McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.1 GOAL PROGRAMMING 1. Opprettholde stabil profitt 2. Øke eller opprettholde gitte markedsandeler

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 200311.1

GOAL PROGRAMMING

1. Opprettholde stabil profitt2. Øke eller opprettholde gitte markedsandeler3. Diversifisering av produkter4. Holde arbeidsstokken på et gitt nivå5. Forurense minst mulig

• Målene kan ofte ikke sammenlignes eller kombineres direkte

• Ulike mål er ofte i konflikt med hverandre



Two approaches

• Weighted goal programming- goals are roughly comparable

• Preemptive goal programming – Hierarchy of priority levels for the different goals


MULTIPLE OBJECTIVESMULTIPLE OBJECTIVESIn many applications, the planner has more than one In many applications, the planner has more than one objective. The presence of multiple objectives is objective. The presence of multiple objectives is frequently referred to as the problem of “combining frequently referred to as the problem of “combining apples and oranges.”apples and oranges.”Consider a corporate planner whose long-range Consider a corporate planner whose long-range goals are to:goals are to:

1.1. Maximize discounted profits Maximize discounted profits

2.2. Maximize market share at the end of the Maximize market share at the end of the planning periodplanning period

3.3. Maximize existing physical capital at the end Maximize existing physical capital at the end of the planning periodof the planning period


It is also clear that the goals are It is also clear that the goals are conflictingconflicting (i.e., (i.e., there are trade-offs in the sense that sacrificing the there are trade-offs in the sense that sacrificing the requirements on any one goal will tend to produce requirements on any one goal will tend to produce greater returns on the others. greater returns on the others.

These goals are not commensurate (i.e., they cannot These goals are not commensurate (i.e., they cannot be be directlydirectly combined or compared). combined or compared).

These models, although not applied as often in These models, although not applied as often in practice as some of the other models (such as linear practice as some of the other models (such as linear programming, forecasting, inventory control, etc.), programming, forecasting, inventory control, etc.), have been found to be especially useful on problems have been found to be especially useful on problems in the public sector.in the public sector.


Several approaches to multiple objective models Several approaches to multiple objective models (also called multi-criteria decision making) have (also called multi-criteria decision making) have been developed:been developed:

Multi-attribute utility theoryMulti-attribute utility theory

Only Goal Programming will be discussed.Only Goal Programming will be discussed.

Developed by Thomas Saaty, AHP helps Developed by Thomas Saaty, AHP helps managers choose between many decision managers choose between many decision alternatives on the basis of multiple criteria.alternatives on the basis of multiple criteria.

Search for Pareto optimal solutions via Search for Pareto optimal solutions via multi-criteria linear programmingmulti-criteria linear programming

Analytic Hierarchy Process (AHP)Analytic Hierarchy Process (AHP)

Goal Programming (GP)Goal Programming (GP)Introduced by A. Charnes and W.W. Cooper. Introduced by A. Charnes and W.W. Cooper. GP is a heuristic approach to the multiple-GP is a heuristic approach to the multiple-objectives model.objectives model.


Goal ProgrammingGoal Programming is an extension of Linear is an extension of Linear Programming that enables the planner to come as Programming that enables the planner to come as close as possible to satisfying various goals and close as possible to satisfying various goals and constraints.constraints.

GOAL PROGRAMMINGGOAL PROGRAMMING

It allows the decision maker, at least in a heuristic It allows the decision maker, at least in a heuristic sense, to incorporate his or her preference system sense, to incorporate his or her preference system in dealing with multiple conflicting goals. in dealing with multiple conflicting goals. GP is sometimes considered to be an attempt to put GP is sometimes considered to be an attempt to put into a mathematical programming context, the into a mathematical programming context, the concept of concept of satisficingsatisficing..

Coined by Herbert Simon, it communicates the idea Coined by Herbert Simon, it communicates the idea that individuals often do not seek optimal solutions, that individuals often do not seek optimal solutions, but rather solutions that are “good enough” or but rather solutions that are “good enough” or “close enough.”“close enough.”


Weighted Goal Programming

• A common characteristic of many management science models (linear programming, integer programming, nonlinear programming) is that they have a single objective function.

• It is not always possible to fit all managerial objectives into a single objective function. Managerial objectives might include:

– Maintain stable profits.

– Increase market share.

– Diversify the product line.

– Maintain stable prices.

– Improve worker morale.

– Maintain family control of the business.

– Increase company prestige.

• Weighted goal programming provides a way of striving toward several objectives simultaneously.


Weighted Goal Programming

• With weighted goal programming, the objective is to– Minimize W = weighted sum of deviations from the goals.

– The weights are the penalty weights for missing the goal.

• Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal.

• The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints:

Level Achieved – Amount Over + Amount Under = Goal


The Dewright Company

• The Dewright Company is one of the largest producers of power tools in the United States.

• The company is preparing to replace its current product line with the next generation of products—three new power tools.

• Management needs to determine the mix of the company’s three new products to best meet the following three goals:

1. Achieve a total profit (net present value) of at least $125 million.

2. Maintain the current employment level of 4,000 employees.

3. Hold the capital investment down to no more than $55 million.


Data for Contribution to the Goals

Unit Contribution of Product

Factor 1 2 3 Goal

Total profit (millions of dollars) 12 9 15 ≥ 125

Employment level (hundreds of employees) 5 3 4 = 40

Capital investment (millions of dollars) 5 7 8 ≤ 55


Penalty Weights

Goal Factor Penalty Weight for Missing Goal

1 Total profit 5 (per $1 million under the goal)

2 Employment level 4 (per 100 employees under the goal)2 (per 100 employees over the goal)

3 Capital investment 3 (per $1 million over the goal)


Data for Contribution to the Goals

Unit Contribution of Product

Factor 1 2 3 Goal

Total profit (millions of dollars) 12 9 15 ≥ 125

Employment level (hundreds of employees) 5 3 4 = 40

Capital investment (millions of dollars) 5 7 8 ≤ 55


Weighted Goal Programming Formulation forthe Dewright Co. Problem

Let Pi = Number of units of product i to produce per day (i = 1, 2, 3),Under Goal i = Amount under goal i (i = 1, 2, 3),Over Goal i = Amount over goal i (i = 1, 2, 3),

Minimize W = 5(Under Goal 1) + 2Over Goal 2) + 4 (Under Goal 2) + 3 (Over Goal 3)subject to

Level Achieved Deviations GoalGoal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125

Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40

Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 55

andPi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)


Weighted Goal Programming Spreadsheet

3456789101112131415

B C D E F G H I J K L M N OGoals

Contribution per Unit Produced Level Amount Amount BalanceProduct 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) Goal

Goal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 48.333333 = 40 8.333333 0 40 = 40Goal 3 (Investment) 5 7 8 55 <= 55 0 0 55 = 55

Product 1 Product 2 Product 3 Penalty Over Under Weighted SumUnits Produced 8.33333333 0 1.66666667 Weights Goal Goal of Deviations

Profit 5 16.66666667Employment 2 4Investment 3

Deviations Constraints


Suppose that we have an educational program Suppose that we have an educational program design model with decision variables design model with decision variables xx11 and and xx22, ,

wherewherexx11 is the hours of classroom work is the hours of classroom work

xx22 is the hours of laboratory work is the hours of laboratory work

Assume the following constraint on total program Assume the following constraint on total program hours:hours:

xx11 + + xx22 << 100 (total program hours) 100 (total program hours)

Two Kinds of ConstraintsTwo Kinds of Constraints In the goal programming In the goal programming approach, there are two kinds of constraints:approach, there are two kinds of constraints:

1.1. System constraintsSystem constraints (so-called hard (so-called hard constraints) that cannot be violated.constraints) that cannot be violated.

2.2. Goal constraintsGoal constraints (so-called soft constraints) (so-called soft constraints) that may be violated if necessary.that may be violated if necessary.


Now, suppose that each hour of classroom work Now, suppose that each hour of classroom work involvesinvolves

12 minutes of small-group experience and12 minutes of small-group experience and

19 minutes of individual problem solving19 minutes of individual problem solving

Each hour of laboratory work involvesEach hour of laboratory work involves29 minutes of small-group experience and29 minutes of small-group experience and

11 minutes of individual problem solving11 minutes of individual problem solving

The total program time is at most 6,000 minutes The total program time is at most 6,000 minutes (100 hr * 60 min/hr).(100 hr * 60 min/hr).

There are two goals: Each student should spend as There are two goals: Each student should spend as close as possible to close as possible to

¼ of the maximum program time working in ¼ of the maximum program time working in small groups and small groups and

¹/¹/33 of the time on problem solving. of the time on problem solving.


These conditions are:These conditions are:

1212xx11 + 29 + 29xx22 1500 (small-group experience) 1500 (small-group experience)~~==

1919xx11 + 11 + 11xx22 2000 (individual problem solving) 2000 (individual problem solving)~~==Where means that the left-hand side is desired to Where means that the left-hand side is desired to be “as close as possible” to the right-hand side.be “as close as possible” to the right-hand side.

~~==

In order to satisfy the system constraint, at least one In order to satisfy the system constraint, at least one of the two goals will be violated.of the two goals will be violated.

To implement the goal programming approach, the To implement the goal programming approach, the small-group experience condition is rewritten as the small-group experience condition is rewritten as the goal constraint:goal constraint:

1212xx11 + 29 + 29xx22 + + uu11 – – vv11 = 1500 ( = 1500 (uu11 >> 0, 0, vv11 >> 0) 0)Where Where uu11 = the amount by which total small-group = the amount by which total small-group

experience falls short of 1500 experience falls short of 1500vv11 = the amount by which total small-group = the amount by which total small-group

experience exceeds 1500 experience exceeds 1500


Deviation VariablesDeviation Variables Variables Variables uu11 and and vv11 are called are called

deviation variables deviation variables since they measure the amount since they measure the amount by which the value produced by the solution by which the value produced by the solution deviates from the goal.deviates from the goal.

Note that by definition, we want either Note that by definition, we want either uu11 or or vv11 (or (or

both) to be zero because it is impossible to both) to be zero because it is impossible to simultaneously exceed and fall short of simultaneously exceed and fall short of 15001500..

In order to make In order to make 1212xx11 + 29 + 29xx22 as close as possible to as close as possible to

15001500, it suffices to make the sum , it suffices to make the sum uu11 + + vv11 small.small.

The individual problem-solving condition is written The individual problem-solving condition is written as the goal constraint:as the goal constraint:

1919xx11 + 11 + 11xx22 + + uu22 – – vv22 = 2000 ( = 2000 (uu22 >> 0, 0, vv22 >> 0) 0)

As before, the sum of As before, the sum of uu22 + + vv22 should be small.should be small.


The complete (illustrative) model is:The complete (illustrative) model is:

1212xx11 + 29 + 29xx22 + + uu11 – – vv11 = 1500 (small-group experience) = 1500 (small-group experience)

1919xx11 + 11 + 11xx22 + + uu22 – – vv22 = 2000 (problem solving) = 2000 (problem solving)

s.t. s.t. xx11 + + xx2 2 << 100 (total program hours) 100 (total program hours)

xx11, , xx2 2 , , uu11, , vv11, , uu22, , vv22 >> 0 0

Now this is an ordinary LP model and can be easily Now this is an ordinary LP model and can be easily solved in Excel. The optimal decision variables will solved in Excel. The optimal decision variables will satisfy the system constraint (total program hours).satisfy the system constraint (total program hours).

Min Min uu11 + + vv1 1 ++ uu22 + + vv22

Note:Note: Both Both uu11 and and vv1 1 can be 0can be 0


Solver will guarantee that either Solver will guarantee that either uu11 or or vv11 (or both) will (or both) will

be zero, and thus these variables automatically be zero, and thus these variables automatically satisfy this desired condition. satisfy this desired condition.

Note that the objective function is the sum of the Note that the objective function is the sum of the deviation variables. deviation variables.

This choice of an objective function indicates that This choice of an objective function indicates that there is no preference among the various deviations there is no preference among the various deviations from the stated goals. from the stated goals.

The same statement holds for The same statement holds for uu22 and and vv22 and in and in

general for any pair of deviation variables.general for any pair of deviation variables.


For example, any of the following three decisions is For example, any of the following three decisions is acceptable:acceptable:

1.1. A decision that overachieves the group A decision that overachieves the group experience goal by 5 minutes and hits the experience goal by 5 minutes and hits the problem-solving goal exactly,problem-solving goal exactly,

2.2. A decision that hits the group experience goal A decision that hits the group experience goal exactly and underachieves the problem-exactly and underachieves the problem-solving goal by 5 minutes, andsolving goal by 5 minutes, and

3.3. A decision that underachieves each goal by A decision that underachieves each goal by 2.5 minutes.2.5 minutes.


There is no preference among the following three There is no preference among the following three solutions because each of these yields the same solutions because each of these yields the same value (i.e., 5) for the objective function.value (i.e., 5) for the objective function.

uu11 = 0 = 0

vv1 1 = 5= 5

uu22 = 0 = 0

vv2 2 = 0= 0

(1)(1) uu11 = 0 = 0

vv1 1 = 0= 0

uu22 = 5 = 5

vv2 2 = 0= 0

(2)(2) uu11 = 2.5 = 2.5

vv1 1 = 0= 0

uu22 = 2.5 = 2.5

vv2 2 = 0= 0

(3)(3)


Weighting the Deviation VariablesWeighting the Deviation Variables Differences in Differences in units alone could produce a preference among the units alone could produce a preference among the deviation variables. deviation variables.

One way of expressing a preference among the One way of expressing a preference among the various goals is to assign different coefficients various goals is to assign different coefficients (weights) to the deviation variables in the objective (weights) to the deviation variables in the objective function. function.

as the objective function. Since as the objective function. Since vv22 (over- (over-

achievement of problem solving) has the smallest achievement of problem solving) has the smallest coefficient, the program designers would rather coefficient, the program designers would rather have have vv22 positive than any of the other deviation positive than any of the other deviation

variables (positive variables (positive vv22 is penalized the least). is penalized the least).

Min 10Min 10uu11 + 2 + 2vv11 + 20 + 20uu22 + + vv22

In the program-planning example, one might selectIn the program-planning example, one might select


With this objective function it is better to be With this objective function it is better to be 99 minutes over the problem-solving goal than to minutes over the problem-solving goal than to underachieve by underachieve by 11 minute the small-group- minute the small-group-experience goal.experience goal.

To see this, note that for any solution in which To see this, note that for any solution in which uu11 >> 1 1, decreasing , decreasing uu11 by by 11 and increasing and increasing vv22 by by 99

would yield a smaller value for the objective would yield a smaller value for the objective function.function.


Goal Interval ConstraintsGoal Interval Constraints Another type of goal Another type of goal constraint is called a constraint is called a goal interval constraintgoal interval constraint. .

Such a constraint restricts the goal to a range or Such a constraint restricts the goal to a range or interval interval rather than a specific numerical value. rather than a specific numerical value.

Suppose, for example, that in the previous Suppose, for example, that in the previous illustration the designers were indifferent among illustration the designers were indifferent among programs for which programs for which

1800 1800 << [minutes of individual problem solving] [minutes of individual problem solving] << 2100 2100

i.e.,i.e., 1800 1800 << 19 19xx11 + 11 + 11xx22 << 2100 2100

In this situation the interval goal is captured with In this situation the interval goal is captured with two goal constraints:two goal constraints:

1919xx11 + 11 + 11xx22 – – vv11 << 2100 ( 2100 (vv11 >> 0) 0)

1919xx11 + 11 + 11xx22 + + uu11 >> 1800 ( 1800 (uu11 >> 0) 0)


When the terms When the terms uu11 and and vv11 are included in the are included in the

objective function, the LP code will attempt to objective function, the LP code will attempt to minimize them. minimize them. Summary of the Use of Goal ConstraintsSummary of the Use of Goal Constraints Each goal Each goal constraint consists of a left-hand side, say constraint consists of a left-hand side, say ggii((xx11, …, , …, xxnn)), and a right-hand side, , and a right-hand side, bbii. .

Goal constraints are written by using nonnegative Goal constraints are written by using nonnegative deviation variables deviation variables uuii, , vvii..

At optimality at least one of the pair At optimality at least one of the pair uuii, , vvii will always will always

be zero.be zero.uuii represents represents underachievementunderachievement; ; vvii represents represents

overachievementoverachievement..Whenever Whenever uuii is used it is is used it is addedadded to to ggii((xx11, …, , …, xxnn))..Whenever Whenever vvii is used it is is used it is subtractedsubtracted from from

ggii((xx11, …, , …, xxnn))..


Only deviation variables appear in the objective Only deviation variables appear in the objective function, and the objective is always to minimize.function, and the objective is always to minimize.

The decision variables The decision variables xxii, , i = 1, …, ni = 1, …, n do not appear in do not appear in

the objective.the objective.

Four types of goals have been discussed:Four types of goals have been discussed:

1.1. TargetTarget. Make . Make ggii((xx11, …, , …, xxnn) as close as possible ) as close as possible

as possible to as possible to bbii. To do this write the goal . To do this write the goal

constraint asconstraint as

ggii((xx11, …, , …, xxnn) + ) + uuii - - vvii = = bbi i ( (uuii >> 0, 0, vvii >> 0) 0)



2.2. Minimize UnderachievementMinimize Underachievement. To do this, write. To do this, write

and in the objective, minimize and in the objective, minimize uuii, the under-, the under-

achievement.achievement.

vvii does not appear in the objective function does not appear in the objective function

and it is only in this constraint, hence, the and it is only in this constraint, hence, the constraint can be equivalently written asconstraint can be equivalently written as

ggii((xx11, …, , …, xxnn) + ) + uuii >> bbi i ( (uuii >> 0) 0)

If the optimal If the optimal uuii is positive, this constraint will is positive, this constraint will

be active, for otherwise be active, for otherwise uuii** could be made could be made

smaller.smaller.If If uuii**>0>0 then, since then, since vvii** must equal zero, it must must equal zero, it must

be true that be true that ggii((xx11, …, , …, xxnn) + ) + uuii* = * = bbi i ..



3.3. Minimize OverachievementMinimize Overachievement. To do this, write. To do this, write

and in the objective, minimize and in the objective, minimize vvii, the over-, the over-

achievement.achievement.

uuii does not appear in the objective function, does not appear in the objective function,

the constraint can be equivalently written asthe constraint can be equivalently written as

ggii((xx11, …, , …, xxnn) - ) - vvii << bbi i ( (vvii >> 0) 0)

If the optimal If the optimal vvii is positive, this constraint will is positive, this constraint will

be active. The argument is analogous to that be active. The argument is analogous to that in item 2.in item 2.


4.4. Goal Interval ConstraintGoal Interval Constraint. In this instance, the . In this instance, the goal is to come as close as possible to goal is to come as close as possible to satisfying satisfying

aaii << g gii((xx11, …, , …, xxnn) ) << bbii

In order to write this as a goal, first “stretch In order to write this as a goal, first “stretch out” the interval by writingout” the interval by writing

aaii - u - uii << g gii((xx11, …, , …, xxnn) ) << bbi i + v+ vi i ((uuii >> 0, 0, vvii >> 0) 0)

which is equivalent to the two constraintswhich is equivalent to the two constraints

ggii((xx11, …, , …, xxnn) ) + u+ ui i >> aai i ggii((xx11, …, , …, xxnn) ) + u+ ui i - - vvii + + aai i ((uuii >> 0, 0, vvii >> 0) 0)^̂ ^̂

ggii((xx11, …, , …, xxnn) ) - u- ui i >> bbi i ggii((xx11, …, , …, xxnn) ) + u+ ui i - - vvii + + bbi i ((uuii >> 0, 0, vvii >> 0) 0)^̂ ^̂

The objective function The objective function uui i + + vvii is minimized. is minimized.

Variables Variables uui i andand vvii are merely surplus and are merely surplus and slack, respectively.slack, respectively.

^̂ ^̂


Weighted vs. Preemptive Goal Programming

• Weighted goal programming is designed for problems where all the goals are quite important, with only modest differences in importance that can be measured by assigning weights to the goals.

• Preemptive goal programming is used when there are major differences in the importance of the goals.

– The goals are liested in the order of their importance.

– It begins by focusing solely on the most important goal.

– It next does the same for the second most important goal (as is possible without hurting the first goal).

– It continues the the following goals (as is possible without hurting the previous more important goals).


In some cases, managers do not wish to express In some cases, managers do not wish to express their preferences among various goals in terms of their preferences among various goals in terms of weighted deviation variables, for the process of weighted deviation variables, for the process of assigning weights may seem too arbitrary or assigning weights may seem too arbitrary or subjective.subjective.

ABSOLUTE PRIORITIESABSOLUTE PRIORITIES

In such cases, it may be more acceptable to state In such cases, it may be more acceptable to state preferences in terms of preferences in terms of absolute prioritiesabsolute priorities (as (as opposed to weights) to a set of goals.opposed to weights) to a set of goals.

This approach requires that goals be satisfied in a This approach requires that goals be satisfied in a specific order. Therefore, the model is solved in specific order. Therefore, the model is solved in stages as a sequence of models.stages as a sequence of models.


Preemptive Goal Programming

• Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal.

• The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints:

Level Achieved – Amount Over + Amount Under = Goal

• Start with the objective of achieving the first goal (or coming as close as possible):

– Minimize (Amount Over/Under Goal 1)

• Continue with the next goal, but constrain the previous goals to not get any worse:

– Minimize (Amount Over/Under Goal 2)subject to

Amount Over/Under Goal 1 = (amount achieved in previous step)

• Repeat the previous step for all succeeding goals.


Preemptive Goal Programming for Dewright

The goals in the order of importance are:1. Achieve a total profit (net present value) of at least $125 million.

2. Avoid decreasing the employment level below 4,000 employees.

3. Hold the capital investment down to no more than $55 million.

4. Avoid increasing the employment level above 4,000 employees.

• Start with the objective of achieving the first goal (or coming as close as possible):

– Minimize (Under Goal 1)

• Then, if for example goal 1 is achieved (i.e., Under Goal 1 = 0), then– Minimize (Under Goal 2)

subject to(Under Goal 1) = 0


Preemptive Goal Programming Formulation forthe Dewright Co. Problem (Step 1)


Minimize (Under Goal 1)subject to

Level Achieved Deviations GoalGoal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 55





Minimize (Under Goal 2)subject to


(Under Goal 1) = (Level Achieved in Step 1)and

Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)




Minimize (Over Goal 3)subject to


(Under Goal 1) = (Level Achieved in Step 1)(Under Goal 2) = (Level Achieved in Step 2)





Minimize (Over Goal 2)subject to


(Under Goal 1) = (Level Achieved in Step 1)(Under Goal 2) = (Level Achieved in Step 2)(Over Goal 3) = (Level Achieved in Step 3)



Preemptive Goal Programming SpreadsheetStep 1: Minimize (Under Goal 1)

1

23456789101112

A B C D E F G H I J K L M N O

Dewright Co. Goal Programming (Preemptive Priority 1: Minimize Under Goal 1)

GoalsContribution per Unit Produced Level Amount Amount Balance

Product 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) GoalGoal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 40 = 40 0 0 40 = 40Goal 3 (Investment) 5 7 8 61.481 <= 55 6.481 0 55 = 55

Minimize (Under Goal 1)Product 1 Product 2 Product 3

Units Produced 3.7037 0 5.3704



Preemptive Goal Programming SpreadsheetStep 3: Minimize (Over Goal 3)

1

23456789101112


Dewright Co. Goal Programming (Preemptive Priority 3: Minimize Over Goal 3)


Product 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) GoalGoal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 48.333 = 40 8.333333 0 40 = 40Goal 3 (Investment) 5 7 8 55 <= 55 0 0 55 = 55

Minimize (Over Goal 3)Product 1 Product 2 Product 3 (Under Goal 1) = 0

Units Produced 8.333 0 1.667 (Under Goal 2) = 0



Preemptive Goal Programming SpreadsheetStep 4: Minimize (Over Goal 2)

1

2345678910111213


Dewright Co. Goal Programming (Preemptive Priority 4: Minimize Over Goal 2)


Product 1 Product 2 Product 3 Achieved Goal Over Under (Level - Over + Under) GoalGoal 1 (Profit) 12 9 15 125 >= 125 0 0 125 = 125Goal 2 (Employment) 5 3 4 48.333 = 40 8.333 0 40 = 40Goal 3 (Investment) 5 7 8 55 <= 55 0 0 55 = 55

Minimize (Over Goal 2)Product 1 Product 2 Product 3 (Under Goal 1) = 0

Units Produced 8.333 0 1.667 (Under Goal 2) = 0(Over Goal 3) = 0



Example: Swenson’s Media Selection ModelExample: Swenson’s Media Selection Model J. R. Swenson is an advertising agency which has J. R. Swenson is an advertising agency which has just completed an agreement with a pharmaceutical just completed an agreement with a pharmaceutical manufacturer to mount a radio and television manufacturer to mount a radio and television campaign to introduce a new product, Mylonal.campaign to introduce a new product, Mylonal.

The total expenditures for the campaign are not to The total expenditures for the campaign are not to exceed $120,000.exceed $120,000.

The client wants to reach several audiences, The client wants to reach several audiences, however, radio and television are not equally however, radio and television are not equally effective in reaching all audiences.effective in reaching all audiences.

Therefore, the agency will estimate the impact of the Therefore, the agency will estimate the impact of the advertisements in terms of rated exposures (i.e., advertisements in terms of rated exposures (i.e., “people reached per month”) on the audiences of “people reached per month”) on the audiences of interest.interest.


The following data represent the number of The following data represent the number of exposures per $1000 expenditure:exposures per $1000 expenditure:

TV RADIOTV RADIO

TotalTotal 14,00014,000 6,0006,000Upper IncomeUpper Income 1,200 1,200 1,2001,200

The following are the campaign goals, listed in order The following are the campaign goals, listed in order of absolute priority.of absolute priority.

1.1. Total exposures will hopefully be at least Total exposures will hopefully be at least 840,000.840,000.

2.2. In order to maintain effective contact with the In order to maintain effective contact with the leading radio station, no more than $90,000 leading radio station, no more than $90,000 will be spent on TV advertising.will be spent on TV advertising.


3.3. The campaign should achieve at least 168,000 The campaign should achieve at least 168,000 upper-income exposures.upper-income exposures.

4.4. If all other goals are satisfied, the total number If all other goals are satisfied, the total number of exposures would come as close as possible of exposures would come as close as possible to being maximized.to being maximized.Note that if all of the $120,000 is spent on TV Note that if all of the $120,000 is spent on TV advertising, then the maximum obtainable advertising, then the maximum obtainable exposures would be 1,680,000 (120*14,000).exposures would be 1,680,000 (120*14,000).

To model the problem, the following notation will be To model the problem, the following notation will be used:used:

xx11 = dollars spent on TV ( in thousands) = dollars spent on TV ( in thousands)

xx22 = dollars spent on radio (in thousands) = dollars spent on radio (in thousands)

The objective function will be to maximize total The objective function will be to maximize total exposures and the other goals will be treated as exposures and the other goals will be treated as constraints.constraints.


Infeasible LP model

Media Selection

TV Radio Maximize

X1 X2 OF

Total Exposures (thousands) 14 6 840

Expenditures (th's) 15 105

Technological

coeffisients LHS RHS

Slack or surplu

s

Max Expenditures (th's) 1 1 120 <= 120 1.6485E-11

Min Exposures (th's) 14 6 840 >= 840 -3.7517E-10

Max TV (th's) 1 15 <= 90 75

Min Upper-income Exposures (th's) 1.2 1.2 144 >= 168 -24


Since there are only two decision variables in this Since there are only two decision variables in this model, the graphical approach can be used. model, the graphical approach can be used.

<

140

x2

x1120

120

140

X1 = 90

X1 + X2 = 120

1200X1 +1200X2 = 168,000<

>

The graph shows that there The graph shows that there are no points that satisfy all are no points that satisfy all the constraints.the constraints.


Swenson’s Goal Programming ModelSwenson’s Goal Programming Model Note that the Note that the first goal (total exposures will be at least 840,000), if first goal (total exposures will be at least 840,000), if violated, will be underachieved.violated, will be underachieved.

The second goal (no more than $90,000 will be spent The second goal (no more than $90,000 will be spent on TV advertising), if violated, will be overachieved, on TV advertising), if violated, will be overachieved, etc.etc.

Employing this reasoning, the goals are restated, in Employing this reasoning, the goals are restated, in descending priority, as:descending priority, as:

1.1. Minimize the underachievement of 840,000 Minimize the underachievement of 840,000 total exposures.total exposures.Min Min uu11 subject to the condition subject to the condition

14,00014,000xx11 + + 6,0006,000xx22 + u + u11 >> 840,000; 840,000; uu1 1 >> 0 0


2.2. Minimize expenditures in excess of $90,000 on Minimize expenditures in excess of $90,000 on TVTVMin Min vv22 subject to the condition subject to the condition

xx11 – v – v22 << 90,000; 90,000; vv2 2 >> 0 0

3.3. Minimize underachievement of 168,000 upper- Minimize underachievement of 168,000 upper-income exposuresincome exposuresMin Min uu33 subject to the condition subject to the condition

1,2001,200xx11 + + 1,2001,200xx22 + u + u33 >> 168,000; 168,000; uu3 3 >> 0 0

4.4. Minimize underachievement of 1,680,000 total Minimize underachievement of 1,680,000 total exposures (the maximum possible)exposures (the maximum possible)Min Min uu44 subject to the condition subject to the condition

14,00014,000xx11 + + 6,0006,000xx22 + u + u44 >> 1,680,000; 1,680,000; uu4 4 >> 0 0


Note that the goals are now stated in terms of either Note that the goals are now stated in terms of either minimizing underachievement (i.e., min. minimizing underachievement (i.e., min. uuii) or ) or

minimizing overachievement (i.e., min. minimizing overachievement (i.e., min. vvii).).In addition, the goals have been expressed as In addition, the goals have been expressed as inequalities. This method will facilitate a graphical inequalities. This method will facilitate a graphical analysis.analysis.

Given that the priorities are formulated correctly, we Given that the priorities are formulated correctly, we must now distinguish betweenmust now distinguish between

1.1. system constraintssystem constraints (all constraints that may (all constraints that may not be violated)not be violated)

2.2. goal constraintsgoal constraints

The only system constraint is: Total The only system constraint is: Total expenditures will be no greater than expenditures will be no greater than $120,000$120,000

xx11 + x + x22 << 120 120


The model can now be expressed as:The model can now be expressed as:

Min PMin P11uu11 + P + P22vv22 + P + P33uu33 + P+ P44uu44

s.t.s.t. x x11 + x + x22 << 120120 (S)(S)

14,000 14,000 xx1 1 + + 6,0006,000xx22 + u + u1 1 >> 840,000840,000 (1)(1)

xx1 1 - v- v2 2 << 9090 (2)(2)

1,200 1,200 xx1 1 + + 1,2001,200xx22 + u + u3 3 >> 168,000168,000 (3)(3)

14,000 14,000 xx1 1 + + 6,0006,000xx22 + u + u4 4 >> 1,680,0001,680,000 (4)(4)

xx11, , xx22, , uu11, , vv2 2 , , uu33, , uu44 >> 0 0

Note that the objective function consists only of Note that the objective function consists only of deviation variables and is of the deviation variables and is of the MinMin form. form.

In the objective function, In the objective function, PP11 denotes the highest denotes the highest

priority, and so on. priority, and so on.


The previous problem statement precisely means:The previous problem statement precisely means:

1.1. Find the set of decision variables that Find the set of decision variables that satisfies the system constraint (satisfies the system constraint (SS) and that ) and that also gives the Min possible value to also gives the Min possible value to uu11

subject to constraint (1) and subject to constraint (1) and xx11, , xx22, , uu11 >> 0 0..

Call this set of decisions FR I (i.e., feasible Call this set of decisions FR I (i.e., feasible region I). region I).

Considering Considering only the highest goalonly the highest goal, all of the , all of the points in FR I are “optimal” and (again points in FR I are “optimal” and (again considering only the highest goal), we are considering only the highest goal), we are indifferent as to which of these points are indifferent as to which of these points are selected.selected.


2.2. Find the subset of points in FR I that gives the Find the subset of points in FR I that gives the Min possible value to Min possible value to vv22, subject to constraint , subject to constraint

(2) and (2) and vv22 >> 0 0. Call this subset FR II. . Call this subset FR II. Considering only the ordinal ranking of the Considering only the ordinal ranking of the two highest-priority goals, all of the points in two highest-priority goals, all of the points in FR II are “optimal,” and in terms of these two FR II are “optimal,” and in terms of these two highest-priority goals, we are indifferent as to highest-priority goals, we are indifferent as to which of these points are selected. which of these points are selected.

3.3. Let FR III be the subset of points in FR II that Let FR III be the subset of points in FR II that minimize minimize uu33, subject to constraint (3) and , subject to constraint (3) and

uu33 >> 0 0..

4.4. FR IV is the subset of points in FR III that FR IV is the subset of points in FR III that minimize minimize uu44, subject to constraint (4) and , subject to constraint (4) and

uu44 >> 0 0. Any point in FR IV is an optimal . Any point in FR IV is an optimal

solution to the model.solution to the model.


Graphical Analysis and Spreadsheet Implementation Graphical Analysis and Spreadsheet Implementation of the Solution Procedureof the Solution Procedure Since there are only two Since there are only two decision variables, we can use the graphical method decision variables, we can use the graphical method of LP.of LP.

1.1. Both the spreadsheet output and the Both the spreadsheet output and the geometry reveal the the Min of geometry reveal the the Min of uu11 s.t. ( s.t. (SS), (1), ), (1),

and and xx11, , xx22, , uu11 >> 0 0 is is uu11** = 0= 0..The important information is that The important information is that uu11 = 0 = 0 which which

tells us that the first goal can be completely tells us that the first goal can be completely attained. attained. Alternative optima for the current model are Alternative optima for the current model are provided by all values of (provided by all values of (xx11, , xx22) that satisfy ) that satisfy

the conditions the conditions xx11 + x + x22 << 120120

14,00014,000xx11 + + 6,0006,000xx22 >> 840,000840,000

xx11, x, x22 >> 00FR IFR I


Media Selection - GP model

TV Radio Minimiz

e

X1 X2 U1 V2 U3 U4 OF

Total Exposures (thousands) 1 1 1 1 54

Expenditures (th's) 120 0 0 30 24 0

Technological coeffisients LHS RHS

Slack or

surplus

Max Expenditures (th's) 1 1 120 <= 120 0

Min Exposures (th's) 14 6 1 1680 >= 840 840

Max TV (th's) 1 -1 90 <= 90 0

Min Upper-income Exposures (th's) 1.2 1.2 1 168 >= 168 0

Exposures Target (th's) 14 6 1 1680 >= 1680 0


Rank

Media Selection -Step 1

TV Radio Minimize


Penalty 1 1 0


Technological coeffisients LHS RHS Slack or surplus


1 Min Exposures (th's) 14 6 1 840 >= 840 2.121E-10

2

3

4


u1 = 0

At any such point, the goal is attained (At any such point, the goal is attained (uu11* = 0* = 0) so ) so

that, in terms of only the first goal, these decisions that, in terms of only the first goal, these decisions are equally preferable. are equally preferable.

Thus FR I is the shaded areaThus FR I is the shaded area ABC ABC..


Rank

Media Selection -Step 2

TV Radio Minimize


Penalty 1 1 0




1 Min Exposures (th's) 14 6 1 840 >= 840 0

2 Max TV (th's) 1 -1 60 <= 90 30

3

4

Value of U1 found in step 1 1 0 = 0 0


u1 = 0

v1 = 0

We see that: Min We see that: Min vv22 such that such that xx in FR I, goal (2) and in FR I, goal (2) and

vv22 >> 0 0 is is vv22* = 0* = 0. . xx11, x, x22 >> 00 Thus, FR II is defined by Thus, FR II is defined by

xx11 + x + x22 << 120120

14,00014,000xx11 + + 6,0006,000xx22 >> 840,000840,000

xx11 << 9090

xx11, x, x22 >> 00

FR IIFR II

The shaded area The shaded area ABDEABDE is a is a subset of FR I and as subset of FR I and as expected, the size of the expected, the size of the feasible region is smaller.feasible region is smaller.


Rank

Media Selection - Step 3

TV Radio Minimize


Penalty 1 1 24





2 Max TV (th's) 1 -1 15 <= 90 75

3 Min Upper-income Exposures (th's) 1.2 1.2 1 168 >= 168 0


Value of V2 found in step 2 1 0 = 0 0

4


FR III is the line segment FR III is the line segment BD.BD. In this case In this case uu33* = 24,000* = 24,000. Although the first . Although the first

two goals were completely attained (since two goals were completely attained (since uu11* = * = vv22* = 0* = 0), the third goal cannot be ), the third goal cannot be

completely attained because completely attained because uu33* > 0* > 0..

xx11 + x + x22 << 120120

14,00014,000xx11 + + 6,0006,000xx22 >> 840,000840,000

xx11 << 9090

1,2001,200xx11 + + 1,2001,200xx22 >> 168,000 – 24,000 = 144,000168,000 – 24,000 = 144,000

FR IIIFR III


Rank

Media Selection - Step 4

TV Radio Minimize


Penalty 1 240





2 Max TV (th's) 1 -1 90 <= 90 0


4 Exposures Target (th's) 14 6 1 1680 >= 1680 0





Recall that the fourth goal is to minimize Recall that the fourth goal is to minimize underachievement of the maximum possible number underachievement of the maximum possible number of exposures, which is 1,680,000. of exposures, which is 1,680,000.

14,00014,000xx11 + + 6,0006,000xx22 + u + u44 >> 1,680,0001,680,000

Thus, we wish to minimize the Thus, we wish to minimize the underachievement underachievement uu44 where where

The unique optimum The unique optimum is is xx11* * = 90= 90,, xx22* * = 30= 30

(i.e., spend $90,000 (i.e., spend $90,000 on TV ads & $30,000 on TV ads & $30,000 on radio ads).on radio ads).

Since Since uu44 = 240,000= 240,000, we , we

achieve achieve 1,680,000 - 1,680,000 - 240,000 = 1,440,000240,000 = 1,440,000 exposures. exposures.


In reviewing the results of the absolute priority In reviewing the results of the absolute priority study, the older members of the Mylonal market study, the older members of the Mylonal market begins to take on importance. begins to take on importance.

COMBINING WEIGHTS AND COMBINING WEIGHTS AND ABSOLUTE PRIORITIESABSOLUTE PRIORITIES

The exposures per $1000 of advertising are: The exposures per $1000 of advertising are:

TV RADIOTV RADIO

50 and over50 and over 3,000 3,000 8,000 8,000

EXPOSURE GROUPEXPOSURE GROUP

Note that radio and TV exposures are not equally Note that radio and TV exposures are not equally effective in generating exposures in this segment of effective in generating exposures in this segment of the population.the population.


If there were no other considerations, then we would If there were no other considerations, then we would like as many 50-and-over exposures as possible.like as many 50-and-over exposures as possible.

Since radio yields such exposures at a higher rate Since radio yields such exposures at a higher rate than TV (8000 > 3000), the maximum possible than TV (8000 > 3000), the maximum possible number of 50-and-over exposures would be number of 50-and-over exposures would be achieved by allocating all of the $120,000 available achieved by allocating all of the $120,000 available to radio.to radio.

Thus, the maximum number of 50-and-over Thus, the maximum number of 50-and-over exposures is exposures is 120 x 8000 = 960,000120 x 8000 = 960,000..

Once the first three goals are satisfied, we would like Once the first three goals are satisfied, we would like to come as close as possible to minimizing to come as close as possible to minimizing underachievement.underachievement.

To resolve this conflict of goals, use a weighted sum To resolve this conflict of goals, use a weighted sum of the deviation variables as the objective in the final of the deviation variables as the objective in the final phase of the absolute priorities approach.phase of the absolute priorities approach.


Rank

Weight

Media Selection - Weighted Step 4

TV Radio Minimize

X1 X2 U1 V2 U3 U4 U5 OF

Penalty 1 3 1065

Expenditures (th's) 15 105 0 0 24 840 75

Technological coeffisients LHS RHS

Slack or

surplus



2 Max TV (th's) 1 -1 15 <= 90 75


4 Exposures Target (th's) 14 6 1 1680 >= 1680 0

4 Exposures > 50 years (th's) 3 8 1 960 >= 960 0





Note that the new objective function has moved the Note that the new objective function has moved the optimal solution from one end of FR III to the other.optimal solution from one end of FR III to the other.This optimal solution is as close as possible to the This optimal solution is as close as possible to the more heavily weighted goal.more heavily weighted goal.

Sensitivity analysis Sensitivity analysis on the weights in the on the weights in the objective function objective function could be used to see could be used to see when the solution when the solution changes from point changes from point BB to point to point DD..


Multi-Objective Decision Making

• Many problems have multiple objectives:– Planning the national budget

• save social security, reduce debt, cut taxes, build national defense

– Admitting students to college

• high SAT or GMAT, high GPA, diversity

– Planning an advertising campaign

• budget, reach, expenses, target groups

– Choosing taxation levels

• raise money, minimize tax burden on low-income, minimize flight of business

– Planning an investment portfolio

• maximize expected earnings, minimize risk

• Techniques– Preemptive goal programming

– Weighted goal programming

Documents

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.1 GOAL PROGRAMMING 1. Opprettholde stabil profitt 2. Øke eller opprettholde gitte markedsandeler