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Multi-objectiveDecision MakingDR MRINALINI SHAH

PROFESSOR

IMT GHAZIABAD

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Multiple Objective programming

When there are multiple objectives, you can proceed in several fundamentalways.

First, you can prioritize your objectives. This is done in goal programming,where the highest priority objective is optimized first, then the second, and so

on.

Second, you can optimize one objective while constraining the others to beno worse than specified values. This approach is used to find trade-off curvesbetween the objectives.

Finally, you can attempt to weight the objectives to measure their importancerelative to one another. This is the approach taken by the Analytic HierarchyProcess.

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GoalProgramming

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Goal Programming

Firms usually have more than one goal. For example,maximizing total profit,

maximizing market share,

maintaining full employment,

providing quality ecological management,minimizing noise level in the neighborhood, and

meeting numerous other non-economic goals.

It is not possible for LP to have multiple goals unless they are all measured in

the same units (such as dollars), a highly unusual situation.

An important technique that has been developed to supplement LP is calledgoal programming.

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Goal ProgrammingGoal programming satisfies,as opposed to LP, which tries to optimize.

Satisfice means coming as close as possible to reaching goals.

The objective function is the main difference betweengoal programming and LP.

In goal programming, the purpose is to minimizedeviational variables,

which are the only terms in the objective function.

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Media Planning of Ad agencyThe Uncommon sense Ad Agency is trying to determine a TV advertising schedule for aclient. The client has three goals (listed here in descending order of importance)concerning

whom it wants its ads to be seen by:

Goal 1: at least 65 million high-income men (HIM)

Goal 2: at least 72 million high-income women (HIW)

Goal 3: at least 70 million low-income people (LIP)

Uncommon sense can purchase several types of TV ads: ads shown on live sports shows,

on game shows, on news shows, on sitcoms, on dramas, and on soap operas. At most INR700,000 total can be spent on ads. The advertising costs and potential audiences (inmillions of viewers) of a one-minute ad of each type are shown in Table. As a matter ofpolicy, the client requires that at least two ads each be placed on sports shows, newsshows, and dramas. Also, it requires that no more than 10 ads be placed on any single typeof show. Uncommon sense wants to find the advertising plan that best meets its clients

goals.

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sports

show game sow news sitcom drama

soap

opera

HIM 7 3 6 4 6 3

HIW 4 5 5 5 8 4

LIP 8 6 3 7 6 5

cost 120000 40000 50000 40000 60000 40000

Number of ads are 2, 1, 2, 1, 5, 1 respectively and cost is 760000 INR.

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Methods to solve Goal programming

Weighted Goal method

Ranked or Priority Goal Method

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Weighted Goal methodDifferent Goal are given different weight.Minimize total weighted underachievement of goals

Drawback of the method:

Is appropriate only if all the goals are measured in same unit

It is not always easy to assign suitable weights for the differentdeviation variables.

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Ranked Goal MethodRank the goals rather then weight the goalMinimize the ranked deviations as objective function.

Solve the model for Rank 1 goal first.

Use the result , to solve the model for Rank 2 goal .

Repeat the process.

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GOAL programming: algorithm

1. Identify the decision variable in the problem

2. Identify any hard constraints in the problem

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Do It YourselfLucys Music Store at present employs five full-time employees and three part-time employees. The normal workload is 40 hours per week for full-timeemployees and 20 hours per week for part-time employees. Each full-timeemployee is paid $6 per hour for work up to 40 hours per week and can sell fiverecordings per hour. A full-time employee who works overtime is paid $10 perhour. Each part-time employee is paid $3 per hour and can sell three recordings

per hour. It costs Lucy $6 to buy a recording, and each recording sells for $9. Lucyhas weekly fixed expenses of $500. She has established the following weeklygoals, in order of priority:

Goal 1: Sell at least 1600 recordings per week.

Goal 2: Earn a profit of at least $2200 per week.

Goal 3: Full-time employees should work at most 100 hours of overtime.

Goal 4: To promote a sense of job security, the number of hours by which eachfull-time employee fails to work 40 hours should be minimized.

Use a goal programming model to determine how many hours per week eachemployee should work?

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XYZ Electric Company

The Company produces two products popular with home renovators: old-fashioned chandeliers and ceiling fans. Both the chandeliers and fans require atwo-step production process involving wiring and assembly. It takes about 2 hoursto wire each chandelier and 3 hours to wire a ceiling fan. Final assembly of thechandeliers and fans requires 6 and 5 hours, respectively. The production

capability is such that only 12 hours of wiring time and 30 hours of assembly timeare available. The profit is Rs 6 for Chandeliers and Rs 7 for Fans.

Goals Harrisons management wants to achieve, each equal in priority:

Goal 1: to produce as much profit above $30 as possible during the productionperiod.

Goal 2: to fully utilize the available wiring department hours.

Goal 3: limit overtime in the assembly department to 10 hours.

Goal 4: to meet a contract requirement to produce at least seven ceiling fans.

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maximize profit = $7X

1 + $6X

2

subject to: 2X1 + 3X2 12 (wiring hours)

6X1 + 5X2 30 (assembly hours)

X1, X2 0 (nonnegative)

X1 = number of chandeliers producedX2 = number of ceiling fans produced

XYZ Electric company:

If each chandelier produced nets the firm $7 and each fan $6, Harrisonsproduction mix decision can be formulated using LP as follows:

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Harrison Electric Revisited

Need a clear definition of deviational variables,

such as :d1

= underachievement of the profit target

d1+ = overachievement of the profit target

d2= idle time in the wiring dept. (underused)

d2+ = overtime in the wiring dept. (overused)

d3= idle time in the assembly dept. (underused)

d3+ = overtime in the wiring dept. (overused)d4

= underachievement of the ceiling fan goal

d4+ = overachievement of the ceiling fan goal

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Stockco

Stockco produces two products. Product 1 requires 4 labour hours andproduct 2, requires 2 hours . revenue contribution of product 1 and 2 is $4and $2 respectively. Stcokco has a goal of $48 as weekly revenue and incur$1 penalty for each dollar it falls short of this goal. A total of 32 hours oflabor available. A $2 penalty incurred for each hour of overtime and a $1penalty incurs for each hour of labour unused. Marketing considerationrequires at least 7 units of product 1, and at least 10 units of product 2 tobe produced. For each unit by which production falls short of demand $5penalty is assessed. Determine how to minimize total penalty.

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Goal 1: do not use any overtime .

Goal 2: meet demand for product1

Goal 3: Meet demand for product 2

Goal 4: Avoid underuse of labour.

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Analytic HierarchyProcess (AHP)

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The Analytic Hierarchy Process (AHP), developed originally by ThomasSaaty, is a powerful tool that can be used to make decisions in situationswhere multiple objectives are present.

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You will graduate in few months time. You are planning to chooseamong the offers by determining how well each job offer meets thefollowing four objectives:

Objective 1: High starting salary

Objective 2: Quality of life in city where job is located

Objective 3: Interest of work

Objective 4: Nearness of job to family

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Applications of AHPAHP in Saudi Arabia

Bahurmoz (2003) designed and implemented a system based on AHP toselect the best candidates to send overseas to do graduate studies and

eventually become teachers at the Dar Al-Hekma womens college inSaudi Arabia.

AHP has been used by companies in many areas, including accounting,finance, marketing, energy resource planning, microcomputer selection,sociology, architecture, and political science. See Zahedi (1986), Goldenet al. (1989), and Saaty (1988) for a discussion of applications of AHP.

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How to build AHP ModelForm a pairwise comparison matrix ( Lets call A). The entry in row iand columnjofA, aij, indicates how much more (or less) important objective iis thanobjectivej to the decision maker. Importance is measured on an integer-valuedscale from 1 to 9, with each number having the interpretation:

Value of ai j Interpretation 1 Objectives iandj are equally important.

3 Objective iis slightly more important thanj.

5 Objective iis strongly more important thanj.

7 Objective iis very strongly more important thanj.

9 Objective iis absolutely more important thanj.

For consistency, it is necessary to set aji= 1/ aij.

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How to build AHP Model1. Obtains a vector of weights that summarizes the relative importance ofthe objectives. Determine weights of each objective as:

a). Create Normalised matrix. Divide each element of matrix by sum ofthe corresponding column.

b). Find weight by averaging the corresponding row of Norm Matrix.( By convention, the weights are always chosen so that they sum to 1.)

2. This matrix measures how well each job compares to other jobs withregard to this objective. For each matrixAi, obtain a vector of scores Si thatsummarizes how the jobs compare in terms of achieving objective iDeterminethe score of each decision alternative on each objective using step 2 a & b.

3. The final step is to combine the scores in the Si vectors with the weightsin the w vector. If you form a matrix S of these score vectors and multiply thismatrix by w, you obtain a vector of overall scores for each job. Determine thebest alternative. assesses a pairwise comparison matrixAi for each objective i.

.

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MO ELING ISSUES

1. AHP can handle a hierarchy of objectives and sub-objectiveshencethe term hierarchy in the name of the procedure.

2. Other than spreadsheet, you can use special-purpose software suchas Expert Choice, for AHP in more complicated situation.

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Thank YouDR MRINALINI SHAH