DECISION MODELING WITH DECISION MODELING WITH MICROSOFT EXCELMICROSOFT EXCEL

Chapter 12Chapter 12Chapter 12Chapter 12

Copyright 2001Prentice Hall Publishers and

Ardith E. Baker

Multi-Objective Decision Making and Heuristics

Multi-Objective Decision Making and Heuristics

Part 2Part 2

MULTIPLE OBJECTIVESMULTIPLE OBJECTIVESIn many applications, the planner has more than one In many applications, the planner has more than one ___________. The presence of multiple objectives is ___________. The presence of multiple objectives is frequently referred to as the problem of “_________ frequently referred to as the problem of “_________ 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__________ Maximize discounted__________

2.2. ____________market share at the end of the ____________market share at the end of the planning periodplanning period

3.3. Maximize existing physical _________at the Maximize existing physical _________at the end of the planning periodend of the planning period

It is also clear that the goals are ____________(i.e., It is also clear that the goals are ____________(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___________ on the others. greater___________ on the others.

These goals are not __________________(i.e., they These goals are not __________________(i.e., they cannot be __________combined or compared).cannot be __________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,_____________, inventory control, programming,_____________, inventory control, etc.), have been found to be especially useful on etc.), have been found to be especially useful on problems in the________________.problems in the________________.

Several approaches to multiple objective models Several approaches to multiple objective models (also called ______________decision making) have (also called ______________decision making) have been developed:been developed:

Multi-attribute ___________theoryMulti-attribute ___________theory

AHP and Goal Programming will be discussed.AHP and 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 _____________on the basis of multiple criteria._____________on the basis of multiple criteria.

Search for _________optimal solutions via Search for _________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 ___________approach to the multiple-GP is a ___________approach to the multiple-objectives model.objectives model.

Goal ProgrammingGoal Programming is an extension of ___________ is an extension of ___________ Programming that enables the planner to come as Programming that enables the planner to come as close as possible to satisfying various _______and close as possible to satisfying various _______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 ___________system sense, to incorporate his or her ___________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 _________________context, the into a mathematical _________________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 “___________” or but rather solutions that are “___________” or “close enough.”“close enough.”

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

wherewherexx11 is the hours of _______________work is the hours of _______________work

xx22 is the hours of _______________work is the hours of _______________work

Assume the following ___________on total program Assume the following ___________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. ___________ ___________constraintsconstraints (so-called hard (so-called hard constraints) that cannot be violated.constraints) that cannot be violated.

2.2. _______ _______constraintsconstraints (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 ______________experience and12 minutes of ______________experience and

19 minutes of _____________problem solving19 minutes of _____________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 _________time is at most 6,000 minutes The total _________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 _____________program time working ¼ of the _____________program time working in small groups and in small groups and

¹/¹/33 of the time on problem_____________. of the time on problem_____________.

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 “__________________” to the right-hand side.be “__________________” 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_____________.of the two goals will be_____________.

To ___________the goal programming approach, the To ___________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

____________________variables 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 ______because it is impossible to both) to be ______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 _______________:as the goal _______________:

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__________.should be__________.

The complete (___________) model is:The complete (___________) 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 ________LP model and can be easily Now this is an ________LP model and can be easily solved in Excel. The optimal decision variables will solved in Excel. The optimal decision variables will _______the system constraint (total program hours)._______the system constraint (total program hours).

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

Note:Note: Both Both uu11 and and vv1 1 can’t be 0can’t be 0

Solver will _________that either Solver will _________that either uu11 or or vv11 (or both) will (or both) will

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

Note that the ___________function is the sum of the Note that the ___________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 _________among the various deviations there is no _________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 ___________variables.general for any pair of ___________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 _______________the group A decision that _______________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 ___________that hits the group experience A ___________that hits the group experience goal exactly and underachieves the problem-goal exactly and underachieves the problem-solving goal by 5 minutes, andsolving goal by 5 minutes, and

3.3. A decision that ______________each goal by A decision that ______________each goal by 2.5 minutes.2.5 minutes.

There is no ___________among the following three There is no ___________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____________.value (i.e., 5) for the objective____________.

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 ____________among units alone could produce a ____________among the deviation variables. the deviation variables.

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

as the_______________. Since as the_______________. Since vv22 (over-achievement (over-achievement

of problem solving) has the smallest coefficient, the of problem solving) has the smallest coefficient, the program designers would rather have program designers would rather have vv22 __________ __________

than any of the other deviation variables (positive than any of the other deviation variables (positive vv22

is _____________the least).is _____________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 _______than to minutes over the problem-solving _______than to ______________by ______________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 ________________constraintconstraint. .

Such a constraint _________the goal to a range or Such a constraint _________the goal to a range or interval interval rather than a specific ___________value. rather than a specific ___________value.

Suppose, for example, that in the previous Suppose, for example, that in the previous illustration the designers were ____________among illustration the designers were ____________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 ___________with In this situation the interval goal is ___________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 ______will attempt to objective function, the LP ______will attempt to ___________them. ___________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 ____________ Goal constraints are written by using ____________ 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________.be________.uuii represents represents underachievementunderachievement; ; vvii represents represents

______________.______________.Whenever Whenever uuii is used it is ___________to is used it is ___________to ggii((xx11, …, , …, xxnn))..Whenever Whenever vvii is used it is ______________from is used it is ______________from

ggii((xx11, …, , …, xxnn))..

Only __________variables appear in the objective Only __________variables appear in the objective function, and the objective is always to___________.function, and the objective is always to___________.

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

the_____________.the_____________.

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

1. ________1. ________. Make . Make ggii((xx11, …, , …, xxnn) as close as ) as close as

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

goal constraint asgoal constraint as

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

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

2.2. _________ _________UnderachievementUnderachievement. To do this, write. To do this, write

and in the__________, minimize and in the__________, 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_____________, hence, the and it is only in this_____________, 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__________, this constraint is__________, this constraint

will be active, for otherwise will 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________, it must equal________, it

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

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

3.3. __________ __________OverachievementOverachievement. To do this, write. To do this, write

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

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_______, this constraint will is_______, this constraint will

be active. The argument is __________to that be active. The argument is __________to that in item 2.in item 2.

4.4. Goal IntervalGoal Interval______________________. In this instance, . In this instance, the goal is to come as close as possible to the 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 “________ In order to write this as a goal, first “________ 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 ____________to the two constraintswhich is ____________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_____________. is_____________.

Variables Variables uui i andand vvii are merely ____________and are merely ____________and slack, respectively.slack, respectively.

^̂ ^̂

In some cases, managers do not wish to express In some cases, managers do not wish to express their __________among various goals in terms of their __________among various goals in terms of weighted deviation variables, for the process of weighted deviation variables, for the process of assigning __________may seem too arbitrary or assigning __________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 ___________________(as preferences in terms of ___________________(as opposed to weights) to a set of goals.opposed to weights) to a set of goals.

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

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_____________.exceed_____________.

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 ____________in reaching all audiences.____________in reaching all audiences.

Therefore, the agency will estimate the _________of Therefore, the agency will estimate the _________of the advertisements in terms of rated exposures (i.e., the 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 ____________per $1000 expenditure:____________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___________.of absolute___________.

1.1. Total exposures will hopefully be at least Total exposures will hopefully be at least ___________.___________.

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

3.3. The campaign should achieve at least _______ The campaign should achieve at least _______ 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___________.to being___________.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 __________ advertising, then the maximum __________ 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 _____( in thousands) = dollars spent on _____( in thousands)

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

The objective function will be to maximize total The objective function will be to maximize total ___________and the other goals will be treated as ___________and the other goals will be treated as _______________._______________.

An Infeasible ModelAn Infeasible Model The formulation and The formulation and spreadsheet solution is shown below:spreadsheet solution is shown below:

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___________________.violated, will be___________________.

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_____________, on TV advertising), if violated, will be_____________, etc.etc.

Employing this reasoning, the goals are restated, in Employing this reasoning, the goals are restated, in _____________priority, as:_____________priority, as:

1.1. ___________the underachievement of 840,000 ___________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 _________in excess of $90,000 on TV Minimize _________in excess of $90,000 on TV

Min 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____________income____________Min 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 ____ Minimize underachievement of 1,680,000 ____ 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 _____________underachievement (i.e., min. _____________underachievement (i.e., min. uuii) or ) or

minimizing _________________(i.e., min. minimizing _________________(i.e., min. vvii).).In addition, the goals have been expressed as In addition, the goals have been expressed as ______________. This method will facilitate a ______________. This method will facilitate a graphical analysis.graphical 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. _________ _________constraintsconstraints (all constraints that (all constraints that may not be violated)may not be violated)

2.2. _________ _________constraintsconstraints

The only system constraint is: Total The only system constraint is: Total expenditures will be no _________than expenditures will be no _________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 __________variables and is of the ______form.__________variables and is of the ______form.

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

_________, and so on. _________, 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 __________(satisfies the system __________(SS) and that ) and that also gives the ____possible value to also gives the ____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____________only the____________, 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 ____________as to which of these points are ____________as to which of these points are selected.selected.

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

constraint (2) and constraint (2) and vv22 >> 0 0. Call this subset FR II. . Call this subset FR II. Considering only the _________ranking of the Considering only the _________ranking of the two highest-priority goals, all of the points in two highest-priority goals, all of the points in FR II are “_______,” and in terms of these two FR II are “_______,” 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 __________________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 ____________(4) and , subject to ____________(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 ________method decision variables, we can use the ________method of LP.of LP.

1.1. Both the spreadsheet output and the Both the spreadsheet output and the __________reveal the the Min of __________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 __________. __________. Alternative __________for the current model Alternative __________for the current model are provided by all values of (are provided by all values of (xx11, , xx22) that ) that

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

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

xx11, x, x22 >> 00FR IFR I

First goal:First goal:

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..

2.2. Now enter the constraints defining FR I, Now enter the constraints defining FR I,together with the new goal constraint (2)together with the new goal constraint (2)

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.

This worksheet shows the third goal.This worksheet shows the third goal.

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

The optimal solution is shown in this worksheet. The optimal solution is shown in this worksheet.

Recall that the fourth goal is to ____________ Recall that the fourth goal is to ____________ underachievement of the maximum possible number underachievement of the maximum possible number of___________, which is 1,680,000. of___________, 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 ______optimum The ______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 _________priority In reviewing the results of the _________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 14,000 14,000 6,000 6,000

EXPOSURE GROUPEXPOSURE GROUP

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

If there were no other considerations, then we would If there were no other considerations, then we would like as many _____________exposures as possible.like as many _____________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 __________all of the $120,000 available achieved by __________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 _______________________._______________________.

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

It is decided that underachievement in the ________ It is decided that underachievement in the ________ (960,000 exposures to the 50-and-over group) is (960,000 exposures to the 50-and-over group) is three times as _________as underachievement in the three times as _________as underachievement in the fourth goal (1,680,000 total exposures).fourth goal (1,680,000 total exposures).

Note that the new objective function has moved the Note that the new objective function has moved the ________solution from one end of FR III to the other.________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_________.more heavily weighted_________.

__________analysis __________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..

ANALYTICAL HIERARCHYANALYTICAL HIERARCHYPROCESSPROCESS

This section deals with the real-world topic of This section deals with the real-world topic of making a decision when there are ________ making a decision when there are ________ objectives or criteria to consider. For example:objectives or criteria to consider. For example:

Choosing which employment offer to accept.Choosing which employment offer to accept.

Picking which computer (or car, etc.) to buy.Picking which computer (or car, etc.) to buy.

Deciding which new product to launch first.Deciding which new product to launch first.

Selecting a site for a new restaurant, hotel, etc.Selecting a site for a new restaurant, hotel, etc.

Rating the best cities in which to live.Rating the best cities in which to live.

Choosing a new software package for your Choosing a new software package for your company.company.

A simple way to attack such a decision would be to A simple way to attack such a decision would be to assign __________to each of the criteria that were to assign __________to each of the criteria that were to be considered in making the decision.be considered in making the decision.

Then, _____each decision alternative on a scale Then, _____each decision alternative on a scale from from 11 (worst) to (worst) to 1010 (best). (best).

Finally, you would _________the weights times the Finally, you would _________the weights times the rankings for each criterion and sum them up.rankings for each criterion and sum them up.

The ___________with the highest score would be the The ___________with the highest score would be the most preferred. most preferred.

For example, you are in charge of purchasing the For example, you are in charge of purchasing the next computer for the office. You have to choose next computer for the office. You have to choose between the following three computers:between the following three computers:

1.1. Model A runs an AMD K6-II chip at 400 MHz Model A runs an AMD K6-II chip at 400 MHz

2.2. Model B runs a Celeron chip at 333 MHz Model B runs a Celeron chip at 333 MHz

3.3. Model C runs a Pentium II chip at 450 MHz Model C runs a Pentium II chip at 450 MHz

The important criteria and their weights are:The important criteria and their weights are:

Criteria Criteria WeightWeight

PricePrice 50% 50% SpeedSpeed 15% 15% Hard-disk SizeHard-disk Size 20% 20%Warranty/Support Warranty/Support 15% 15%

Now, rank each of the three models on these four Now, rank each of the three models on these four _______. Rank them on a scale from 1 to 10 as _______. Rank them on a scale from 1 to 10 as described earlier.described earlier.

Model B has the highest weighted _______and thus Model B has the highest weighted _______and thus would be the best computer to purchase.would be the best computer to purchase.

This approach is quite ___________and there are This approach is quite ___________and there are difficulties in setting the ranking scales on such difficulties in setting the ranking scales on such different criteria.different criteria.

Analytic hierarchy process (AHP)Analytic hierarchy process (AHP) also uses a also uses a weighted ________approach idea, but it uses a weighted ________approach idea, but it uses a method for assigning ratings (or rankings) and method for assigning ratings (or rankings) and weights that is considered more ___________and weights that is considered more ___________and consistent.consistent.

(AHP) is based on ________comparisons between (AHP) is based on ________comparisons between the decision alternatives on each of the criteria. the decision alternatives on each of the criteria.

Then, a similar set of ______________are made to Then, a similar set of ______________are made to determine the relative importance of each criterion determine the relative importance of each criterion and thus produces the___________.and thus produces the___________.

The basic procedure is as follows:The basic procedure is as follows:

1.1. Develop the __________for each decision Develop the __________for each decision alternative for each criterion byalternative for each criterion by

• developing a pairwise comparison _____ developing a pairwise comparison _____ for each criterionfor each criterion

• __________the resulting matrix__________the resulting matrix

• _________the values in each row to get _________the values in each row to get the corresponding ratingthe corresponding rating

• calculating and checking the __________ calculating and checking the __________ ratioratio

2.2. Develop the ________for the criteria by Develop the ________for the criteria by

• developing a pairwise comparison matrix developing a pairwise comparison matrix for each___________for each___________

• normalizing the __________matrixnormalizing the __________matrix

• averaging the values in each _____to get averaging the values in each _____to get the corresponding ratingthe corresponding rating

• calculating and ________the consistency calculating and ________the consistency ratioratio

3.3. Calculate the _______average rating for Calculate the _______average rating for each decision alternative. Choose the one each decision alternative. Choose the one with the __________score.with the __________score.

Consider the following example:Consider the following example:

Sleepwell Hotels is looking for some help in Sleepwell Hotels is looking for some help in selecting the “best” revenue management software selecting the “best” revenue management software package from among several vendors. The director package from among several vendors. The director of revenue management for this chain of hotels has of revenue management for this chain of hotels has been given this task.been given this task.

Three vendors have been identified whose software Three vendors have been identified whose software meets the following basic needs:meets the following basic needs:

Revenue Technology Corporation (RTC)Revenue Technology Corporation (RTC)PRAISE Strategic Solutions (PSS)PRAISE Strategic Solutions (PSS)

El Cheapo (EC)El Cheapo (EC)

The important criteria are:The important criteria are:

1.1. The ____________of the installed system The ____________of the installed system

2.2. The follow-up _________provided over the The follow-up _________provided over the coming yearcoming year

3.3. The sophistication of the ___________math The sophistication of the ___________math enginesengines

4.4. The amount of _____________for Sleepwell The amount of _____________for Sleepwell

The first step in the AHP procedure is to make The first step in the AHP procedure is to make pairwise ___________between the vendors for each pairwise ___________between the vendors for each criterion. Here is the ________scale for making criterion. Here is the ________scale for making these comparisons:these comparisons:

DESCRIPTIONDESCRIPTION

11 Equally preferredEqually preferred33 Moderately preferredModerately preferred55 Strongly preferredStrongly preferred77 Very strongly preferredVery strongly preferred99 Extremely strongly preferredExtremely strongly preferred

RATINGRATING

Values Values 22, , 44, , 66, or , or 88 may also be assigned and may also be assigned and represent ___________halfway between the integers represent ___________halfway between the integers on either side.on either side.

Start with the total cost __________and generate the Start with the total cost __________and generate the following data in a spreadsheet:following data in a spreadsheet:

The _______in the row is being compared to the The _______in the row is being compared to the vendor in the column.vendor in the column.

A value between A value between 11 and and 99 indicates that the vendor in indicates that the vendor in the row is __________to the vendor in the column.the row is __________to the vendor in the column.

If the vendor in the ______is preferred to the vendor If the vendor in the ______is preferred to the vendor in the row, then the inverse of the rating is given.in the row, then the inverse of the rating is given.

The next step is to _________the matrix. This is The next step is to _________the matrix. This is done by totaling the numbers in each column.done by totaling the numbers in each column.

Each entry in the column is then ___________by the Each entry in the column is then ___________by the column sum to yield its normalized score.column sum to yield its normalized score.

Now, calculate the _________________and check its Now, calculate the _________________and check its value. The purpose for doing this is to make sure value. The purpose for doing this is to make sure that the original preference ratings were__________.that the original preference ratings were__________.

There are 3 steps to arrive at the consistency ratio:There are 3 steps to arrive at the consistency ratio:

1.1. Calculate the consistency ________for each Calculate the consistency ________for each vendor.vendor.

2.2. Calculate the consistency ________(CI). Calculate the consistency ________(CI).

3.3. Calculate the consistency ________(CI/RI Calculate the consistency ________(CI/RI where RI is a random index).where RI is a random index).

To calculate the consistency measure, we can take To calculate the consistency measure, we can take advantage of Excel’s matrix _____________function advantage of Excel’s matrix _____________function =MMULT()=MMULT()..

Multiply the average ______for each vendor times Multiply the average ______for each vendor times the scores in the first row one-at-a-time, sum these the scores in the first row one-at-a-time, sum these products up and divide this _____by the average products up and divide this _____by the average rating for the first vendor.rating for the first vendor.

RANDOM INDEXRANDOM INDEX

22 0.000.00 33 0.580.58 44 0.900.90 55 1.121.12 66 1.241.24 77 1.321.32 88 1.411.41 99 1.451.451010 1.511.51

NN

If we are perfectly___________, then the consistency If we are perfectly___________, then the consistency measures will equal measures will equal nn and therefore, the CIs will be and therefore, the CIs will be equal to ______and so will the consistency_______.equal to ______and so will the consistency_______.

If this ratio is very ________(Saaty suggests > 0.10), If this ratio is very ________(Saaty suggests > 0.10), then we are not consistent enough and the best then we are not consistent enough and the best thing to do is go back and _______the comparisons.thing to do is go back and _______the comparisons.

Now, continue for the other three criteria. You can Now, continue for the other three criteria. You can easily do this by copying the “________” sheet into easily do this by copying the “________” sheet into three other sheets (“Service,” “Sophistication,” and three other sheets (“Service,” “Sophistication,” and “Custom”) and then simply changing the _______ “Custom”) and then simply changing the _______ comparisons. comparisons.

Consistency ratio for “Service.”Consistency ratio for “Service.”

Consistency ratio for “Sophistication.”Consistency ratio for “Sophistication.”

Consistency ratio for “Customization.”Consistency ratio for “Customization.”

In all three cases, the CR value ranges from 0.0 to In all three cases, the CR value ranges from 0.0 to 0.047 which means that we are being___________.0.047 which means that we are being___________.

Note also that PSS is the winner on the Service Note also that PSS is the winner on the Service criterion, RTC and PSS are tied for the best in terms criterion, RTC and PSS are tied for the best in terms of Sophistication, and PSS is considered the best on of Sophistication, and PSS is considered the best on Customization.Customization.

All of this work concludes the first step in the All of this work concludes the first step in the procedure. The next step is to use similar ______ procedure. The next step is to use similar ______ comparisons to determine the appropriate _______ comparisons to determine the appropriate _______ for each of the criteria.for each of the criteria.

The process is the same in that we make The process is the same in that we make ______________, except that now we make the ______________, except that now we make the comparisons between the criteria not the vendors.comparisons between the criteria not the vendors.

Consistency ratio for weights on criterion.Consistency ratio for weights on criterion.

The final step is to ________the weighted average The final step is to ________the weighted average ratings of each decision alternative and use the ratings of each decision alternative and use the ________to decide from which vendor to purchase ________to decide from which vendor to purchase the software.the software.

These results are pulled from all the other These results are pulled from all the other ___________. From these results, we find that RTC ___________. From these results, we find that RTC barely edges out PSS for the software____________.barely edges out PSS for the software____________.