2Overview Background Problem solving and decision making Quantitative analysisModels of cost, revenue, and profitIron Works examplePonderosa example
3Background Management science Operations research Decision science Business analyticsThese terms are often used interchangeablyOriginated during WWII – military strategic and tactical problems (choosing supply routes, storing material, getting the proper supplies from one point to another, etc.)
4Background Simplex method for solving linear programming problems Modern computing power and lots of data have resulted in all kinds of applicationsAirline industryProduction and manufacturingVehicle routing (mail, delivery, etc.)
5Problem solving and decision making 7 Steps of Problem Solving(First 5 are the process of decision making)1. Identify and define the problem2. Determine the set of alternative solutions3. Determine the criteria for evaluating alternatives4. Evaluate the alternatives5. Choose an alternative (make a decision)6. Implement the selected alternative7. Evaluate the results
6Problem solving and decision making This seems trivial and intuitive – why do we even need to discuss this?
7Quantitative analysis and decision making Further breaking down the decision-making processProblems with single objective (one criterion) are single-criterionProblems with multiple objectives (more than one criteria) are multicriteria
8Quantitative analysis and decision making Focus on being able to numerically measure the data associated with the problemDevelop mathematical expressions that describe objectives, constraints and other relationships in the problemUse some type of quantitative methods approach to make a recommendation
9Quantitative analysis and decision making Quantitative analysis process (4 steps)Model developmentData preparationModel solutionGenerate output or reports
10Model development A model is some type of representation of reality There are three (3) forms of models:Iconic models – physical replicas of real objects (i.e. model car, a model of a building, etc.)Analog models – physical form, but do not physically represent object being modeled (speedometer, thermometer, graph of social network, etc.)
11Model developmentMathematical 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
12Some different models Some types of models Maps (2 dimensions) Music scoresArchitectural drawingsData flow diagramsMathematical modelsMax P = 18x1 + 12x2Subject to0.16x x2 ≤ 0 (Cutting)0.47x x2 ≤ 0 (Sewing)0.40x x2 ≤ 0 (Decorating)x1, x2 ≥ 0 (Non negativity)
13Why use models? Experimenting with models generally: requires less timeis less expensiveinvolves less riskcan enable you to investigate a situation that cannot be represented in realityThe more closely the model represents the real situation, the more accurate the conclusions and predictions will be
14Why use models? Models are attempts to represent reality “Essentially, all models are wrong, but some are useful” quote attributed to statistician George BoxIn practice, models rarely capture “the exact” or “full” reality of a given situation
15Mathematical modelsObjective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing costP=10x (from slide #11)10x is the OBJECTIVE FUNCTION
16Mathematical modelsConstraints – a set of restrictions, limitations, or assumptions such as a limit in production capacity or a fixed number of labor hoursTake 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 405x ≤ 40 is the LABOR CONSTRAINT
17Mathematical modelsUncontrollable Inputs – environmental factors that are not under the control of the decision makerIn 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
18Mathematical modelsDecision Variables – controllable inputs; choices made by the decision maker, such as the number of units of a product to produceIn our model, the decision maker (or manager) has one choice – how much of x to produce
19Mathematical modelsOur complete mathematical model for the simple production problems is:Maximize Profit 10x (objective function)Subject to: ) 5x ≤ 40 (labor constraint)2) x ≥ 0 (non-negativity constraint)
20Mathematical models Mathematical models can be: Deterministic - if all uncontrollable inputs (profit, labor hours, etc.) to the model are known with certainty and cannot varyFor this class, we commonly make this assumption, although in reality this is RARELY the case
21Mathematical modelsStochastic (or probabilistic)- if any uncontrollable inputs (profit, labor hours, etc.) are uncertain and subject to variationFor 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 5Obviously the “answer” to the problem (how many units of x should we produce?) will change if the production time changes
22Mathematical modelsCost/benefit, usability, whether managers understand the model, etc. – these are considerations that must be made when selecting a modelIt 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
23Transforming inputs to outputs $10 profit per unit5 labor hours per unit40 – maximum labor hours / weekUncontrollable Inputs(Environmental Factors)Our value for production quantity (say x=8)Profit = 80Labor hours = 40ControllableInputs(DecisionVariables)Output(ProjectedResults)MathematicalModelMax P: 10(8)s.t (8) ≤ 408 ≥ 0
24Data preparation Data preparation is not a trivial step in modeling TimeData errorsUnderstanding of dataA model with 50 decision variables (x1, x2, … x50) and 25 constraints could have over 1,300 data elementsMight need IS expertise
25Model solutionThe analyst attempts to identify the alternative (the decision variable values) that provides the “best” output for the modelThe “best” output is the optimal solutionIf the alternative does not satisfy all of the model constraints, it is rejected as being infeasible, regardless of the objective function valueIf the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution
26Model solutionOn page 12, the book illustrates a trail-and-error solutionProductionProjectedTotal HoursFeasibleQuantityProfitof ProductionSolutionYes220104406603088010050No12120
27Model testing and validation Goodness/accuracy of a model may not be assessed until solutions are generatedSmall test problems having known, or at least expected, solutions can be used for model testing and validationIf the “test” model generates expected solutions, use the model on the full-scale problem
28Model testing and validation If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action such as:Collection of more-accurate input dataModification of the model
29Reports / outputA managerial report, based on the results of the model, should be preparedReport should be easily understood by the decision maker and should contain the analyst’s interpretation of the resultsrecommended decisionother pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model)
30Implementation and follow-up Successful implementation of model results is of critical importanceSecure as much user involvement as possible throughout the modeling processContinue to monitor the contribution of the modelIt might be necessary to refine or expand the modelThings change over time, and so should most models
31Iron Works, Inc. exampleIron Works, Inc. manufactures two products made from steel and just received this month's allocation of b pounds of steel. It takes a1 pounds of steel to make a unit of product 1 and a2 pounds of steel to make a unit of product 2.Let x1 and x2 denote this month's production level ofproduct 1 and product 2, respectivelyLet p1 and p2 denote the unit profits for products 1 and 2, respectivelyIron 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
32Iron Works, Inc. example Write the mathematical model What is our objective function verbally?What is our objective function mathematically?
33Iron Works, Inc. example Write the mathematical model What are our constraints verbally?
34Iron Works, Inc. example Write the mathematical model What are our constraints mathematically?
35Iron Works, Inc. exampleWrite the mathematical modelFull problem
36Iron Works, Inc. example Solve for a particular situation Suppose b = 2000, a1 = 2, a2 = 3, m = 60, u = 720, p1 = 100, and p2 = 200Rewrite the model with these specific values for the uncontrollable inputs
37Iron Works, Inc. example Uncontrollable Inputs $100 profit per unit Prod. 1$200 profit per unit Prod. 22 lbs. steel per unit Prod. 13 lbs. Steel per unit Prod. 22000 lbs. steel allocated60 units minimum Prod. 1720 units maximum Prod. 20 units minimum Prod. 260 units Prod. 1units Prod. 2Max 100(60) + 200(626.67)s.t. 2(60) + 3(626.67) < 2000> 60< 720>Profit = $131,333.33Steel Used = 2000Controllable InputsOutputMathematical Model
38Iron Works, Inc. exampleWhat is a potential issue with the solution for the example problem?
39Ponderosa 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 $115,000Land for each cottage costs $55,000 and lumber, supplies, and other materials run another $28,000 per cottageTotal labor costs are approximately $20,000 per cottage
40Ponderosa Development Corp. example Ponderosa leases office space for $2,000 per monthThe cost of supplies, utilities, and leased equipment runs another $3,000 per monthThe one salesperson of PDC is paid a commission of $2,000 on the sale of each cottagePDC has seven permanent office employees whose monthly salaries are given on the next slide
41Ponderosa 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
42Ponderosa Development Corp. example Identify all the fixed costs components and the marginal cost and revenue for each cottage
43Ponderosa Development Corp. example Write out the monthly cost functionWrite out the monthly revenue functionWrite out the monthly profit function
44Ponderosa Development Corp. example What is the breakeven point for monthly cottage sales?
45Ponderosa Development Corp. example What is the monthly profit if 12 cottages per month are built and sold?
46Ponderosa Development Corp. example 1200Total Revenue =115,000x1000800Thousands of Dollars600Total Cost =40, ,000x400200Break-Even Point = 4 Cottages12345678910Number of Cottages Sold (x)
47Summary Background Problem solving and decision making Quantitative analysisDifferent types of modelsHow to set up a mathematical modelModels of cost, revenue, and profitIron Works examplePonderosa example