Chapters 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

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-.99 N )(30Min)/(60min) = 1–½ (1-.99 N )

Rel, Av E(N) trans/sec Number of processors, N E(N) DSC = (SC)(E(N))(Av(N)) Rel(N) Av(N) TPS Reliability, Availability, and Performance

Number of processors, N DSC(N) = E(N)*Av(N) TPS Delivered System Capability

Goals as Constraints 1.Can’t afford availability <.98 –Choose N to maximize E(N) = 80N (11-N) subject to AV(N) >.98 2.Can’t afford delivered capacity < 1800 TR/sec –Choose N to maximize AV(N) subject to E(N) > Comm. Line limits E(N) to < 1500 TR/sec; value of TR/sec = $500 (a); $1000(b)

Optimal Solution for TV = 0.5E E=TV NCNC E < E(N) Av > 0.98 NV = 180 NV = 150 E < 1500 NV = 190 NV = 44.5E – 40N – 450 = 200NV =

E=TV NCNC E < E(N) Av > 0.98 NV = 870 E < 1500 NV = 930 NV = 910 Optimal Solution for TV = E E – 40N – 450 = 970NV =

General Optimal Decision Problem With Constraints Choose values of the decision variables –X 1, X 2, …, X n So as to maximize the objective function –f(X 1, X 2, …, X n ) Subject to the constraints –g 1 (X 1, X 2, …, X n ) < b, –g 2 (X 1, X 2, …, X n ) < b 2 – … –g m (X 1, X 2, …, X n ) < b m

Optimal Solution: Necessary and Sufficient Conditions The optimal solution (X 1, X 2, …, X n ) max And the optimal value V max Are characterized by the necessary and sufficient conditions 1.(X 1, X 2, …, X n ) max is a feasible point on the isoquant f(X 1, X 2, …, X n ) = V max 2.If V> V max, then its isoquant f(X 1, X 2, …, X n ) = V does not contain any feasible points

Objective function isoquants g 1 (x 1, …, x n ) = b 1 g 3 (x 1, …, x n ) = b 3 g 2 (x 1, …, x n ) = b 2 x1x1 (Decision variable) xnxn (Decision variable) Decision space Optimal solution Infeasible point Feasible point Feasible set f(x 1, …, x n ) = v 1 = v 2 = v 3 = v 4 = v max = v 5 Geometric View

The Linear Programming Problem Choose X 1, X 2, …, X n So as to maximize –C 1 X 1 + C 2 X 2 + … + C n X n Subject to the costraints –a 11 X 1 + a 12 X 2 + … + a 1n X n < b 1 –a 21 X 1 + a 22 X 2 + … + a 2n X n < b 2 –… –a m1 X 1 + a m2 X 2 + … + a mn X n < b m –X 1 > 0, X 2 > 0, …, X n > 0

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?

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?

X2X2 Feasible Set 6x 1 + 2x 2 3x 1 + 3x 2 2x 1 + 4x 2 X1X1 Feasible Set: Universal Software

Feasible set 20x x 2 = 60 20x x 2 = x x 2 = x x 2 = 150 Optimal solution Optimal Solution: Universal Software

Mathematical Optimization 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. System Analysis [Quade68] Iteration

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

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

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