Decisions Involving Multiple Objectives

The truly excellent textbook “Decision Analysis for Management Judgment” by Paul Goodwin and George Wright (1991) explains a number of decision methodologies, one of which is for decisions involving multiple objectives. Commonly called multi-criteria decision analysis (MCDA), multi-attribute decision analysis (MADA) or multi-objective decision analysis (MODA) it involves a decision problem with a number of objectives that are often conflicting.

The textbook (chapter 2) provides a worked example of the methodology and this blog shows how to model this in Promax.

An Office Location Problem

Goodwin & Wright explain the problem thus:

A small printing and photocopying business must move from its existing office because the site has been acquired for redevelopment. The owner of the business is considering seven possible new offices, all of which would be rented. Details of the location of these offices and the annual rent payable are given below.

Location of office	Annual rent ($)
Addison Square (A)	30 000
Bilton Village (B)	15 000
Carlisle Walk (C)	5 000
Denver Street (D)	12 000
Elton Street (E)	30 000
Filton Village (F)	15 000
Gorton Square (G)	10 000

While the owner would like to keep his costs as low as possible, he would also like to take other factors into account. For example the Addison Square office is in a prestigious location and also close to potential customers, but it is expensive to rent. It is also an old, dark building which will not be comfortable for staff to work in. In contrast, the Bilton Village office is a new building which will provide excellent working conditions, but it is several miles from the center of town, where most potential customers are to be found. The owner is unsure how to set about making his choice, given the number of factors involved.

An Overview of the Analysis

There are seven stages required:

Stage 1: Identify the alternative courses of action. Options. In the example these are the office locations.

Stage 2: Identify the factors that are relevant to the decision. Criteria. In the example above some have already been mentioned such as annual rent, image, closeness to customers and staff comfort.

Stage 3: Determine how to measure the criteria. Criteria properties. In the example above, “Annual Rent” already has a measurement scale in $. However, “Image” doesn’t one readily available.

Stage 4: Determine the weight for each factor. Weight. This reflects the importance of each criterion compared to each other.

Stage 5: Assign scores on how well each option performs on each criterion. Scoring.

Stage 6: Make a provisional decision. Priorities

Stage 7: Perform sensitivity analysis to see how robust the decision is to changes in the data used.

Constructing a Value Tree

Start by creating two headings – one for costs and one for benefits. The objectives here are (a) to minimize costs and (b) to maximize benefits. Then, for each heading, identify the criteria that contribute to meeting those twin objectives. Goodwin & Wright established the following:

Minimize Costs

·       Annual Rent

·       Cost of Electricity (for heating, lighting, operating equipment, etc.)

·       Cost of Cleaning

Maximize Benefit

·       Increase Turnover

·       Improve Working Conditions

Increasing the turnover of the business will be achieved by being close to customers, being visible to the customers and having inviting premises.

Improving the working conditions for staff will be achieved by having large premises that are comfortable with adequate car parking.

The value tree that results can be represented thus

To create this in Promax use the “Criteria” tab. Double click in the workspace to add a box. Drag the boxes onto the “Value Tree” box. This turns all the boxes into “Criteria”. To create a “Topic” or “Node” (the red headings in the value tree) right click a box and turn it into a “Topic”.

Measuring How Well The Options Perform On Each Criteria

Having created the value tree the next step is to determine how best to measure each criterion.

Fixed Mapping

For the cost criteria, this is relatively simple. Costs for rent have already been provided and estimates for electricity and cleaning can be obtained. All of these will be measured in $ over the entire year and all three can be added together to give a total cost per year in $. It is also true in this case that $100 is valued at twice $50 and this relationship holds throughout.

In Promax, the measurement scales are added through “Criteria Properties”. In the criteria tab, choose the Total Cost criteria right click and choose properties. This brings up a box as below:

There are a number of areas to explain more about the criteria but we’ll focus on just a couple.

First, choose the radio button for “Cost”. This is essential.

Next, we’ll choose the “Fixed” mapping (second from the left) and then set the range – the left hand number should be zero and the right hand number 35000 (which represents the largest total cost figure). In fact, as long as this number is larger than the maximum it doesn’t matter. What this does is to create a straight-line mapping with the real-world scale (of $0 - $35000) on the x-axis and the value assigned to the scale on the y-axis (0 – 100). So $17,500 is worth 50 points.

Repeat for all the cost criteria with the scale going from zero to the largest in all cases. Note, that at this stage you won’t necessarily know what is the largest cost so you will need to go back and check.

Custom Mapping

In a lot of cases the relationship between the real-world scale and what it is valued is not a straight-line. Goodwin & Wright use the example of “Floor Area”.

Let us now consider the benefit attributes which can be represented by easily quantified variables. First we need to measure the owner’s relative strength of preference for offices if different sizes. The floor area of the offices is shown below:

Location of office	Floor area (ft2)
Addison Square (A)	1000
Bilton Village (B)	550
Carlisle Walk (C)	400
Denver Street (D)	800
Elton Street (E)	1500
Filton Village (F)	400
Gorton Square (G)	700

Now it may be that an increase in area from 500ft² to 1000 ft² is very attractive to the owner, because this would considerably improve working conditions. However, the improvements to be gained from an increase from 100 ft² to 1500 ft² might be marginal and make this increase less attractive. Because of this we need to translate the floor area into values. This can be achieved as follows.

The owner judges that the larger the office, the more attractive it is. The largest office, Elton Street, has an area of 1500ft2 so we can give 1500 ft² a value of 100. …… Similarly, the smallest offices (Carlisle Walk and Filton Village) both have areas of 400 ft² so we can attach a value of 0 to this area.

The technique is to then identify the mid-point. That is what floor area is about halfway between most preferred and least preferred. Then do the quarter points in between until you have the scales mapped.

In Promax, select the criteria “Area” and right click to get to the properties box. The "Benefit" button is the default. Choose “Custom” as the mapping type. In the top row add in the floor areas and in the bottom row add in the values. In this case you have 400 and 0 on the left and 1500 and 100 on the right. Then complete the intermediate points. If you need more points, go to one of the existing points and right click to insert an additional point to the right or left of the one highlighted. Click OK to accept.

Preference Scales

In many cases it is difficult (or impossible) to use real-world scales and so preference scales are often used. "Image" is one such criterion.

In Promax, go to the Image box and right click to get the properties box. Choose the “Preference” mapping type (the one in the middle). This produces a straight-line mapping of the preference on the x-axis of 0 – 100 and the value on the y-axis of 0 – 100.

The process is actually carried out in the score window. First rank the offices from the most preferred to the least preferred:

1.     Addison Square

2.     Elton Street

3.     Filton Vilage

4.     Denver Street

5.     Gorton Square

6.     Bilton Village

7.     Carlisle Walk

Then give the most preferred a score of 100 and the least preferred a score of 0.

Goodwin & Wright go on to explain that:

The owner is now asked to rate the other locations in such a way that the space between the values he gives to the offices represents his strength of preference for one office over another in terms of image.

So in the example, moving from Carlisle Walk to Gorton Square is twice as preferable to moving from Carlisle Walk to Bilton Village. Similarly moving from Carlisle Walk to Addison Square is ten times more preferable than moving from Carlisle Walk to Bilton Village.

Note that since this is an interval scale you can’t say that Gorton Square is twice as preferable as that of Bilton Village.

Determining the Weights of the Criteria

Goodwin & Wright have an excellent section on weighting and the problems associated with the typical approach of weighting by importance.

An intuitively appealing way ….. is to attach weights to each of the attributes which reflect their importance to the decision maker. For example, he might consider office area to be less important than distance from customers and therefore give a weight of only 1 to office area and a weight of 5 to distance from customers. The problem with this approach is that it may not take into account how large the range is between the most preferred and the least preferred range on each attribute.

A better way is to use swing weights. In this method you are asked to compare a change from the least preferred to the most preferred value on one criteria to a similar change on another criteria.

The process is to imagine the worst office possible – the one which scores the least on all criteria. Then ask “if you could move just one criteria to its best score which one would it be?” In this example the owner selects “Closeness to Customer”. This is assigned a weight of 100 and is the yardstick. You then compare all the other criteria against this one.

In Promax is to go to the “Weights” tab and select “Swing Weighting”. Select two criteria from the left hand side, “Closeness” and “Visibility”. This will duplicate them on the right hand side. Right click on “Closeness” and select “Set as Yardstick”.

Drag the “Visibility” part up and down until the most preferred is equivalent to the yardstick. In this case the decision maker believes that the best in visibility is worth about 80% of the best in closeness to customers.

This is carried out for all criteria until all the weights have been done.

Note that in the example, weighting has been done only on the benefit criteria. This is so you can compare costs with benefits in a specific way. It is possible to include costs in weighting but as Goodwin & Wright explain

Our owner does have difficulties in judging the cost-benefit trade-off.

Aggregating the benefits using the additive model

Goodwin & Wright explain the calculations involved which is adding an office’s weighted value scores together to obtain a measure of the overall benefits, which that office has to offer.

Promax does all the arithmetic for you of course.

Firstly here are the scores used.

Costs

Benefits

Scores

Second, here are the mapped and mapped & weighted values.

Mapped Values

Mapped & Weighted Values

Results Chart
After choosing the criteria, weighting them and the scoring each alternative you can look at which one is preferred.

Click the “Priorities” tab to give you the results. Addison Square provides the best benefit.

By default, Promax places the options in order. If you want to look at the result alphabetically sort alphabetically.

Strengths & Weaknesses

Another way of looking at the results (not discussed by Goodwin & Wright) is strengths & weaknesses. Choose this from the “Priorities” tab and select the options you want to compare.

The length of the white background line is the weight of each criterion and the coloured lines reflect how well each option performs against each criterion.

Trading Benefits Against Costs

Goodwin & Wright use a graph to compare costs against benefits. You can do the same in Promax.

Firstly we need to create a new weight set where costs have a weight (at the moment, costs are weighted as zero as we’re only comparing the benefits).

Go to the “Weights” tab and select “Add” from the weight sets block. Give it a name and copy from “Initial Weights”. You’ll end up with a table with an additional row. Put in a weight for Total Cost and make sure all the other costs are zero weighted. Note, it doesn’t matter what value you pick for a cost weight since costs and benefits will be kept separate. It does matter that the other costs are zero weighted since, if they weren’t, you’d be double counting some of the costs.

Then to look at the graph go to the Priorities tab and select “Scatter” from the drop down on the left hand side.

By default, the display plots costs on the x-axis with the cheapest costs (better) to the left. Benefits are on the y-axis with the greatest benefits towards the top.

To mimic the Goodwin & Wright graph you need to make the cheapest cost go to the right so click “Reverse Cost Axis” in the ribbon. Click the frontier button and you end up with the following:

Note, only the mapped values are shown for costs rather than the actual $ amounts.

Clicking on table and then each of the options on the frontier, highlights each option.

The B/C ratio gives the benefit score (mapped & weighted) divided by the costs, which are positively mapped (i.e. a high cost is a high score).

The graph provides some interesting insights. The Addison Square office provides the greatest benefits but also has the highest cost. The only options worth considering are those on the frontier i.e. Addison Square, Gorton Square and Carlisle Walk. If (and only if) all the options meet the minimum benefit required then the decision maker should choose the option with the best benefit to cost ratio – in this case Carlisle Walk.

Sensitivity Analysis

Gordon & Wright’s definition of sensitivity analysis is:

Sensitivity analysis is used to examine how robust the choice of an alternative is to changes in the figures used in the analysis.

This is a fine definition but they then go on to describe the variation of only one element of the data – weights. Clearly there are other factors can also vary, especially the scores (uncertainty in the costs for instance) and in the value mapping (uncertainty in whether a floor area of 1000ft² is valued at 75% that of 1500ft².

However, to look at the uncertainty of weights on the results choose the “Sensitivity” graph from the Priorities tab. By default, it will plot the top options at different weights. As it suggests in their example the owner is most concerned about the effect of turnover and whether a change in weight would affect the choice.

Select “Turnover” as the filter and the graph below will appear. The current weight is shown by vertical the red line (81). The top lines in yellow and blue reflect the top options as the weight of “Turnover” changes. It can be seen that the top option is Addison Square until the weight drops to around 50 whereupon Elton Street is the most preferred. Choosing the “What If” button in the ribbon allow you to see exact figures and test out different weights. In this case it would take a substantial change in weights for a different option to become more favoured and the decision can be considered robust.

This needs to be done for all criteria to determine whether the result is sensitive to a small change in one of them.

Future blogs will consider:

• How to use scales other than preference scales, for difficult to measure criteria
• Integrating cost criteria into weighting using swing weights, and
• Considering additional uncertainties in scores and value mapping.