# OR-Notes

## J E Beasley

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.

#### Decision trees tutorial class solution

The decision tree for the problem is shown below.

This decision tree illustrates a reactive situation - we can either carry out some action now (i.e. invest 750K) or we can "wait and see" (don't invest). If we choose to wait and see then we may react depending upon what our competitor Y does.

Below we carry out step 1 of the decision tree solution procedure which involves calculating the total profit for each of the paths from the initial node to the terminal nodes. We have also numbered all the nodes in the decision tree from 1 to 21.

Note here that we have included in the decision tree alternative 5 (no program). This is because it may not be worthwhile undertaking the crash/normal programs of minor improvements.

#### Step 1

• path to terminal node 9 - X invests 750K, Y improves, sales increase 8% on sales of 500,000

Profit contribution (pc) change = +0.08 x 500 x 40
Total cost = 750
Total pc change = 850 (all figures in £K)

Note here that, instead of total profit which we had in the M997 example, we focus here on the profit contribution (pc) change based on the £40 per unit profit contribution.

• path to terminal node 10 - X invests 750K, Y doesn't improve, sales increase 15%

Profit contribution (pc) change = +0.15 x 500 x 40
Total cost = 750
Total pc change = 2250

• path to terminal node 11 - X invests 750K, Y doesn't improve, sales increase 10%

Profit contribution (pc) change = +0.10 x 500 x 40
Total cost = 750
Total pc change = 1250

• path to terminal node 12 - X invests 750K, Y doesn't improve, sales increase 5%

Profit contribution (pc) change = +0.05 x 500 x 40
Total cost = 750
Total pc change = 250

• path to terminal node 13 - X doesn't invest, Y improves, X has crash program costing 600K which is successful and results in a sales increase of 7%

Profit contribution (pc) change = +0.07 x 500 x 40
Total cost = 600
Total pc change = 800

• path to terminal node 14 - X doesn't invest, Y improves, X has crash program costing 600K which fails and sales decrease 15%

Profit contribution (pc) change = -0.15 x 500 x 40
Total cost = 600
Total pc change = -3600

• path to terminal node 15 - X doesn't invest, Y improves, X has crash program costing 600K which fails and sales decrease 10%

Profit contribution (pc) change = -0.10 x 500 x 40
Total cost = 600
Total pc change = -2600

• path to terminal node 16 - X doesn't invest, Y improves, X has crash program costing 600K which fails and sales decrease 5%

Profit contribution (pc) change = -0.05 x 500 x 40
Total cost = 600
Total pc change = -1600

• path to terminal node 17 - X doesn't invest, Y improves, X has normal program costing 400K resulting in a sales increase of 5%

Profit contribution (pc) change = +0.05 x 500 x 40
Total cost = 400
Total pc change = 600

• path to terminal node 18 - X doesn't invest, Y improves, X does nothing and sales decrease 15%

Profit contribution (pc) change = -0.15 x 500 x 40
Total cost = 0
Total pc change = -3000

• path to terminal node 19 - X doesn't invest, Y improves, X does nothing and sales decrease 10%

Profit contribution (pc) change = -0.10 x 500 x 40
Total cost = 0
Total pc change = -2000

• path to terminal node 20 - X doesn't invest, Y improves, X does nothing and sales decrease 5%

Profit contribution (pc) change = -0.05 x 500 x 40
Total cost = 0
Total pc change = -1000

• path to terminal node 21 - X doesn't invest, Y doesn't improve, sales unchanged

Profit contribution (pc) change = 0
Total cost = 0
Total pc change = 0

Hence we can form the table below indicating for each branch, the total pc change involved in that branch from the initial node to the terminal node.

```Terminal node  Total pc change (£K)
9              850
10             2250
11             1250
12             250
13             800
14             -3600
15             -2600
16             -1600
17             600
18             -3000
19             -2000
20             -1000
21             0```

We can now carry out the second step of the decision tree solution procedure where we work from the right-hand side of the diagram back to the left-hand side.

#### Step 2

Consider chance node 7 (with branches to terminal nodes 14, 15 and 16 emanating from it). The expected monetary value (EMV) for this chance node is given by

```0.8 x (-3600) + 0.1 x (-2600) + 0.1 x (-1600) = -3300
node 14         node 15         node 16```

Consider chance node 5, the EMV for this chance node is given by

```0.9 x (800) + 0.1 x (-3300) = 390
node 13       chance node 7```

Hence the EMV for the crash program decision is 390K.

Consider chance node 6 (with branches to terminal nodes 18, 19 and 20 emanating from it). The EMV for this chance node is given by

```0.8 x (-3000) + 0.1 x (-2000) + 0.1 x (-1000) = -2700
node 18         node 19         node 20```

Then for the decision node relating to whether to do a crash/normal/no program we have the three alternatives:

• crash program EMV = 390 (calculated above)
• normal program EMV = 600 (terminal node 17)
• no program EMV = -2700 (calculated above)

It is clear that, in £ terms, the normal program alternative is the most attractive alternative and so we can discard the other two alternatives, giving the revised decision tree shown below.

We can now continue the process. The EMV for chance node 4 is given by

```0.6 x (2250) + 0.3 x (1250) + 0.1 x (250) = 1750
node 10        node 11        node 12```

The EMV for chance node 2 is therefore given by

```0.5 x (850) + 0.5 x (1750)  = 1300
node 9        chance node 4```

Hence the EMV for the decision "X invests 750K" is 1300K.

The EMV for chance node 3 is given by

```0.5 x (600)       +     0.5 x (0) = 300
program decision node   node 21```

Hence at the initial decision node relating to whether to invest 750K or not we have the two alternatives

• invest 750K EMV = 1300
• don't invest 750K EMV = 300

Therefore our recommendation is that X should invest £750K now with an expected monetary value of £1300K.

Note however that should we follow this recommendation the actual monetary outcome will be dependent upon chance events but will be one of [850, 2250, 1250, 250] corresponding to terminal nodes 9, 10, 11 and 12 respectively. Therefore, based on our figures, we are always going to show a profit of at least 250K on our investment of £750K and our profit could go as high as £2250K. Hence for this decision the downside is quite healthy.