1. Chapters 16-18 Multiple-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
AV(N) =
1 –
= 1 – (1-.99N)(30Min)/(60min) = 1–½ (1-.99N)

3. TPS Reliability, Availability, and Performance1 2 3 4 5 6
0.94
0.95
0.96
0.97
0.98
0.99
1.00
Rel, Av
400
800
1200
1600
2000
2400
E(N)
trans/sec
Number of processors, N
DSC =
(SC)(E(N))(Av(N))
Rel(N)
Av(N)

4. TPS Delivered System Capability1 2 3 4 5 6
400
800
1200
1600
2000
2400
Number of processors, N
DSC(N) =
E(N)*Av(N)
1426
796
1892
2196
2340
2328

5. Goals as Constraints Can’t afford availability <.98
Choose N to maximize E(N) = 80N (11-N) subject to AV(N) > .98
Can’t afford delivered capacity < 1800 TR/sec
Choose N to maximize AV(N) subject to
E(N) > 1800
Comm. Line limits E(N) to < 1500 TR/sec; value of TR/sec = $500 (a); $1000(b)

6. Optimal Solution for TV = 0.5E
E=TV
E < E(N)
1680
Av > 0.98
1620
.5E – 40N – 450 = 200
NV =
NV = 190
1560
1500
NV = 180
E < 1500
1440
NV = 150
NV = 44
1380
E < E(N) N C 2
530 3
570 4
610

7. Optimal Solution for TV = E
E=TV
E < E(N)
1680
Av > 0.98
1620
E – 40N – 450 = 970
NV =
1560
NV = 930
1500
E < 1500
1440
NV = 910
NV = 870
1380
E < E(N) N C 2
530 3
570 4
610

8. 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) < b,
g2 (X1, X2, …, Xn) < b2 …
gm (X1, X2, …, Xn) < bm

9. Optimal Solution: Necessary 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 V> Vmax, then its isoquant f(X1, X2, …, Xn) = V does not contain any feasible points

10. Geometric View xn g1(x1, … , xn) = b1 = v5 = v4 = vmax = v3 = v2
Objective function
isoquants
g1(x1, … , xn) = b1
g3(x1, … , xn) = b3
g2(x1, … , xn) = b2 x1
(Decision variable) xn
(Decision
variable)
Decision
space
Optimal
solution
Infeasible point
Feasible point
Feasible set
f(x1, … , xn) = v1
= v2
= v3
= v4 = vmax
= v5
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 < b1
a21X1 + a22X2 + … + a2nXn < b2 …
am1X1 + am2X2 + … + amnXn < bm
X1 > 0, X2 > 0, …, Xn > 0

12. Universal Software, Inc.
16 analysts, 24 programmers, 15 hr/day computer
Text-processing systems
2 analysts, 6 prog’rs, 3 hr/day comp $20K profit
Process control systems
4 analysts, 2 prog’rs, 3 hr/day comp $30K profit
How many of each should Universal develop to maximize profit?

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

14. Feasible Set: Universal SoftwareX2 6 4
Feasible
Set 2 6 8 2 4 X1
6x1 + 2x2
3x1 + 3x2
2x1 + 4x2

15. Optimal Solution: Universal Software4 3
20x1 + 30x2 = 150
20x1 + 30x2 = 120
20x1 + 30x2 = 130 2
Feasible
set
20x1 + 30x2 = 60 1 1 2 3 4 5 6

16. 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

17. 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
#ALT’S
2-10
2-3
2-20
2-5
5-30

19. 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