Ahp Lecture

8/3/2019 Ahp Lecture

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Analytic Hierarchy Process

Developed by Thomas L. Saaty, published in hisbook The Analytic Hierarchy Process in 1980

A method for ranking decision alternatives andselecting the best one when the decision maker

has multiple objectives or criteria on which tobase the decision.

Goal programming gives us a mathematicalquantity that aims to satisfy a set of goals. The

output of GP answers the question how much? The output of AHP is a prioritized ranking of

decision alternatives.


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Steps in AHP

1. Decision maker develops a graphicalrepresentation of the problem. The hierarchyreveals the factors to be considered as well asthe various alternatives in the decision.

2. Two alternatives are compared according to acriterion and one is preferred. The followingpreference scale is used. This process is known

as Pairwise Comparison.


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Preference Scale forPairwise Comparison


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Steps in AHP3. Prioritize alternatives with respect to each

criterion. This step in AHP is referred to assynthesization. The mathematical procedure

for synthesization is very complex andbeyond the scope of this course. Instead, wewill use an approximation method forsynthesization that provides a reasonablygood estimate of preference scores for eachdecision in each criterion.


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Steps in AHP4. The next step in AHP is to determine

the relative importance, or weight,of the criteria that is, to rank the

criteria from most important to leastimportant. This is accomplished thesame way we ranked the sites withineach criterion: using pairwisecomparisons


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Steps in AHP

5. Multiply the output of step 3 withstep 4. The output of this step is anoverall ranking of each alternative.


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Steps in Developing GoalProgramming Model

3. Define the decision variables.

4. Formulate the constraints in theusual linear programming fashion.

5. For each goal, develop a goalequation, with the RHS specifyingthe target value for the goal.

Deviation variables di+ and di- areincluded in each goal equation toreflect the possible deviationsabove or below the target value.


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Steps in Developing GoalProgramming Model

6. Write the objective function in termsof minimizing a prioritized functionof the deviation variables.


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Extension to Equally ImportantMultiple Goals

Now Harrisons management wants to achieveseveral goals of equal in priority

Goal 1:to produce a profit of $30 if possibleduring the production period

Goal 2:to fully utilize the available wiringdepartment hours

Goal 3:to avoid overtime in the assemblydepartment

Goal 4:to meet a contract requirement to produceat least seven ceiling fans


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Extension to Equally ImportantMultiple Goals

The deviational variables are

d1 = underachievement of the profit target

d1+ = overachievement of the profit target

d2

= idle time in the wiring department (underutilization)d2

+ = overtime in the wiring department (overutilization)

d3 = idle time in the assembly department (underutilization)

d3+ = overtime in the assembly department (overutilization)

d4

= underachievement of the ceiling fan goald4

+ = overachievement of the ceiling fan goal


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Ranking Goals with Priority Levels

In most goal programming problems, one goalwill be more important than another, which will inturn be more important than a third

Goals can be ranked with respect to their

importance in managements eyes Higher-order goals are satisfied before lower-

order goals

Priorities (Pis) are assigned to each deviational

variable with the ranking so thatP1 is the mostimportant goal,P2 the next most important,P3 thethird, and so on


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Harrison Electric has set the following prioritiesfor their four goals

GOAL PRIORITY

Reach a profit as much above $30 as possible P1

Fully use wiring department hours available P2

Avoid assembly department overtime P3

Produce at least seven ceiling fans P4


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This effectively means that each goal is infinitelymore important than the next lower goal

With the ranking of goals considered, the newobjective function is

Minimize total deviation =P1d1 +P2d2

+P3d3+ +P4d4

The constraints remain identical to the previousones


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Solving Goal Programming ProblemsGraphically

We can analyze goal programmingproblems graphically

We must be aware of three characteristics

of goal programming problems1. Goal programming models are allminimization problems

2. There is no single objective, but multiplegoals to be attained

3. The deviation from the high-priority goalmust be minimized to the greatest extentpossible before the next-highest-priority goalis considered


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Recall the Harrison Electric goal programmingmodel

Minimize total deviation =P1d1 +P2d2

+P3d3+ +P4d4

subject to 7X1 + 6X2 +d1d1+ = 30 (profit )2X1 + 3X2 +d2

d2+ = 12 (wiring )

6X1 + 5X2 +d3d3

+ = 30 (assembly )

X2 +d4d4

+ = 7 (ceiling fans)

AllXi,di variables 0 (nonnegativity)where

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


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To solve this we graph one constraint at a timestarting with the constraint with the highest-priority deviational variables

In this case we start with the profit constraint as

it has the variabled1 with a priority ofP1 Note that in graphing this constraint the

deviational variables are ignored

To minimized1 the feasible area is the shaded

region


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Analysis ofthe first goal 7

6

5

4

3

2

1

0

X1

X2

| | | | | |

1 2 3 4 5 6

Minimize Z =P1d1

7X1 + 6X2 = 30

d1+

d1

Figure 11.4


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The next graph is of the second priority goal ofminimizingd2

The region below the constraint line 2X1 + 3X2 =12 represents the values ford2

while the region

above the line stands ford2+ To avoid underutilizing wiring department hours

the area below the line is eliminated

This goal must be attained within the feasible

region already defined by satisfying the first goal


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Analysis offirst andsecond goals

Minimize Z =P1d1 +P2d2

d2

7

6

5

4

3

2

1

0

X1

X2

| | | | | |

1 2 3 4 5 6

7X1 + 6X2 = 30

d1+

Figure 11.5

d2+

2X1

+ 3X2

= 12


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The third goal is to avoid overtime in theassembly department

We wantd3+ to be as close to zero as possible

This goal can be obtained

Any point inside the feasible region bounded bythe first three constraints will meet the threemost critical goals

The fourth constraint seeks to minimized4

To do this requires eliminating the area belowthe constraint lineX2 = 7 which is not possiblegiven the previous, higher priority, constraints


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Analysis ofall fourpriority goals

MinimizeZ =P1d1 +P2d2

+P3d3 +P4d4

d3

Figure 11.6

7

6

5

4

3

2

1

0

X1

X2

| | | | | |

1 2 3 4 5 6

7X1 + 6X2 = 30

d1+

d2+

2X1 + 3X2 = 12

d3

+

6X1 + 5X2 = 30

d4

d4+

A

B

C

D

X2 = 7


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The optimal solution must satisfy the first threegoals and come as close as possible tosatisfying the fourth goal

This would be point A on the graph with

coordinates ofX1 = 0 andX2 = 6 Substituting into the constraints we find

d1 = $0 d1

+ = $6

d2 = 0 hours d2

+ = 6 hours

d3 = 0 hours d3+ = 0 hours

d4 = 1 ceiling fan d4

+ = 0 ceiling fans

A profit of $36 was achieved exceeding the goal


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Graphical Solution ProcedureGoal Programming Model

1. Identify the feasible solution points;these are the ones that satisfy theproblem constraints.

2. Identify all feasible solutions thatachieve the highest priority goal; ifthere are no feasible solutions that

will achieve the highest-prioritygoal, identify the solution(s) thatcomes closest to achieving thehighest priority goal.


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Ahp Lecture

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