1. 1 BA 555 Practical Business Analysis Linear Programming (LP) Sensitivity Analysis Simulation Using @Risk Agenda

2. 2 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?

3. 3 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.

4. 4 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).

5. 5 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.

6. 6 LINDO: The Model and Report

7. 7 EXCEL: The Model

8. 8 EXCEL: The Answer Report

9. 9 EXCEL: The Sensitivity Report Dual Prices in LINDO

10. 10 EXCEL: The Limit Report 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. 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.

11. 11 Simulation (pp. 81 – 104) Uncertainty

12. 12 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.

13. 13 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. In short, a probability distribution tells us (1) what possible outcomes of a random experiment are, and (2) how likely each outcome occurs.

14. 14 Game 1 Expected Payoff (p.82)

15. 15 Game 2 Expected Payoff (p.82)

16. 16 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?”

17. 17 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)

18. 18 Simulation Using Excel Functions (p.82)

19. 19 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).

20. 20 Game 1 Simulation Using @Risk (p.83)

21. 21 Game 2 Simulation Using @Risk (p.84)

22. 22 Key Idea: Use Probability Distributions to Describe Uncertainty/Summarize Experience Probability Distributions

23. 23 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})

24. 24 NPV: Simulation Results

25. 25 Other @Risk Functions