Chapters 16-18 Multiple-Goal Decision Analysis II

1
Chapters 16-18Multiple-Goal Decision Analysis II

• Goals as constraints
• TPS example
• Constrained optimization
• Linear programming
• System Analysis
• Dealing with unquantifiable goals
• Preference table
• Screening matrix
• Presentation techniques for mixed criteria

2
TPS Decision Problem 5

• Option A operating system
• If one processor fails, system fails
• Processor reliability 0.99/hour
• Mean time to repair 30 min.
• System reliability for N processors
• Rel (N) (0.99)N
• System availability for N processors
• (prob. of
  failure per time period)(Avg. down time)
• Length of time period

1
AV(N)
1 (1-.99N)(30Min)/(60min) 1½ (1-.99N)
3
TPS Reliability, Availability, and Performance
4
TPS Delivered System Capability
5
Goals as Constraints

• Cant afford availability lt.98
• Choose N to maximize E(N) 80N (11-N) subject to
  AV(N) gt .98
• Cant afford delivered capacity lt 1800 TR/sec
• Choose N to maximize AV(N) subject to
• E(N) gt 1800
• Comm. Line limits E(N) to lt 1500 TR/sec value of
  TR/sec 500 (a) 1000(b)

6
Optimal Solution for TV 0.5E
ETV
1680
1620
NV 190
1560
1500
NV 180
1440
NV 150
NV 44
1380
N C
2 530
4 610
3 570
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Optimal Solution for TV E
ETV
1680
1620
1560
NV 930
1500
1440
NV 910
NV 870
1380
N C
2 530
4 610
3 570
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General Optimal Decision Problem With Constraints

• Choose values of the decision variables
• X1, X2, , Xn
• So as to maximize the objective function
• f(X1, X2, , Xn)
• Subject to the constraints
• g1 (X1, X2, , Xn) lt b,
• g2 (X1, X2, , Xn) lt b2
• gm (X1, X2, , Xn) lt bm

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Optimal SolutionNecessary and Sufficient
Conditions

• The optimal solution (X1, X2, , Xn)max
• And the optimal value Vmax
• Are characterized by the necessary and sufficient
  conditions
• (X1, X2, , Xn)max is a feasible point on the
  isoquant
• f(X1, X2, , Xn) Vmax
• If Vgt Vmax, then its isoquant f(X1, X2, , Xn)
  V does not contain any feasible points

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Geometric View
11
The Linear Programming Problem

• Choose X1, X2, , Xn
• So as to maximize
• C1X1 C2X2 CnXn
• Subject to the costraints
• a11X1 a12X2 a1nXn lt b1
• a21X1 a22X2 a2nXn lt b2
• am1X1 am2X2 amnXn lt bm
• X1 gt 0, X2 gt 0, , Xn gt 0

12
Universal Software, Inc.

• 16 analysts, 24 programmers, 15 hr/day computer
• Text-processing systems
• 2 analysts, 6 progrs, 3 hr/day comp 20K profit
• Process control systems
• 4 analysts, 2 progrs, 3 hr/day comp 30K profit
• How many of each should Universal develop to
  maximize profit?

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Solution Steps

1. What objective are we trying to optimize?
2. What decisions do we control which affect the
   objective?
3. What items dictate constraints on our range of
   choices?
4. How are the values of the objective function
   related to the values of the decision variables?
5. What decision provides us with the optimal value
   of the objective function?

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Feasible Set Universal Software
X2
6
4
Feasible Set
2
6
8
2
4
X1
6x1 2x2
3x1 3x2
2x1 4x2
15
Optimal Solution Universal Software
Optimal solution
4
20x1 30x2 150
3
20x1 30x2 120
20x1 30x2 130
2
Feasible set
20x1 30x2 60
1
1
2
3
4
5
6
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Mathematical Optimization
System Analysis Quade68
Formulation What objectives are we trying to
optimize or satisfy? Search What decisions do we
control which affect our objectives? What items
dictate constraints on our range of
choices? Evaluation What criteria should we use
to evaluate the alternatives? How are the values
of the criterion function related to the values
of the decision variables which define the
alternatives? What choice provides us with the
best criterion value? Interpretation How
sensitive is the decision to assumptions made
during the analysis? Are there alternative
decisions providing satisfactory results with
less sensitivity to these assumptions?
Formulation Clarifying the objectives, defining
the issues of concern, limiting the
problem. Search Looking for data and
relationships, as well as alternative programs of
action that have some chance of solving the
problem. Evaluation Building various models,
using them to predict the consequences that are
likely to follow from each choice of
alternatives, and then comparing the alternatives
in terms of these consequences.
Interpretation Using the predictions obtained
from the models, and whatever other information
or insight is relevant, to compare the
alternatives further, derive conclusions about
them, and indicate a course of action.
Iteration
Iteration
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TPS Decision Problem 6

• Cost of option B OS with switchover, restart
  150K
• Cost of option B OS from vendor 135K
• Which should we choose?
• Key personnel availability
• Staff morale and growth
• Controllability
• Ease of maintenance

18
Presentation Techniques

• Unquantifiable criteria
• Criterion summaries
• Preference table
• Screening matrix
• Mixed Criteria
• Tabular methods
• Cost vs. capability graph
• Polar graph
• Bar charts

CRIT ALTS
2-10 2-3
2-20 2-5
5-30 2-10
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Summary Goals As Constraints II

• Adding constraints can simplify multiple-goal
  decision problems
• System analysis approach very similar to
  constrained optimization
• 6-step approach
• Sensitivity analysis
• Satisficing
• Unquantifiable goals require subjective
  resolution
• But effective presentation techniques can help
  decision process