OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Answer FOUR questions.

All questions carry equal marks.

If more than four questions are answered only the first four answered will
be marked.

Marks will be deducted in this examination if there is insufficient written
explanation.

**Question 1**

A company investigating switching between varying types of clothes washing products has identified three generic types - Tablets, Powder and Liquid. Data has been collected from a number of shoppers at a supermarket on a Saturday as to their switching behaviour, as below:

To Tablets Powder Liquid From Tablets 0.90 0.05 0.05 Powder 0.15 0.55 0.30 Liquid 0.10 0.20 0.70

For example, a shopper who bought Tablets last month has a probability of 0.90 of buying Tablets the following month, 0.05 of buying Powder the following month and 0.05 of buying Liquid the following month. Currently Tablets, Powder and Liquid have 20%, 70% and 10% of the market respectively.

- Draw the state transition diagram.
- Calculate the market share for Tablets in two months time. C
- Calculate the long-run market share for Tablets, Powder and Liquid.

** Question 2**

The Mayor of London, Ken Deadrock, has been concerned about the relative efficiency of a number of tube stations. Differing stations have widely varying numbers of staff and widely varying passenger numbers. As a preliminary study seven different tube stations have been taken and the following data gathered as to passenger and staff numbers.

Station Arriving passengers Entering passengers Staff (thousands) (thousands) South Kensington 5 4 1 Gloucester Road 7 5 2 Richmond 3 1 6 Wimbledon 4 1 5 Earls Court 14 4 5 Knightsbridge 6 3 5 Green Park 4 3 6

For example at South Kensington at 9am on a Monday morning (the day/time the data was gathered) there were 5 staff on duty, 4000 passengers arrived on tube trains and exited the station and 1000 arrived at the station though the entry gates to board tube trains.

What insight can you offer as a result of this data and what further information would be useful to improve your analysis?

**Question 3**

A company uses 250 units per month of a particular component X and 755 units per month of another component Y. Both X and Y are bought from the same supplier . Each unit of X that is purchased costs £3.45 and each unit of Y that is purchased costs £8.75. The company estimate that the cost of placing an order with their supplier is £25 and currently interest rates are 4%. Storing one unit of X costs £2.80 per year and storing one unit of Y is estimated to cost 10% of the purchase price per year.

- If the company currently orders monthly from its supplier what would be the associated total cost per year?
- Is there a better order policy that the company can adopt - if so what is it and what would be the associated total cost per year?
- If the supplier is prepared to reduce the cost of one unit of Y to £8.25 provided that each order is at least 2000 units then what order policy would you now recommend and what would be the associated cost per year?

**Question 4**

The demand for a particular product in each of the last six weeks is shown below.

Week 1 2 3 4 5 6 Demand 123 183 210 243 237 283

- Calculate a forecast for week 7 using a two week moving average.
- Calculate a forecast for week 7 using exponential smoothing with a smoothing constant of 0.4.
- Which of these forecasts for week 7 do you prefer and why?

The following table gives the coordinates (x,y) of seven points in the Euclidean plane.

Point x-coordinate y-coordinate 1 2 8 2 3 4 3 4 2 4 6 3 5 2 7 6 1 4 7 6 4

If these points are to be connected together suggest two possible connection networks (that are of low total length) and evaluate the total length of the connection network in each case.

**Question 5**

The following table defines the various activities in a small project:

Activity Completion time (days) A 5 B 4 C 7 D 9 E 4 F 7 G 8 H 3 I 6 J 7

The immediate precedence relationships are:

Activity Activity J must be finished before D can start B J,G C A,E H I,J,F A G

In addition there must be a time lag of at least 3 days between the end of activity F and the start of activity D.

- Draw the network diagram.
- Calculate the critical path, the overall project completion time and the float times for each activity.
- Suppose now that activity H could be split into two separate activities H1 and H2, with H1 taking one day and H2 taking 4 days. The precedence relationship relating to H in this case would change to H1 must be finished before H2,J and F can start; and H2 must be finished before I can start. Do you think that this split would be worthwhile or not and why?

**Question 6**

In a factory concerned with the production of soft drinks you have been asked to analyse the production of two drinks, X and Y. Each litre of X that is produced can (if required) be further processed into Y. Producing one litre of X costs £0.12 and the further processing to produce Y costs an additional £0.03. Wholesalers have in the forthcoming month ordered 150,000 litres of X and 175,000 litres of Y.

The production capacity allocated to X and Y for the next month means that no more than 250,000 litres (X and Y combined) can be produced. There is none of X or Y currently in stock. A subcontractor can supply X to you for £0.40 per litre and Y for £0.63 per litre. Technological restrictions mean that the ratio of the number of litres of X produced to the number of litres of Y produced must be at least 0.33.

Formulate this decision problem as a linear program.

Assuming that the factory cannot store excess production how much of each drink should it produce next month?