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MBA 691 Introduction to Modeling and Linear Programming

MBA 691 Introduction to Modeling and Linear Programming. Dr. Michael F. Gorman. Lecture Outline. Introduction to the Analytic approach to business problem solving Analytical and experiential based decision making A framework for approaching business problem solving

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MBA 691 Introduction to Modeling and Linear Programming

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  1. MBA 691 Introduction to Modeling and Linear Programming Dr. Michael F. Gorman

  2. Lecture Outline • Introduction to the Analytic approach to business problem solving • Analytical and experiential based decision making • A framework for approaching business problem solving • Introduction to Management Science/Operations Research/Decision Sciences/Production Operations Management • Why study it? • Introduction to modeling • What is a model? • What are attributes of a good model? • What types of models will we cover in this class? • Introduction to Mathematical Programming • General structure of a math program • Linear Programming • Integer Programming • Non-linear (quadratic) programming

  3. Analytic Approach to Business Decision Making • This course presents methods and techniques which support: • Quantitative • Analytical • Approach to business problem solving and decision making • Quantitative • Fact/data based support for decision making • Analytical Approach • Breaks down problem to its component parts to allow better understanding of the behavior of the component interactions and thus, the whole problem

  4. Experiential approach to business decision making • Experiential approach: • Historical experience: What happened last time? • Expert opinion: What will happen this time? • Intuition: what is the current feel for how things work together? • Can analytical and Experiential approaches be reconciled? • Trick is to balance the “domain expertise” of the expert with the objectivity and thoroughness of the analytic approach • The model is just another opinion at the table • Although experiential and analytical approaches often conflict, their strengths can be combined to arrive at improved business decisions • Often the “business expert” (old head) and “technical” (propeller head) expert are different people, follow different career paths. • Move the process from combative to cooperative

  5. Analytical (Pros) Objective Exhaustive/Thorough consideration of options Forces logical considerations of component interactions Analytical (Cons) Can misinterpret (or ignore) extreme information to draw errant conclusions Difficult to calibrate the “non-quantifiables” Experiential (Cons) Subjective: bias, perception Limited Options Explored Can oversimplify/misstate component interactions Subject to organizational pressure Experiential (Pros) Can handle outliers well Handles “non-quantifiables” well Analytic vs. Experiential We try to balance the “insights” analytics bring to decision making from the “insanity” that can result when methods are misapplied.

  6. A framework for the decision making process Observe/ Collect Data Implement Evaluate Problem Definition Formulate Model Verify Model Select Alternative Present Results Feedback Loop

  7. What is - Management Science - Operations Research - Decision Science?What Operations Research (Preferred term) Is:In a nutshell, operations research (O.R.) is the discipline of applying advanced analytical methods to help make better decisions.By using techniques such as mathematical modeling to analyze complex situations, operations research gives executives the power to make more effective decisions and build more productive systems based on: - More complete data - Consideration of all available options - Careful predictions of outcomes and estimates of risk - The latest decision tools and techniques

  8. In research, these are highly technical methods for solving complicated problems (often in business and engineering)In this class, these topics are described as the use of quantitative methods and analytical tools to solve business problems.

  9. Why study Management Science/Operations Research/Decision Sciences? • Build analytical thinking skills - focus on relevant variables • Modeling/spreadsheet skills are valuable to employers- for both “producer” and “consumer” of model results • Leverage “dominant” technology - supply chain planning/execution, yield mgmt, asset management • Exploit computer revolution - easier, cheaper and more powerful tools- ubiquitous data – what to do with it? • Combine the MBA and the techie into one person- eliminate the “expert” vs. “analytical” debate • Integrates other disciplines (marketing/finance/operations)

  10. Introduction to Modeling: • A model is • A simplified representation or approximation of a real-world system • Designed to give insight into interrelationships of key interrelating variables • Used to assist in drawing conclusions on the real-world system • Examples of models: • Spreadsheet model – revenue/cost/profit of a business • Weather model – prediction of precipitation, temperature, wind • Engineering model – structural integrity of a physical structure • Sports model – Sagarin team rankings • Physiology – life habits (diet/sleep/exercise) and body response (weight/body fat/longevity) • Statistical – relationship of random variables • What is a good model?A good model is as simple as possible, but no simpler. • A good model is useful model: • Provides insights into proper decisions • Fits into business decision-making process • A good model is a simple model: • Captures key attributes/relations of the real world system, but no more

  11. The role of assumptions • An assumption removes some complexity of the problem in order for it to be solvable. (AKA - simplifying assumption) • Assumption of linear costs • Assumption of no price impacts • Assumption of no interaction between two variables • Assumption of forecast accuracy • Assumptions are your friends! • Assumptions are necessary to solve problems • Without them, we are stuck with “real-world” complexity that is essentially unsolvable. • A good assumption: • Simplifies the problem statement/model significantly • Is intuitively plausible to the “domain experts” • Has negligible impact on the model results

  12. Models covered in this class:Prescriptive and Descriptive • Prescriptive models make a recommendation on a course of action • Mathematical Programming models • Also known as “Optimization Models”: Linear programming, Integer, quadratic • Given known inputs interaction and limitations • Prescribe (or recommend) an optimal strategy • Decision-making models • Decision making under uncertainty (probabilistic models) • Game-theoretic models (considering the reaction of others to a decision) • Descriptive models describe and outcome of a course of action • Statistical Models • Specifically - Regression analysis • Given a number of variables with unknown quantitative relationship, • Estimate the impact of one set of variables on another • Simulation Models • Given random or unspecified variation in inputs and dynamic interactions between inputs • Describe likely outcomes of a given set of inputs (“What-if” analysis)

  13. An example of a business problem which requires multiple model types • Optimal pricing for a suite of interrelated products • Interrelated by their cost structures – cost of one affects the cost of another • Interrelated by their demand – consumption of one affects the consumption of another • Example: Rail freight transportation services • Regression analysis • Estimates the cost curves, demand curves • Optimization • Recommends optimal prices, given costs and demands • (may be a math program or decision-making/game theoretic model) • Simulation • Evaluates “robustness” of recommendations against future unknown variations (e.g. customer response to price change, shifts in demand)

  14. Introduction to Math Programming • A Math Programming Model has the following ingredients: • An Objective – • a mathematical expression • to be maximized or minimized • By changing values of decision variables • E.g. Maximize Profits; Minimize Total Distance, etc. • And Constraints - • Relationships among the decision variables which somehow limits their use • “Subject To:” – used to identify the constraints • E.g. • Minimum output produced must be greater than or equal to customer demand • Maximum used must be less than or equal to total available inputs • Production of one product must equal production of another

  15. Types of Math programming • Math programming has different “functional” forms: • Linear Program (LP) – • Objective function and constraints are linear • This means decision variables are multiplied only by a constant, never raised to a power or multiplied times each other • Constraints are one of: ≥, ≤, or = • All decision variables are continuous (can take on a fractional value) • Integer Program (IP) • Like an LP, but (some) decision variables can take on only integer values • E.g. How many jobs are assigned to a machine • Special case IP: variables take on values of only 0 or 1 (binary) • E.g. Should a warehouse be open or not? (yes or no; nothing in between) • Quadratic Program (QP) • Like an LP, but the objective can take on a squared value • E.g. Pricing maximization, portfolio variance minimization

  16. An example: Tables and Chairs • A furniture manufacturer is deciding on tables and chairs production for the upcoming quarter. • Each chair sold nets the manufacturer $20; Each table makes $30 in profit • The manufacture has a supply of 500 board feet each week and 100 labor hours to allocate • Each chair takes 10 board feet of wood; each table takes 20 board feet • Each chair requires 4 labor hours; each table takes 2 hours of labor • The manufacturer wants to produce no more than 40 chairs and no more than 20 tables • What should the manufacturer do?

  17. Identifying the Key ingredients to the math model • Decision Variables: • Manufacturer must decide how many tables and chairs to produce • Call them “T” and “C” • The objective and constraints must be functions of these decision variables • Objective: • The manufacturer’s objective? Unstated in problem • If it is to minimize costs – should produce 0 units • If it is to maximize T or C, should produce T=25 or C=20 • It may be safe to assume (or clarify with the business owner) that this manufacturer wants to MAXIMIZE PROFITS • Maximize Profit = $20*C + $30*T • Constraints • Total board feet used = 20*T + 10*C ≤ 500 • Total labor used = 4*C + 2* L ≤ 100 • Total Chairs = C ≤ 40 • Total Tables = T ≤ 20 • (Implied that T, C are both >= 0)

  18. Algebraic Formulation:Tables and Chairs Problem • Objective: • Max Profit = 20C + 30T • Subject to: • 4C + 2T <= 100 (Labor) • 10C + 20T <= 500 (Wood) • T <=20 (Tables) • C<= 40 (Chairs)

  19. Example in a Picture: Tables or Chairs? Table Constraint Problem: Maximize Profit Wood Supply = 500 board feet Labor Supply = 100 hours Chair uses 10 wood and 4 labor; Table uses 20 wood and 2 labor; Chair = $20 profit; Table = $30 profit Customer orders: 20 Tables, 40 chairs Problem Statement: Max Profit = 20C + 30T Subject to 4C + 2T <= 100 (Labor) 10C + 20T <= 500 (Wood) T <=20 C<= 40 Chairs 50 Chair Constraint 40 Wood Constraint 25 (16.67T,16.67C) (20T,10C) Feasible Solution Space Labor constraint 25 20 50 Tables

  20. Linear Programming ModelHow does it work? • The “Simplex” method • Like knowing how internal combustion works in order to drive a car • Nice to know, but not necessary… • Linear Programs often start with a feasible solution • Meets all constraints • Example: produce 0 units (T=0, C=0) • Model Looks at trade-offs between variables • Benefit produced by increasing one output (increased profit) • Compared to the cost of giving up an alternative output (each chair means you give up some capacity for building tables) • Change current solution by moving in profit-improving directions • Continually try different combinations of outputs • Until no profit-improving possibilities exist • Efficiently checks only the intersections of constraints • Because of the linear nature of the problem, we know that the optimal solution will end up at a corner of two constraints

  21. Tables And Chairs: Excel Formulation

  22. Example in a Picture: Tables or Chairs? Table Constraint Chairs 50 Chair Constraint 40 Wood Constraint 25 Optimum Solution: Approx. (16.67T,16.67C) S0’: Profit = 500 (20T,10C) Labor constraint S0: Profit=$0 25 20 S3: Profit = $833 Tables S2: Profit = 800 S1: Profit = 600

  23. Adding Solver Add-InTools – Add Ins --- Solver Add-in

  24. If Solver Add-In is not Found in Tools – Add Ins You can find SOLVER.XLA in the directory above. Browse to this directory, then double-click to include.

  25. Special Case LP Results that can occur: • Unbounded • You want to maximize something, and there are no constraints on the decision variables • The optimal solution is to produce infinite!! • Unbounded problems are usually missing important constraints, or that you have reversed a sign that makes a bad thing good • Infeasible • By the time you implement all the constraints, the feasible region is null (nada, zip, zilch, nothing, empty) • The problem has no solution • Usually the constraints are implemented incorrectly (units, etc.) • Sometimes a cost is represented as a constraint

  26. Sensitivity Analysis • Sometimes we want to understand how much an optimum solution will change if some input data changed • Remember, we have assumed perfect data and information to this point • Examples: • How much would the Table constraint have to be reduced before it became “important” – has an effect on the solution? • Currently, maximum tables is set at 20; if it were set at 16.67 or less (a reduction of 3.33), it would become important to (or change) the solution • What if we could sell tables for a higher price (more profit); Would that change the optimal mix? • This is like changing the slope of the isoprofit curve • What would it be worth to expand labor or wood constraints? • A parallel shift of one of the constraints

  27. Sensitivity Analysis:Making a loose table constraint tight Table Constraint Chairs 50 Chair Constraint 40 Reducing the max table Constraint by 3.3 makes it Tight; further reductions reduces the objective function as the optimal solution moves along the labor constraint. Wood Constraint 25 Optimum Solution: Approx. (16.67T,16.67C) Labor constraint 25 16.67 Tables

  28. Sensitivity analysis: Price of chairs falls by $5 Table Constraint Wood constraint forces tradeoff Between tables and chairs of 2:1 The profit ratio of tables and chairs is 3:2 So we produce more chairs. If tables fetched $10 more, or Chairs $5 less, then we would Change the optimum to (20T, 10C) Chairs 50 Chair Constraint 40 Wood Constraint Optimum Solution: Approx. (16.67T,16.67C) OR (20T, 10C) Or anywhere in between 25 S0’: Profit = 375 (20T,10C) Labor constraint S0: Profit=$0 20 25 S3: Profit = $750 Tables S2: Profit = 750 S1: Profit = 600

  29. Sensitivity analysis:One more unit of labor Table Constraint Chairs 50 Chair Constraint One more unit of labor allows 1/3 more chairs; 1/6 less tables (as dictated by wood constraint) (17C, 16.5 T) +6.67 in Chair profit -5.00 in Table profit Net profit change of $1.67 40 Wood Constraint 25 Optimum Solution: Approx. (16.5T,17C) Labor constraint 25 20 S3: Profit = $835 Tables

  30. Presentation of Results • We recommend 16.67 chairs and 16.67 tables be produced per week in order to earn $833 in profit per week • All labor and wood inputs are fully utilized with this solution • We suggest if it is possible, that we expand labor as much as 70 hours per week in order to gain $1.66 in profit per week increase per labor hour (116 total potential)

  31. Sensitivity analysis terms:Adjustable Cells • In Excel, two tables are produced in a sensitivity analysis: • “Adjustable Cells” and “Constraints” • Adjustable Cells Sensitivity analysis • Final Value – optimal solution for the decision variable • E.g. 16.67 Tables in our base solution • Reduced cost – amount the coefficient in the objective function on a decision has to change before it appears in the solution • No example in our problem – both Tables and Chairs have a positive value in the final solution • Allowable increase/decrease – amount an objective function coefficient can change before changing the optimal combination of outputs • E.g. Chair profit can go down $5, or Table profit can go up $10, without changing the chair/table mix of 16.67 apiece • Objective coefficient – the input data value of each decision variable to the objective function • E.g. Tables = $30; Chairs = $20

  32. Sensitivity analysis terms:Constraints • Constraints: • Final value - the final value of the use of a resource in a constraint • E.g. we use all 100 units of labor and 500 units of wood in the base solution • Shadow Price – the amount the objective function would grow if the constraint were expanded by one unit • E.g. The objective function increased by $1.67 by expanding the labor constraint one unit • Allowable increase or decrease • Amount a constraint can change before the shadow price changes • In the case of the Table constraint, can decrease the maximum table constraint by 3.33 before the shadow price changes from 0 • Constraint R.H. (Right Hand) side • Original value (input data) of the constraint maximum or minimum • Called R.H. side because typically constraints are stated as:f(decision variables) <= maxand “max” is on the right hand side of the inequality.

  33. Problem with Chairs and Tables Solution! • A marketing expert exasperatedly informs us that it is crazy to produce equal numbers of chairs and tables • First of all, we sell chairs at a ratio of no less than 4:1 over tables • (that make sense) • And chairs break more often than tables, so we have a thriving replacement business • (We can exceed 4:1, but we better not fall below it) • Going back to our operations expert, we inquire if this is true. • Of course, he indicates, “&%*$ yes, it’s true, why do you think I told you we never produce more than 20 tables or 40 chairs??” • We now realize we never had a well-defined problem… so we go back and reformulate the problem with this new constraint

  34. Revised Table and Chairs Problem Table Constraint Chairs 50 Chair >= 4* Table With the new 4:1 Chair to table constraint, more chair must be produced. Labor is used up more quickly, and we have excess wood. 40 Wood Constraint New Optimum: (5.56T,22.24C) 25 Feasible Solution Space (16.67T,16.67C) Labor constraint 25 50 Tables

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