Building systems using models, decision analysis and statistical techniques

Building systems using models, decision analysis and statistical techniques Using systems to “hide” complex algorithms from management while still providing powerful results

DSS Humans: Decision Making Process Data: Facts pertinent to the decision at hand. DSS Algorithms: Math/Flow Chart stuff that helps the tools help the humans make decisions. Tools: Computers and IT. VB, VBA, Excel, InterDev, Etc.

A Modeling Approach to Decision Making • mental model - picturing in your “minds eye” • visual model - blueprints, schematics, maps • physical models - scale models and prototypes

A Modeling Approach to Decision Making • mathematical models - mathematical symbols are used to represent decision variables which are then related by the appropriate math functions to describe a real system or decision problem • spreadsheet models - cells are used to represent decision variables which are then related by functions to describe the decision problem

Benefits of Models • less costly to analyze than real system or problem • can be analyzed more rapidly • facilitate “what if” and “what’s best” analysis • provide insight about real problem or system

The Modeling Process • The process of building a model begins with an understanding of the real world system. Management must be able to answer questions such as:

The Modeling Process What drives the system under consideration? What are the key factors which predict the behavior of the system? What problems are relevant to management? What are the appropriate measures of success?

The Modeling Process • The next step in the modeling process involves simplifying and abstracting from the real world system. This model formulation step is exceptionally important.

The Modeling Process Should the model be too complex it may be difficult if not impossible to analyze. If the model is too simple management may not be willing to trust results obtained with the model fearing that the model is unrealistic.

The Modeling Process • Once a model is formulated an appropriate method of analysis must be identified.

The Modeling Process • That results are available at this point does not mean that the managerial questions have been answered. These results must be interpreted. Interpretation may be thought of as a reversal of the formulation step.

The Modeling Process Formulation required that the broad description of the real world system be narrowed and abstracted. Interpretation requires that the narrow meaning of the results be broadened and applied to the actual situation.

The Modeling Process • This set of steps; understanding, formulation, analysis and interpretation should provide new insights on the real world system. This insight may well provide new understanding of the problem and restart the modeling process. Thus, the modeling process should be an iterative one.

Start with Excel, build to VBA, and beyond… Let’s build a sheet from scratch: Thom Pearman has a new house and a new child. He feels he needs more insurance. His current policy has a $40,000 death benefit and is paid in full for life. Thom is trying to decide whether or not to keep this policy, or pay premiums on a new policy which has a death benefit of $350,000.

Insurance Decision Problem Thom’s current policy can be cashed in for $6,000. Premiums for the next ten years for the new policy are:

Insurance Decision Problem Thom would like to cash in his old policy, and use after-tax interest income from the $6,000 to pay the premiums for the new policy. Thom’s marginal tax rate is 28%. Is this realistic? In particular, what is the smallest interest rate Thom’s investments would have to earn to accomplish this?

Let’s build a more complex system: Optimization (more later, but just watch this for now). VBA.

Constructing the Insurance System. • Build the spreadsheet first. • Build the VBA “front-end”. • What inputs/outputs are necessary? • Flexibility. • Graphical capability.

How much should/can we make? • Tiger’s Golf Factory makes three products: clubs, bags, and balls. Clubs sell for $50 each, bags sell for $105 each, and balls sell for $2 each. Variable costs for making each product are $24, $20, and $1, respectively. Fixed costs are $1,000. • TGF has limited resources. In particular, each club takes .4 worker hours and .3 machine hours to make. Each bag takes 1.5 worker hours and .8 machine hours, and each ball take .001 worker and .005 machine hours to produce. • Problem is, TGF only has workers for 150 hours per week, and capacity of 200 machine hours per week. TGF wants to make as much profit as possible. Assuming there is demand for anything made, how much of each product should TGF make to get the largest profit?

Linear Programming Linear Programming: finding the best (or “optimal”) value for a certain linear objective given a set of linear constraints. Linear: a mathematical expression that can be written with no variables multiplied together or raised to any power. A “flat” equation.

Linear Programming Models • Elements: • Objective • Decision Variables • Constraints • Parameters

Simple Illustrative Example: • The Leghorn Toy Company makes a profit of $4 on each stuffed bear they sell, and $6 on each stuffed pig. 2 hours of labor and 3 square yards of fabric are required to make a pig; the same numbers for a bear are 1.5 hours and 2.5 square yards. Daily labor available is 16 hours (two workers), and daily material available is 25 square yards. How many bears and how many pigs should Leghorn make each day to maximize profit? • Decision Variables: ?? • Objective Function: ?? • Constraints: ??

When Can I Use LP? What assumptions are made? • All equations must be linear. • The answers to the decision variables don’t have to be integers. • The parameters are known, and they stay the same.

Important!!! • If you satisfy and are comfortable with the assumptions, AND CAN CORRECTLY FORMULATE THE PROBLEM, the computer will find the answer for you. • Complex DSS/MSS/Intelligent systems which are “manager-friendly” can be built on top of these algorithms!

LP Constraint Examples: • “We must produce 5 times as many Bears as Pigs.” • “The ratio of Pigs to Bears must be at least 2:1.” • “Forty percent of production should be Bears.” • Etc.

General Form of an LP Model

General Form of an LP Model where the c’s, a’s and b’s are constants determined from the problem and the x’s are the decision variables

In Solver: • Decision Variables <==> “Changing cells”

Constructing the Tiger System. • Build the spreadsheet first. • What questions and answers does management want/need? • Build the VBA “front-end”.

Sensitivity Analysis The Blue Blocker Company produces two types of sunglasses: Blueblockers and Blueblocker Imitations. Blueblockers yield a profit of $10, while the Imitations yield only $1. Blueblockers require 3 units of plastic per week, Imitations require 1, and there are 12 total units of plastic available per week. Each product requires two labor hours to produce, and there are two workers who each work 8 hours per week for Blue Blocker. Finally company policy states that Blue Blockers produced be at most 2 more than the number of Imitations produced.

After you get the “answer” (values of the Decision Variables)... • THREE VERY IMPORTANT LP CONCEPTS: • Range of Optimality: The range for each DV coefficient in the Objective Function for which the DV values will not change.

Shadow Price: The amount the objective function value will increase (decrease) for each unit increase in the right-hand-side of the associated constraint (as long as that r-h-s remains within its “Range of Feasibility”). • Range of Feasibility: The range for the r-h-s of a constraint where the Shadow Price is applicable (where you can use it). Also: range where the “binding” status stays the same.

Range of Optimality • Examples of Managerial Questions (Blue Blocker example): • "How much would the profit made on Imitations have to increase to change the production mix?" • "What would the maximum profit be if a profit of only $8 were achieved on the Blueblockers?"

Shadow Price • Examples of Managerial Questions (Blue Blocker example): • "If Blue Blocker changed company policy and allowed the number of Blueblockers produced to be at most 4 more than the number of Imitations produced, what effect would this action have on maximum profit?” • "If Blue Blocker hired 3 more workers, how much would their profit increase?”

Range of Feasibility • Examples of Managerial Question (Blue Blocker example): • “Vendor 1 will give us 4 extra units of plastic per week for $14, and Vendor 2 will give us 6 extra units of plastic per week for $17. Which vendor should we use?”

Make Vs Buy Decisions“Outsourcing” The Human Resources department of ABC Company is responsible for responding to employee inquiries on three functions: payroll, 401(k), and vacation accrual. HR has a $750,000 annual budget for these functions. Each inquiry for a function requires a certain amount of time from an ABC technical specialist and lawyer. The following table summarizes the relevant data:

Make Vs Buy Decisions

Make Vs Buy Decisions ABC does not have enough technical specialists and lawyers to respond to all these inquiries. The company has 5 full time technical specialists (10,000 hrs per yr.) and 2 full time lawyers and one part time lawyer (5,000 hours per yr.). ABC is considering outsourcing some of the work to a consulting firm, and has determined that Firm XYZ has the best reputation and prices in this area. The unit-cost of handling inquiries in house vs. having XYZ take care of them are given in the following table:

Make Vs Buy Decisions

Outsourcing LP example Let I1 = Number of Employee Payroll Inquiries to handle "in house". I2 = Number of Employee 401(k) Inquiries to handle "in house". I3 = Number of Employee Vacation Accrual Inquiries to handle "in house". O1 = Number of Employee Payroll Inquiries to outsource. O2 = Number of Employee 401(k) Inquiries to outsource. O3 = Number of Employee Vacation Accrual Inquiries to outsource. Min 50I1 + 83I2 +130I3 + 61O1 + 97O2 + 145O3 Subject to: I1+O1 = 3000 I2 + O2 = 2000 I3 + O3 = 900 2I1 + 1.5I2 + 3I3  10000 I1 + 2I2 + I3  5000 I1-3, O1-3  0

Examples of Managerial Questions: For $4,000 more in salary, ABC's part time attorney will become full time (move from 1000 hrs./yr. To 2000 hrs./yr.). Do you authorize this decision? Why or why not? Through hard work and excellent planning, ABC is able to reduce the cost of handling a 401(k) inquiry from $83/inquiry to $80/inquiry. Will this new lower cost reduce the number of 401(k) inquiries that ABC outsources? Will it affect ABC's overall cost? Through extra hard work and most excellent planning, ABC is able to reduce the cost of handling a 401(k) inquiry from $83/inquiry to $80/inquiry and reduce the cost of handling of vacation accrual inquiry from $130/inquiry to $100/inquiry. Will this new lower cost reduce the overall number of inquiries that ABC outsources? Will it affect ABC's overall cost?

Constructing the Outsourcing System. • Build the spreadsheet first. • What questions and answers does management want/need? • Build the VBA “front-end”. • What inputs/outputs? • Flexibility • What if and goal seeking

American Auto Rental American Auto Rental rents one-way to any major US. city. They presently have an imbalance in the number of cars available and the number required in various cities. There are 12 extra cars in Los Angeles, 6 in Miami, and 5 in New York. There are too few cars in Atlanta (3), Boston (8), Chicago (3) and Detroit (8). Using the following mileage chart, they wish to determine the lowest mileage arrangement to remedy their imbalance.

American Auto Rental

American Auto Rental LP example Let Xij = Number of cars to ship from location i to location j, Where i=1 to 3 and j=1 to 4, and for i, LA=1, Miami=2, and NY=3 for j, Atlanta=1, Boston=2, Chicago=3, and Detroit=4 Min 23X11 + 31X12 + 22X13 + 24X14 + 7X21 + 16X22 + 14X23 + 14X24 + 9X31 + 2X32 + 8X33 + 6X34 Subject to: X11 + X21 + X31 = 3 X12 + X22 + X32 = 8 X13 + X23 + X33 = 3 X14 + X24 + X34 = 8 X11 + X12 + X13 + X14  12 X21 + X22 + X23 + X24  6 X31 + X32 + X33 + X34  5 Xij  0

Examples of Managerial Questions: What happens if one fewer car is available from New York? What happens to the objective function if one more car is requested in Boston? What about if one less car is requested in Boston? How much will profit change if one more car is needed in Chicago?

Constructing the Auto Rental System. • Build the spreadsheet first. • What questions and answers does management want/need? • Build the VBA “front-end”. • What inputs/outputs? • Flexibility • What if and goal seeking

Building systems using models, decision analysis and statistical techniques