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

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)

TPS Reliability, Availability, and Performance 1 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)

TPS Delivered System Capability 1 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

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)

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

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

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

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

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

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

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 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?

Feasible Set: Universal Software X2 6 4 Feasible Set 2 6 8 2 4 X1 6x1 + 2x2 3x1 + 3x2 2x1 + 4x2

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

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

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 2-10 2-3 2-20 2-5 5-30

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