Decision-Making

November 23, 2012

Question 2: Makingfirecrackers

A firecracker factory can produce up to 70 000 firecrackers. The materials needed to produce 10 000 firecrackers cost 100 000 yuan. You need at least two workers to operate the plant, and for every 10 000 firecrackers beyond the basic production of 30 000, you need to hire another worker.

You can sell firecrackers for 72 yuan each. But customers will only buy firecrackers in the month before New Year, and the total number you can sell is somewhere between 25 000 and 50 000. You can store unsold firecrackers for sale in subsequent years, but it will cost 200 000yuan/year to store each 10 000 firecrackers. It is not possible to wait till next New Year's to find how many you can sell. If you decide to store some firecrackers for a year, you will pay the storage costs at the beginning of that year. Your MARR is 20%.

Do a scenario analysis for various levels of production and various levels of customer demand. On the basis of this analysis, you have several choices to make. First, should you buy the production apparatus at all? Secondly, if you do buy it, how many firecrackers should you produce?

Suppose a market research study would allow you to find out exactly how many firecrackers the customers would buy each year. What would it be worth paying for the results of such a study?

First, should you buy the production apparatus at all?Secondly, if you do buy it, how many firecrackers should you produce?

Suppose a market research study would allow you tofind out exactly how many firecrackers the customers would buy each year. What would it be worth paying for the results of such a study?

Total MadeFixed cost

Labour cost

Material cost

Total Present Cost

Cost/Cracker

Cost/Cracker, Year2

Cost/Cracker, Year3

Unit Price 1

Unit Price 2

Unit Price 3

Profit1

Profit2

Profit 3

1 1 000 200 100 1 300 1 300 1 467 1 606 600 500 417 -700 -967-

1189

2 1 000 200 200 1 400 700 867 1 006 600 500 417 -100 -367 -589

3 1 000 200 300 1 500 500 667 806 600 500 417 100 -167 -389

4 1 000 300 400 1 700 425 592 731 600 500 417 175 - 92 -314

5 1 000 400 500 1 900 380 547 686 600 500 417 220 - 47 -269

6 1 000 500 600 2 100 350 517 656 600 500 417 250 - 17 -239

7 1 000 600 700 2 300 329 495 634 600 500 417 271 5 -217

Demand (10 000): 2.5 3.0 3.5 4.0 4.5 5.0 Average

Total Made

2 - 200 - 200 - 200 - 200 - 200 - 200 -200

3 167 300 300 300 300 300 278

4 300 433 567 700 700 700 567

5 433 567 700 833 967 1 100 767

6 344 700 833 967 1 100 1 233 863

7 256 611 967 1 100 1 233 1 367 922

First, should you buy the production apparatus at all?Secondly, if you do buy it, how many firecrackers should you produce?

Suppose a market research study would allow you tofind out exactly how many firecrackers the customers would buy each year. What would it be worth paying for the results of such a study?

Given these results, what do we decide to do?

We can apply the ideas of game theory, treating our choice ofproduction level as our move in a game, and the level of demandas the opponent’s response.

Theory of Games and Economic Behavior, John von Neumannand Oskar Morgernstern, 1944

The Minimax Strategy: Assume things will turn out for the worst,plan to minimize your losses (or maximize your gains)under these circumstances.

The Maximax Strategy: Assume things will turn out for the best,plan to minimize your losses (or maximize your gains)under these circumstances.

The Principle of Minimax Regret

…also known as the Savage Principle, is based on the psychologically plausible premise that, if you make a plan based on the assumption that A will happen, and then B happens instead, you will regret losing the benefits you could have had if you'd been smart enough to guess right. So you adopt the strategy of minimizing the regret you might otherwise be obliged to feel.

To apply this, you first construct a regret matrix…

Demand 2.5 3.0 3.5 4.0

Total Made

5 0 133 267 267

6 89 0 134 133

7 177 89 0 0

The Laplace Principle, or Principle of Insufficient Reason:

In the absence of information to the contrary, assume all outcomes areequally likely, and choose the one with the highest expected value.

Qualitative Analysis

How to take non-monetary factors into accountin decision-making

Argument: the only way non-monetary factors should influence decision-making is as a constraint:

``We’re not going to use baby seals as araw material, however profitable it is.’’

If, on the contrary, we allow trade-offs between our non-monetary factors and cost, then there’s an implicit

rate of exchange between the factor and a dollar amount,and we can go back to optimizing the final dollar amount.

…but in fact we don’t usually make decisions this way.Consider the question of getting married, for example.

Accepting that we are going to base our decisions on multiplecriteria that can’t always be traded off against each other, thereare some ways of simplifying the problem:

Suppose we have N criteria and that A and B are two possible courses of action. Then if:

Criterioni(A ) ≥ Criterioni(B) for 1 ≤ i ≤ N

we say A dominates B, and we can eliminate B from our listof possible actions.

If A is not dominated by any other course of action, A isefficient. So a first step is to reduce our alternatives to an efficient set.

Example: which choices are dominated with respectto the two criteria of hard-working and intelligent?

HW

IQ

Bob

Andy

Mary

Angus

Sally

Peter

Amy

Jill

Moe

Once we have eliminated the dominated choices, weare left with an efficient set.

HW

IQ

Bob

Mary

Sally

Decision matrices:

Having reduced the problem to an efficient set,we can give each option a weighted score againsta number of different criteria, thus forming adecision matrix.

Algorithm for setting up decision matrices:

1. Pick your criteria (e.g., industriousness, intelligence, charisma)

2. Give each a weight to indicate its importance. Let the weights sum to 10

3. Rank each alternative (Bob, Sally and Mary) against each criterion, with 10 being the highest.

4. Find the weighted totals for each candidate and pick the highest.

5. Do sensitivity analysis to see if the decision will change with minor changes in our preferences.

Criterion Weight Bob Sally Mary

Industry 5 1 3 7

Intelligence

3 5 7 9

Charisma 2 9 7 2

Score 10 38 50 66

How do we get the individual scores?

Individual scores can be obtained by normalization or by subjective evaluation.

For example, to get a 1-10 score for intelligence, we couldtake the individual’s IQ score and divide by 20.

This normalises with respect to the total population, butwe might also normalise with respect to the candidatepopulation.

For subjective evaluation, we might want to average theevaluations from two or three evaluators.

A variant on the decision matrix is to look at the productof the individual scores as well as the sum. What is the effect of this, and why might we do it?

Criterion Bob Sally Mary

Industry 1 3 7

Intelligence 5 7 9

Charisma 9 7 2

Sum 15 17 18

Product 45 147 126

Some observations about decision-making:

It’s easier to make pairwise comparisons:

``Is Andy smarter than Bob?’’

versus

``How smart is Andy?’’

Some observations about decision-making:

It’s easier to compare specific criteria:

``Is Andy taller than Bob?’’

versus

``Is Andy a better person than Bob?’’

From these observations comes the Analytic Hierarchy Process:

Make pairwise comparisons between candidates with respectto a criterion.

Convert the results of these comparisons into numerical scores

Make pairwise comparisons between the strength of the criteria

Convert the results of these comparisons into numerical scores

Combine the numerical scores to get an overall ranking of candidates

Apply sensitivity analysis

Example: choice of an electricity-generating technology for BC:

Criteria: Cost, Carbon Emissions, Unsightliness, Risk

Candidates: Hydro, nuclear, natural gas, oil, coal.

Pairwise comparison: How does hydro compare to coal with respect to price?

The preferred candidate gets a score of n, 1 ≤ n ≤ 9

The other candidate gets a score of 1/n

In this case, Hydro gets a `2’ compared to coal.

We record this in a pairwise comparison matrix.

Hydro Coal-fired

Hydro 1 2

Coal-fired 1/2 1

Comparison with respect to cost

Carry on and complete the matrix

Comparison with respect to cost

Hydro Nuclear Natural Gas

Oil Coal

Hydro 1 3 2 4 2

Nuclear 1/3 1 1/2 2 1

Natural Gas

1/2 2 1 2 1

Oil 1/4 1/2 1/2 1 1/2

Coal 1/2 1 1 2 1

Next, normalize the matrix: sum each column, then divide eachentry by its column sum.

Comparison with respect to cost

Hydro Nuclear Natural Gas

Oil Coal

Hydro 1 0.387 3 2 4 2

Nuclear 1/3 0.129

1 1/2 2 1

Natural Gas

½ 0.193 2 1 2 1

Oil ¼ 0.097 1/2 1/2 1 1/2

Coal ½ 0.193 1 1 2 1

Sums 2.583 1

Next, normalize the matrix: sum each column, then divide eachentry by its column sum.

Comparison with respect to cost

Hydro NuclearNatural Gas

Oil Coal

Hydro 0.387 0.400 0.400 0.364 0.364

Nuclear0.129 0.133 0.100 0.182 0.182

Natural Gas

0.193 0.267 0.200 0.182 0.182

Oil 0.97 0.067 0.100 0.091 0.091

Coal 0.193 0.133 0.200 0.182 0.182

Now work out the average value of each row:

Comparison with respect to cost

CostAvg.

0.383

0.145

0.205

0.089

0.178

Hydro NuclearNatural Gas

Oil Coal

Hydro 0.387 0.400 0.400 0.364 0.364

Nuclear0.129 0.133 0.100 0.182 0.182

Natural Gas

0.193 0.267 0.200 0.182 0.182

Oil 0.97 0.067 0.100 0.091 0.091

Coal 0.193 0.133 0.200 0.182 0.182

Go through the same process for each criterion to create a priority matrix.

Priority Matrix

CostAvg.

0.383

0.145

0.205

0.089

0.178

CarbonAvg.

0.256

0.177

0.149

0.212

0.251

Generation

Hydro

Nuclear

Gas

Oil

Coal

SightlinessAvg.

0.166

0.067

0.319

0.112

0.311

RiskAvg.

0.066

0.237

0.209

0.322

0.111

Cost Carbon Unsightliness Risk

Cost 1 2 4 1

Carbon ½ 1 2 ½

Unsightliness ¼ ½ 1 ¼

Risk 1 2 4 1

We ask, ``How much more important is cost than risk?’’, and similar questions, to create a pairwise comparisonmatrix for the criteria themselves.

Cost Carbon Unsightliness

Risk

Cost 1 0.364

2 4 1

Carbon ½ 0.182

1 2 ½

Unsightliness ¼ 0.091

½ 1 ¼

Risk 1 0.364

2 4 1

Then we normalise the columns and average the rows to createa vector of preferences.

Column Total 2.75 1

Criterion Average

0.412

0.123

0.098

0.435

Now we multiply the priority matrix by the preference vector...

CriterionAverage

0.412

0.123

0.098

0.435

CostAvg.

0.383

0.145

0.205

0.089

0.178

Generation

Hydro

Nuclear

Gas

Oil

Coal

CarbonAvg.

0.256

0.177

0.149

0.212

0.251

SightlinessAvg.

0.166

0.067

0.319

0.112

0.311

RiskAvg.

0.066

0.237

0.209

0.322

0.111

…and the result gives us our final preferences.

CriterionAverage

0.412

0.123

0.098

0.435

CostAvg.

0.383

0.145

0.205

0.089

0.178

Weighted score

Hydro 0.234

Nuclear 0.191

Gas 0.224

Oil 0.214

Coal 0.183

CarbonAvg.

0.256

0.177

0.149

0.212

0.251

SightlinessAvg.

0.166

0.067

0.319

0.112

0.311

RiskAvg.

0.066

0.237

0.209

0.322

0.111

× =

It is psychologically possible for a person to prefer coffee to hot chocolate, hot chocolate to tea, and tea to coffee.

Such people do not make good subjects for this procedure.Fortunately, there is a test to eliminate them.

If the decision-maker is consistent, each column inthe pairwise-comparison matrix will be a multipleof every other.

If this is the case, the PCM will have a single non-zeroeigenvalue, λ. And λ ≈ n, where n is the rank of the matrix.

Coal Oil Gas

Coal 1 2 4

Oil 0.5 1 2

Gas 0.25 0.5 1

(not λ > n as in the text)

So to determine whether the decision-maker is consistent, calculate the eigenvalues of the PCM and find the largest,λmax.

Then calculate (λmax-n)/(n-1), the consistency index, or CI.

If the decision-maker is perfectly consistent, this shouldbe zero.

Compare this with the same statistic for a random matrix.These have been helpfully tabulated by previous researchers.

Compare this with the same statistic for a random matrix.These have been helpfully tabulated by previous researchers.

Size of Matrix Random Index (RI)

2 0

3 0.58

4 0.90

5 1.12

6 1.24

7 1.32

8 1.41

9 1.45

10 1.49

Now compare CI for the PCM with RI for a randommatrix.

If CI/RI < 0.1

then the PCM is acceptably consistent.