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BUSI4489

BUSINESS SCHOOL

A LEVEL 4 MODULE, AUTUMN SEMESTER 2020-2021

MANAGEMENT SCIENCE FOR DECISION SUPPORT

1. Answer any FIVE parts of this question.  All parts carry equal marks.

a) Using your own LP model with two variables and minimisation objective function, illustrate graphically how you can find the shadow price of a binding constraint.

b) Explain how ranked goal programming works and briefly explain why this approach may be preferred over the weighted goal programming approach.

c) Why are integer linear programming models (as compared with standard linear programming models) in general more difficult to solve?

d) Explain what is meant by the reduced cost of a decision variable and briefly comment why this information may be of interest to a decision maker.

e) What is the difference between decision making under certainty, decision making under risk and decision making under uncertainty?

f) Describe two applications of the shortest path problem.

g) A box contains 8 red balls, 10 green balls and 2 white balls.  A ball is removed from the box and replaced; the balls are mixed and then another ball is removed. What is the probability of:

1. two green balls being drawn?

2. a red ball on the second draw?

3. A red or white ball on the first draw, and a green or white ball on the second draw?

4. a red ball on the second draw, given that a white ball was drawn on the first?

5. Re-examine part (1) in case the first ball is not replaced in the box.

h) Nottingham University must purchase 1100 computers from three potential suppliers. Supplier 1 charges £500 per computer plus a (fixed) delivery charge of £5000. Supplier 2 charges £350 per computer plus a (fixed) delivery charge of £4000. Supplier 3 charges £250 per computer plus a (fixed) delivery charge of £6000. Supplier 1 will sell at most 500 computers; supplier 2, at most 900; and supplier 3, at most 400. Formulate an integer linear programming model to minimise the cost of purchasing the needed computers. Clearly define the decision variables and explain the objective function and constraints.

Section A: Candidates should answer one question from this section

2. HAL produces two types of computers: PCs and VAXes. The computers are produced in two locations: New York and Los Angeles. New York can produce up to 800 computers and Los Angeles up to 1,000 computers. HAL can sell up to 900 PCs and 900 VAXes. The profit associated with each production site and computer sale is as follows: New York – PC, $600; VAX, $800; Los Angeles – PC, $1000; VAX, $1300. The skilled labour required to build each computer at each location is as follows: New York – PC, 2 hours; VAX, 2 hours; Los Angeles – PC, 3 hours; VAX, 4 hours. A total of 4,000 hours of labour are available. Labour is purchased at a cost of $20 per hour. Let

X_NP = PCs produced in New York,

X_LP = PCs produced in Los Angeles,

X_NV = VAXes produced in New York,

X_LV = VAXes produced in Los Angeles,

L = labour hours used.

HAL has formulated the following linear programme to determine the optimal product mix.

Max 600X_NP + 1000X_LP + 800X_NV + 1300X_LV – 20L

Subject to:

2X_NP + 3X_LP + 2X_NV + 4X_LV – L <= 0 (Constraint 1)

X_NP + X_NV <= 800 (Constraint 2)

X_LP + X_LV <= 1000 (Constraint 3)

X_NP + X_LP <= 900 (Constraint 4)

X_NV + X_LV <= 900 (Constraint 5)

L <= 4000 (Constraint 6)

All variables >= 0

After solving this model with Excel Solver, the following answer and sensitivity reports were obtained.

Using the information above, answer each of the following questions. Each question is independent of the others.

a) Interpret (i.e., what is the meaning of) the objective function and the different constraints of the LP model. Discuss also the optimal solution and objective function value and indicate what constraints are binding. [15%]

b) Interpret the shadow prices of constraints 5 and 6. If 3000 hours of skilled labour were available, what would be HAL’s profit? [15%]

c) Suppose an outside contractor offers to increase the capacity of New York to 850 computers at a cost of $5000. Should HAL hire the contractor? (Your answer should be ‘yes’ or ‘no’ with explanation.) [15%]

d) What would be the new profit if the profit of VAXes decreases by $50? Will the optimal solution be different? [15%]

e) HAL is considering the production of laptops. Laptops can be produced in New York only. There is no limitation on the amount of laptops that can be sold in the market. The total production capacity (including laptops) at New York remains at 800 units. A laptop requires 1.5 hours of skilled labour and would generate a profit of $600. Should HAL produce laptops? (Your answer should be ‘yes’ or ‘no’ with explanation.) [25%]

f) Suppose that the total PC production should be at least twice as large as the total VAX production. Does the current production plan satisfy this condition? If not, how can HAL find the new optimal production plan? [15%]

3.  Braneast Airlines must determine how many airplanes should serve the Boston -New York – Washington air corridor and which flights to fly. Braneast may fly any of the daily flights shown in the table below. Each flight can be selected at most once. Braneast cannot fly on any other time than those shown in the table. It is possible, though, for an airplane to stay put in the same city for an hour or more. The fixed cost of operating an airplane is $1000 per day.

Answer the following questions:

(a) Formulate a minimum cost network flow model that can be used to maximise Braneast’s daily profit. Draw a time-expanded network and clearly explain the meaning of the different nodes and arcs. Define the variables, and write down the objective function and one example of each different type of constraint. [80%]

(b) By inspecting your drawing, find a feasible (but not necessarily optimal) solution to the problem. Show this solution on your network. Briefly discuss the flight schedule and the resulting profit. [20%]

Section B: Candidates should answer one question from this section

4. a)  Describe the main components of a queuing system. [40%]

b) Consider the following two queuing systems:

o system 1 has two separate queues each with a single server with exponential service times 12 customers/hour and in each queue, customers arrive at a rate 10 customers/hour (Poisson distributed)

o system 2 has a single queue but two identical parallel servers, each server operating at an exponential rate 12 customers/hour, and customers arrive in the queue at a rate 20 customers/hour (Poisson distributed)

Using appropriate queuing models, evaluate the performance of both systems and calculate the average waiting time for a customer; the average number of customers in a queue and in the system, and the average server utilisation. Which system performs best? [30%]

c) In the past few years, the traffic problems into Nottingham have become worse. Now, Derby Road is congested about half of the time. The normal travel time to work for Luc is only 15 minutes when he takes Derby Road and there is no congestion. With congestion, however, it takes Luc 40 minutes to get to work using Derby Road. If Luc decides to take Wollaton Road, it takes 30 minutes, regardless of the traffic conditions. Luc’s utility for travel time is U(15 minutes) = 0.9; U(30 minutes) = 0.7 and U(40 minutes) = 0.2. Determine (i) the route that minimises Luc’s expected travel time; (ii) the route that maximises Luc’s utility; (iii) whether Luc is a risk seeker or risk avoider when it comes to travel time (Hint: plot the utility curve). [30%]

5. a) Explain why random numbers are commonly needed in simulation, and briefly outline how these are used. [15%]

b) Ashcroft Airlines flies a six-passenger commuter flight once a day to Gainesville, Florida. A non-refundable one-way fare with a reservation costs $129. The daily demand for this flight is given in the table below, along with the probability distribution of no-shows. A no-show has a reservation but does not arrive on time at the gate and forfeits the fare. Ashcroft currently overbooks at most three passengers per flight. If there are not enough seats for all the passengers at the gate, each passenger that cannot board the flight is refunded the passenger’s fare and also $150 voucher good on any other trip. The fixed cost for a flight is $450.

             i) Set up a flow chart showing the logical sequence of events for simulating Ashcroft’s expected profit for this flight. Provide all the details of the formulas used for relevant calculations. [20%]

ii) Using the two-digit random numbers below (in the order as they appear), calculate Ashcroft’s profit per flight and replicate your calculations 10 times. Organise all your calculation in a table. Calculate the expected profit, the occupancy rate of the plane and the probability that Ashcroft profit per flight is higher than £100? Briefly comment on the reliability of your results. [50%]

Random number sequence: 69  56  30  32  66  79  55  24  80  35  10  98  92  92  88  82  13  04  86  31  12  23  40  93  13  42  51  16  17  29  62  08  59  41  47  72  25  96  58  14  68  15  18  99  13  05 03  83  34  78  50  89  98  93  70  11

iii) Explain how you could use this model to investigate Ashroft’s overbooking strategy (no calculations required). [15%]