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.

**Below are sample examination questions, adapted from past papers
to reflect the course as taught in Spring term 2000.**

**Sample compulsory questions (40% of the total mark)**

**Question**

A project is specified by the following activities:

Activity Immediate Duration Predecessor(s) (days) A - 11 B A 9 C A,B 11 D A 19 E B 7 F C,D,E 15 G E,F 4

- Construct an activity on node network to represent the above project.
- Calculate the earliest start, latest start, earliest finish and latest finish times for each activity. Also calculate the minimum project completion time and identify the critical path.
- What is the total float associated with each of the non-critical activities?
- What effect, if any, will each of the following changes have on the completion time of the project :
- Activity D is delayed by 3 days;
- Activity C is finished 1 day early.
- Ignoring the changes to activities D and C given above, suppose now that activity F cannot start until at least 2 days after activity D is finished. How is the project duration and critical path affected as a result of including this dependency in the project ?

**Question**

The table below defines the activities within a small project.

Activity Immediate Normal Normal Crash Crash Predecessor(s) Time Cost Time Cost (days) (£000’s) (days) (£000’s) A - 6 24 4 34 B - 4 12 3 22 C A 5 20 3 28 D A 7 29 4 47 E B 6 26 5 34 F B 8 34 5 52 G C,E 10 27 6 47 H D,F 9 34 7 48

- Draw an activity on node network to represent the above project.
- Using normal duration times, calculate the earliest start, latest start, earliest finish and latest finish activity times. Also calculate the minimum project completion time and identify which activities are critical.
- What is the total float associated with each of the non-critical activities?
- Suppose that activity H cannot start until at least 2 days after activity F is finished. Using normal durations, what effect (if any) will this have on the project duration and critical path?
- Let variable x
_{i}represent the earliest start time for activity i (>=0, integer). Using such variables for each activity in the project network above (including the condition that activity H cannot start until at least 2 days after activity F is finished), formulate the problem of deciding the optimal completion time for each activity so as to ensure that the project is completed within 15 days at minimum cost as a linear integer programming model. What assumptions have you made in producing this model?

**Sample other questions (each 30% of the total mark)**

**Question**

Consider the following project (all times are in days).

Activity Immediate Optimistic Most Pessimistic Predecessor(s) Time Likely Time Time A - 1 2 3 B - 2 3 4 C - 1 3 5 D A 1 2 3 E B 1 1 1 F B 1 2 3 G B 2 3 4 H C 3 5 7 I C 1 3 5 J A 2 3 4 K D,E 2 3 4 L F,K 2 4 6 M G,H 3 4 5 N I 1 3 5 O J,L,M,N 1 2 3

- Construct an activity on arc network for the above project.
- Determine the minimum expected completion time of the project. Identify the critical path.
- Calculate the probability that the project can be completed within 17 days or less.
- Calculate the probability that the project cannot be completed within 21 days.

**Question**

Consider the following network for a small maintenance project (all times are in days).

Activity Immediate Optimistic Most Pessimistic Predecessor(s) Time Likely Time Time A - 2 3 4 B - 5 6 7 C - 5 6 7 D A 3 4 5 E A 2 3 4 F C 3 4 5 G C 8 10 16 H B,E,F 5 6 7 I B,E,F 7 11 15 J B,E,F 2 3 4 K G,J 3 4 5 L D,H 7 11 15

- Draw an activity on node network diagram for the project.
- Calculate the minimum expected completion time of the project. Identify the critical path.
- Estimate the total float, free float and independent float for activities F and G. Explain the meaning of these values.
- Calculate the probability that the project would be completed within 25 days.
- Calculate the probability that the project cannot be completed within 30 days.
- Using an example, briefly explain how PERT can be used in project management.

**Question**

The Research and Development (R&D) division of a company has been developing four possible new product lines. The Project Manager must now make a decision as to which of these four products will actually be produced and at what levels. Therefore, he has to formulate a mathematical model to find the most profitable product mix.

The start-up cost associated with beginning the production of each product and the marginal net revenue from each unit produced are shown below.

Product 1 Product 2 Product 3 Product 4 Start-up cost (£) 40000 30000 60000 50000 Marginal revenue (£) 80 70 100 90

All product lines require the use of two specific resources R1 and R2. The requirements of each resource per unit produced are shown below. In either case, resource availability is limited to 7000 units.

Units of Resource Required per Unit of Product Produced Product 1 Product 2 Product 3 Product 4 Resource R1 6 4 7 5 Resource R2 5 7 4 6

Only 800 units of product 1 could be sold whereas all units that could be produced of the other three products could be sold.

The following policy constraints have been imposed by management on the production levels of each product.

- No more than two of the products can be produced.
- Either product 3 or 4 can be produced only if either product 1 or 2 is produced.

Formulate the problem of deciding which products to produce, and at what levels, as a linear integer program..