Introduction to quantitative methods

Transcription

1 Introduction to quantitative methods BSAD 30 Fall 2018 Agenda Introduction to quantitative methods Dave Novak Source: Anderson et al., 2015 Quantitative Methods for Business 13 th edition some slides are directly from 1 J. Loucks 2013 Cengage Learning 2 Overview Introduction to quantitative methods Background Problem solving and decision making Quantitative analysis Models of cost, revenue, and profit Iron Works example Ponderosa example 3 Background quant methods 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 4 from one point to another, etc.) Background quant methods 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.) Background quant methods Data Information Population Sample 5 6 1

2 Problem solving and decision making 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 Why I m bringing this up 7 8 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 Quantitative analysis and decision making Quantitative analysis involving 1. Determining modeling approach (what solution method are you going to use?) 2. Preparing data 3. Develop modeling solution 4. Generate output 5. Interpret results from output 9 10 Model development 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.) 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 12 2

3 Regular (X2) 8/28/2018 Some different models Some types of models Maps (2 dimensions) Music scores Architectural drawings Data flow diagrams Max P = 18x x Subject to 200 1) 0.16x x 2 0 (Cutting) 150 2) 0.47x x 2 0 (Sewing) 100 3) 0.40x x 2 0 (Decorating) 50 4) x 1, x 2 0 (Non negativity) Deluxe (X1) Constraint 1 (Cutting) Constraint 2 (Sewing) Why use models? Can t conduct trial and error experiments in real life 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 14 Why use models? Why use models? The more closely the model represents the actual situation (reality), the more accurate the conclusions and predictions will be Models are attempts to represent reality Essentially, all models are wrong, but some are useful statement attributed to statistician George Box In practice, models rarely capture the exact or full reality of a given situation Much of this class is focused on basic mathematical and statistical models Example: 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 17 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 18 3

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

5 Transforming inputs to outputs Our value for production quantity (say x=8) Controllable Inputs (Decision Variables) 25 $10 profit per unit 5 labor hours per unit 40 maximum labor hours / week Uncontrollable Inputs (Environmental Factors) Mathematical Model Max P: 10(8) s.t. 5(8) Profit = 80 Labor hours = 40 Output (Projected Results) 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 26 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 27 the best solution Model solution On page 12, the book illustrates a trail-anderror solution Production Projected Total Hours Feasible Quantity Profit of Production Solution Yes Yes Yes Yes Yes No No 28 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 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

6 Implementation and followup 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 31 models 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 32 What is our objective function verbally? What are our constraints verbally? What is our objective function mathematically? What are our constraints mathematically? What are our constraints mathematically?

7 Full problem 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 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 What is a potential issue with the solution for the example problem? 60 units Prod units Prod. 2 Controllable Inputs 39 Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < > < > 0 Mathematical Model Profit = $131, Steel Used = 2000 Output 40 Corporation (PDC) is a small real estate developer that builds only one style cottage. PDC wants to know how many cottages it must sell per month to break even (where do revenues = costs?) Once the breakeven point is identified, PDC can determine a variety of things, such as: 1) How many cottages must be built per month to generate profits of some predetermined amount ($x) 2) Gains and losses based on sales, etc. 41 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 42 7

8 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 43 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, Identify all the fixed cost components Identify all the variable cost components Write out the total cost function What is the breakeven point for cottage sales? Write out the revenue function Write out the profit function

9 Thousands of Dollars 8/28/2018 What is the profit if 12 cottages per month are built and sold? Break-Even Point = 4 Cottages Number of Cottages Sold (x) 50 Total Revenue = 115,000x Total Cost = 40, ,000x Summary Background on quant methods 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 51 9