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BA 555 Practical Business Analysis Agenda Linear Programming (LP) Sensitivity Analysis Simulation Using @Risk 1 Sensitivity Analysis (p.70) How will a change in a coefficient of the objective function affect the optimal solutions? How will a change in the right-hand-side value for a constraint affect the optimal solution? MAX 3 A + 4 B SUBJECT TO 2) 2 A + 2 B <= 3) 2 A + 4 B <= END 80 120 2 Range of Optimality (p.70) The range of values over which an objective function coefficient may vary without causing any change in the values of the decision variables in the optimal solution. MAX 3 A + 4 B SUBJECT TO 2) 2 A + 2 B <= 3) 2 A + 4 B <= END VARIABLE A B CURRENT COEF 3.000000 4.000000 80 120 OBJ COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 1.000000 1.000000 2.000000 1.000000 3 Range of Feasibility (p.70) The range of values over which a right-hand side may vary without changing the value and interpretation of the dual price (shadow price). MAX 3 A + 4 B SUBJECT TO 2) 2 A + 2 B <= 3) 2 A + 4 B <= END OBJECTIVE FUNCTION VALUE 80 1) 120 VARIABLE A B 140.0000 VALUE REDUCED COST OBJECTIVE FUNCTION VALUE 20.000000 .000000 20.000000 .000000 1) 140.0000 VARIABLE VALUE REDUCED ROW SLACK OR SURPLUS DUAL PRICES A 20.000000 2) .000000 1.000000.0 B 20.000000 3) .000000 .500000.0 ROW 2 3 CURRENT RHS 80.000000 120.000000 RIGHTHAND SIDE RANGES ALLOWABLE INCREASE 40.000000 40.000000 ROW SLACK OR SURPLUS 2) .000000 ALLOWABLE 3) .000000 DECREASE 20.000000 40.000000 DUAL P 1.0 .5 4 Reduced Cost (p.70) The amount by which an objective function coefficient would have to improve (increase for a maximization problem, decrease for a minimization problem), before it would be possible for the corresponding variable to assume a positive value in the optimal solution. OBJECTIVE FUNCTION VALUE 1) VARIABLE A B ROW 2) 3) 140.0000 VALUE 20.000000 20.000000 SLACK OR SURPLUS .000000 .000000 REDUCED COST .000000 .000000 DUAL PRICES 1.000000 .500000 5 LINDO: The Model and Report LP OPTIMUM FOUND AT STEP MAX 30 C + 40 D Objective: (carpentry) (varnishing) (demand for desks) (non-negativity) OBJECTIVE FUNCTION VALUE 1) 240 6 C + 4 D <= 36 VARIABLE C 4 C + 8 D <= 40 D D <= 8 C >= 0 D >= 0 ROW VALUE 4.000000 3 s.t. 2) 3) 4) LP OPTIMUM FOUND AT STEP 240 VARIABLE C D VALUE 4.000000 3 ROW 2) 3) 4) VARIABLE REDUCED COST .000000 .000000 SLACK OR SURPLUS .000000 .000000 5.000000 NO. ITERATIONS= DUAL PRICES 2.500000 3.750000 .000000 2 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJECTIVE FUNCTION VALUE 1) REDUCED COST .000000 .000000 SLACK OR SURPLUS .000000 .000000 5.000000 NO. ITERATIONS= 2 2 DUAL PRICES 2.500000 3.750000 .000000 C D ROW 2 3 4 CURRENT COEF 30.000000 40.000000 OBJ COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 30.000000 10.000000 20.000000 20.000000 CURRENT RHS 36.000000 40.000000 8.000000 RIGHTHAND SIDE RANGES ALLOWABLE INCREASE 24.000000 26.666670 INFINITY ALLOWABLE DECREASE 16.000000 16.000000 5.000000 2 6 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES EXCEL: The Model 7 EXCEL: The Answer Report Microsoft Excel 11.0 Answer Report Worksheet: [LP Example 10.xls]Example10 Report Created: 11/7/2006 11:03:04 AM Target Cell (Max) Cell Name $D$11 Maximize profit Original Value 240 Final Value 240 Adjustable Cells Cell Name $B$6 Production C $C$6 Production D Original Value 4 3 Final Value Constraints Cell Name $D$19 Carpentry Total hours $D$20 Varnishing Total hours $C$6 Production D Cell Value 4 3 Formula Status Slack 36 $D$19<=$F$19 Binding 0 40 $D$20<=$F$20 Binding 0 3 $C$6<=$C$8 Not Binding 5 8 EXCEL: The Sensitivity Report Microsoft Excel 11.0 Sensitivity Report Worksheet: [LP Example 10.xls]Example10 Report Created: 11/7/2006 11:03:04 AM Adjustable Cells Cell Name $B$6 Production C $C$6 Production D Final Reduced Objective Value Cost Coefficient 4 0 30 3 0 40 Allowable Allowable Increase Decrease 30 10 20 20 Constraints Cell Name $D$19 Carpentry Total hours $D$20 Varnishing Total hours Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 36 2.5 36 24 16 40 3.75 40 26.66666667 16 Dual Prices in LINDO 9 EXCEL: The Limit Report Microsoft Excel 11.0 Limits Report Worksheet: [LP Example 10.xls]Limits Report 1 Report Created: 11/7/2006 10:26:28 AM Target Cell Name $D$11 Maximize profit Adjustable Cell Name $B$6 Production C $C$6 Production D The values in the Lower Limit column indicate the smallest value each decision variable can assume while the values of all other decision variables remain Constant and all the constraints are satisfied. Value 240 Value 4 3 Lower Target Limit Result 0 120 0 120 Upper Target Limit Result 4 240 3 240 The values in the Upper Limit column indicate the largest value each decision variable can assume while the values of all other decision variables remain constant and all the constraints are satisfied. 10 Simulation (pp. 81 – 104) Uncertainty 11 Simulation: Preparation (p.81) An experiment is the process by which an observation (or measurement) is obtained. Flipping a fair coin 5 times to observe the total number of Heads (H) or Tails (T). An event is the outcome of an experiment. 3 H’s and 2 T’s in 5 trials. A variable X is a random variable if the value it assumes, corresponding to the outcome of an experiment, is a chance or random event. It may be defined as a specification or description of a numerical result from a random experiment. X = total number of T in 5 trials. Probability shows you the likelihood or chances for each of the various potential future events, based on a set of assumptions about how the world works. Probability tells you what the data will be like when you know how the world is. (Cf. Statistics helps you figure out what the world is like after you have seen some data that it generated.) Pr( X = 5 ) = 0.03125. 12 Probability Distributions (p.81) The pattern of probabilities for a random variable is called its probability distribution. It can be represented by a formula, table, or graph. A Probability Distribution Plot Probability Distribution of X 0.4 Probability A Probability Table Variable X Probability 0 0.03125 1 0.15625 2 0.31250 3 0.31250 4 0.15625 5 0.03125 0.3 0.2 0.1 0 0 1 2 3 4 5 X = Total Number of T in 5 trials A Probability Density Function 5 1 1 P( X k ) 1 k 2 2 k 5 k In short, a probability distribution tells us (1) what possible outcomes of a random experiment are, and (2) how likely each outcome occurs. 13 Game 1 Expected Payoff (p.82) Game 1. Flip a fair coin once. You get $1 if T occurs. Expected Payoff = $1 P(T) + $0 P(H) = $1 (0.5) + $0 (0.5) = $0.50 Game 1 Probability Distribution Probability 0.6 0.4 0.2 0 H T 14 Game 2 Expected Payoff (p.82) Game 2. Flip a fair coin 5 times. You get $1 for every T. Expected Payoff 5 = x P( x k ) Game 2 Probability Distribution k 0 5 (0.5) k (1 0.5) 5 k x k 0 k 5 5! = x (0.5) k (1 0.5) 5 k k!(5 k )! k 0 =… =… = $2.5 5 0.4 Probability = 0.3 0.2 0.1 0 0 1 2 3 4 Number of Tails 5 15 Simulation Simulation is a method for learning about a real system by experimenting with a model that represents the system. In other words, a simulation model is a model that imitates a real-life situation. How does a computer “flip coins?” 16 Excel Function: =Rand() Returns an evenly distributed random number greater than or equal to 0 and less than 1. A new random number is returned every time the worksheet is calculated. To generate a random real number between a and b, use: RAND()*(b - a) + a Uniform Distribution (0.0, 1.0) 0 1 17 Simulation Using Excel Functions (p.82) Formula in cells B2, B6:B10: =if(rand() < 0.5, “H”, “T”) Formula in cell B11: =countif(B6:B10, “T”) Formula in Cell C2: =1 * B2 Formula in cell C6: =1 * B11 Problem ? hard to keep track of results. 18 A Simulation Model (p.77) A simulation model contains the mathematical expressions and logical relationships that describe how to compute the value of the output given the values of the inputs (both controllable and probabilistic inputs). Controllable Inputs (values are selected by decision makers) Model (mathematical expressions and logical relationships) Output Probabilistic Inputs (values are randomly generated) 19 Game 1 Simulation Using @Risk (p.83) 20 Game 2 Simulation Using @Risk (p.84) 21 Key Idea: Use Probability Distributions to Describe Uncertainty/Summarize Experience Probability Distributions 22 Estimated Unit Sales Summarize your experience/knowledge on unit sales using: =RiskUniform(0.08, 0.12) =RiskNormal(0.10, 0.02) =RiskNormal(0.10, 0.001) =RiskPert(0.08, 0.10, 0.12) =RiskTriang(0.08, 0.10, 0.12) =RiskDiscrete({0.08,0.10, 0.12},{0.1, 0.7, 0.2}) 23 NPV: Simulation Results 24 Other @Risk Functions 25