There are 600 hours available in the joinery per week.

Each chair takes 1 hour to polish and each desk takes 2 hours to polish. There are a
maximum of 400 hours available per week for polishing.

Management states that there should be maximum production limits of 300 chairs and 60
desks per week.

Each chair yields a profit of $35 and each desk yields a profit of $60.

Other information is as follows:
                                                               Desk            Chair
Apprentice training time (hrs)                                 8               2
Government subsidy                                             $22             $20

The problem has the following goals per week:
       Goal 1: Profit should try to be at least $15 000.
       Goal 2: Government subsidy should try to be at least $6 300.
       Goal 3: Apprentice training hours should try to be at least 1000.

Formulate the problem as one in goal programming.

Question 10
A computer store manager must decide how many computers to order. He sells each
computer for $2500 and the average demand is 100 computers per month. There is an
order cost of $25 each time an order is placed and the holding costs are 15c per dollar of
inventory per annum.

For the problem given above assume that discounts are given for large quantity purchases
as shown in the table below.

Price Range            0-19               20-39           40-49               50+
Price ($)              1500               1400            1300                1200

   a) Consider all the inventory models that you have studied in both STAT279 and
      STAT379. What is the inventory model that is most appropriate for this problem?

   b) Write down the parameters of the problem.

   c) Determine how many computers the manager should order to minimise total cost.

   d) What is the total annual cost if the number in c) are ordered.

Question 11
A shop owner needs to place an order for dresses to sell for the next summer season. The
dresses will cost the shop owner $80 each to buy from the manufacturer and will be sold
to customers for $150 each. Any dresses left unsold at the end of the season will be sold
off to a discount operator for $30 each. If dresses are sold out the shop owner values each
dress short at $40.

Demand for the dresses has the following discrete probability distribution:

No of dresses       220            230             240            250                260
Probability         0.10           0.25            0.30           0.20               0.15

a) Consider all the inventory models that you have studied in both STAT279 and
   STAT379. What is the inventory model that is most appropriate for the above

b) Write down the parameters of the problem.

c) Calculate the expected demand of the dresses.

d) Determine the optimal number of dresses to be ordered.

e) Calculate the expected shortage if the optimal number of dresses are ordered.

f) What is the probability of a shortage? Give reason for your answer.

g) What is the expected surplus if the optimal number of dresses are ordered?

h) Calculate the total expected cost.


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