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Introduction BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

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Overview Background Problem solving and decision making Quantitative analysis Models of cost, revenue, and profit Iron Works example Ponderosa example

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Background Management science Operations research Decision science Business analytics These terms are often used interchangeably Originated during WWII – military strategic and tactical problems (choosing supply routes, storing material, getting the proper supplies from one point to another, etc.)

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Background Simplex method for solving linear programming problems Modern computing power and lots of data have resulted in all kinds of applications Airline industry Production and manufacturing Vehicle routing (mail, delivery, etc.)

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Problem solving and decision making 7 Steps of Problem Solving (First 5 are the process of decision making) 1. Identify and define the problem 2. Determine the set of alternative solutions 3. Determine the criteria for evaluating alternatives 4. Evaluate the alternatives 5. Choose an alternative (make a decision) 6. Implement the selected alternative 7. Evaluate the results

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Problem solving and decision making This seems trivial and intuitive – why do we even need to discuss this?

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Quantitative analysis and decision making Further breaking down the decision-making process Problems with single objective (one criterion) are single-criterion Problems with multiple objectives (more than one criteria) are multicriteria

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Quantitative analysis and decision making Quantitative analysis Focus on being able to numerically measure the data associated with the problem Develop mathematical expressions that describe objectives, constraints and other relationships in the problem Use some type of quantitative methods approach to make a recommendation

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Quantitative analysis and decision making Quantitative analysis process (4 steps) 1. Model development 2. Data preparation 3. Model solution 4. Generate output or reports

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Model development A model is some type of representation of reality There are three (3) forms of models: 1. Iconic models – physical replicas of real objects (i.e. model car, a model of a building, etc.) 2. Analog models – physical form, but do not physically represent object being modeled (speedometer, thermometer, graph of social network, etc.)

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Model development 3. Mathematical models – using mathematical formulas and expressions to represent a real world problem (total profit, total cost, etc.) Profit (P) = 10x, where x is the number of units sold and $10 is the profit from each unit

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Some different models Some types of models Maps (2 dimensions) Music scores Architectural drawings Data flow diagrams Mathematical models Max P = 18x x 2 Subject to 1)0.16x x 2 ≤ 0 (Cutting) 2)0.47x x 2 ≤ 0 (Sewing) 3)0.40x x 2 ≤ 0 (Decorating) 4)x 1, x 2 ≥ 0 (Non negativity)

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Why use models? Experimenting with models generally: requires less time is less expensive involves less risk can enable you to investigate a situation that cannot be represented in reality The more closely the model represents the real situation, the more accurate the conclusions and predictions will be

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Why use models? Models are attempts to represent reality “Essentially, all models are wrong, but some are useful” quote attributed to statistician George Box In practice, models rarely capture “the exact” or “full” reality of a given situation

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Mathematical models Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost P=10x (from slide #11) 10x is the OBJECTIVE FUNCTION

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Mathematical models Constraints – a set of restrictions, limitations, or assumptions such as a limit in production capacity or a fixed number of labor hours Take our production problem P=10x (from slide #11) Assume 5 hours of labor are required to produce each unit of x, and the total hours of labor available each week are 40 5x ≤ 40 is the LABOR CONSTRAINT

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Mathematical models o Uncontrollable Inputs – environmental factors that are not under the control of the decision maker In our simple model, the profit per unit ($10), the production time per unit (5 hours), and the production capacity (40 hours) are environmental factors not under the control of the manager or decision maker

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Mathematical models Decision Variables – controllable inputs; choices made by the decision maker, such as the number of units of a product to produce In our model, the decision maker (or manager) has one choice – how much of x to produce

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Mathematical models Our complete mathematical model for the simple production problems is: Maximize Profit 10x (objective function) Subject to: 1) 5x ≤ 40 (labor constraint) 2) x ≥ 0 (non-negativity constraint)

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Mathematical models Mathematical models can be: Deterministic - if all uncontrollable inputs (profit, labor hours, etc.) to the model are known with certainty and cannot vary For this class, we commonly make this assumption, although in reality this is RARELY the case

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Mathematical models Stochastic (or probabilistic)- if any uncontrollable inputs (profit, labor hours, etc.) are uncertain and subject to variation For example, if the number of hours of production time per unit of x could vary between 3 and 6 hours – right now, we assume this value is fixed at 5 Obviously the “answer” to the problem (how many units of x should we produce?) will change if the production time changes

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Mathematical models Cost/benefit, usability, whether managers understand the model, etc. – these are considerations that must be made when selecting a model It could be that a less complicated and less precise model is more appropriate than a more complicated model due to cost and ease of use considerations

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Transforming inputs to outputs Uncontrollable Inputs (Environmental Factors) Uncontrollable Inputs (Environmental Factors) ControllableInputs(DecisionVariables)ControllableInputs(DecisionVariables) Output(Projected Results) Results)Output(Projected MathematicalModelMathematicalModel $10 profit per unit 5 labor hours per unit 40 – maximum labor hours / week Our value for production quantity (say x=8) Max P: 10(8) s.t. 5(8) ≤ 40 8 ≥ 0 Profit = 80 Labor hours = 40

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Data preparation Data preparation is not a trivial step in modeling Time Data errors Understanding of data A model with 50 decision variables (x 1, x 2, … x 50 ) and 25 constraints could have over 1,300 data elements Might need IS expertise

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Model solution The analyst attempts to identify the alternative (the decision variable values) that provides the “best” output for the model The “best” output is the optimal solution If the alternative does not satisfy all of the model constraints, it is rejected as being infeasible, regardless of the objective function value If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution

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Model solution On page 12, the book illustrates a trail-and- error solution ProductionProjected Total Hours Feasible QuantityProfit of Production Solution 0 00Yes Yes Yes Yes Yes No No

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Model testing and validation Goodness/accuracy of a model may not be assessed until solutions are generated Small test problems having known, or at least expected, solutions can be used for model testing and validation If the “test” model generates expected solutions, use the model on the full-scale problem

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Model testing and validation If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action such as: Collection of more-accurate input data Modification of the model

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Reports / output A managerial report, based on the results of the model, should be prepared Report should be easily understood by the decision maker and should contain the analyst’s interpretation of the results recommended decision other pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model)

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Implementation and follow- up Successful implementation of model results is of critical importance Secure as much user involvement as possible throughout the modeling process Continue to monitor the contribution of the model It might be necessary to refine or expand the model Things change over time, and so should most models

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Iron Works, Inc. example Iron Works, Inc. manufactures two products made from steel and just received this month's allocation of b pounds of steel. It takes a 1 pounds of steel to make a unit of product 1 and a 2 pounds of steel to make a unit of product 2. Let x 1 and x 2 denote this month's production level of product 1 and product 2, respectively Let p 1 and p 2 denote the unit profits for products 1 and 2, respectively Iron Works has a contract calling for at least m units of product 1 this month. The firm's facilities are such that at most u units of product 2 may be produced monthly

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Iron Works, Inc. example Write the mathematical model What is our objective function verbally? What is our objective function mathematically?

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Iron Works, Inc. example Write the mathematical model What are our constraints verbally?

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Iron Works, Inc. example Write the mathematical model What are our constraints mathematically?

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Iron Works, Inc. example Write the mathematical model Full problem

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Iron Works, Inc. example Solve for a particular situation Suppose b = 2000, a 1 = 2, a 2 = 3, m = 60, u = 720, p 1 = 100, and p 2 = 200 Rewrite the model with these specific values for the uncontrollable inputs

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Iron Works, Inc. example Uncontrollable Inputs $100 profit per unit Prod. 1 $200 profit per unit Prod. 2 2 lbs. steel per unit Prod. 1 3 lbs. Steel per unit Prod lbs. steel allocated 60 units minimum Prod units maximum Prod. 2 0 units minimum Prod. 2 $100 profit per unit Prod. 1 $200 profit per unit Prod. 2 2 lbs. steel per unit Prod. 1 3 lbs. Steel per unit Prod lbs. steel allocated 60 units minimum Prod units maximum Prod. 2 0 units minimum Prod units Prod units Prod units Prod units Prod units Prod units Prod. 2 Controllable Inputs Profit = $131, Steel Used = 2000 Profit = $131, Steel Used = 2000 OutputOutput Mathematical Model Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < > > < < > > 0 Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < > > < < > > 0

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Iron Works, Inc. example What is a potential issue with the solution for the example problem?

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Ponderosa Development Corp. example Ponderosa Development Corporation (PDC) is a small real estate developer that builds only one style cottage. The selling price of the cottage is $115,000 Land for each cottage costs $55,000 and lumber, supplies, and other materials run another $28,000 per cottage Total labor costs are approximately $20,000 per cottage

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Ponderosa Development Corp. example Ponderosa leases office space for $2,000 per month The cost of supplies, utilities, and leased equipment runs another $3,000 per month The one salesperson of PDC is paid a commission of $2,000 on the sale of each cottage PDC has seven permanent office employees whose monthly salaries are given on the next slide

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Ponderosa Development Corp. example Employee Monthly Salary President $10,000 VP, Development 6,000 VP, Marketing 4,500 Project Manager 5,500 Controller 4,000 Office Manager 3,000 Receptionist 2,000

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Ponderosa Development Corp. example Identify all the fixed costs components and the marginal cost and revenue for each cottage

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Ponderosa Development Corp. example Write out the monthly cost function Write out the monthly revenue function Write out the monthly profit function

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Ponderosa Development Corp. example What is the breakeven point for monthly cottage sales?

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Ponderosa Development Corp. example What is the monthly profit if 12 cottages per month are built and sold?

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Ponderosa Development Corp. example Number of Cottages Sold (x) Thousands of Dollars Break-Even Point = 4 Cottages Total Cost = Total Cost = 40, ,000x 40, ,000x Total Cost = Total Cost = 40, ,000x 40, ,000x Total Revenue = Total Revenue = 115,000x 115,000x Total Revenue = Total Revenue = 115,000x 115,000x

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Summary Background Problem solving and decision making Quantitative analysis Different types of models How to set up a mathematical model Models of cost, revenue, and profit Iron Works example Ponderosa example

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