OO-CH3_handouts.pdf - 1 Introduction to Linear Programming...

Unformatted text preview: 1 Introduction to 
 Linear Programming Chapter 3 © L.Vanhaverbeke 2 Introduction •Linear programming •Programming means planning •Model contains linear mathematical functions •An application of linear programming •Allocating limited resources among competing activities in the best possible way Applies to wide variety of situations • © L.Vanhaverbeke 20 pc/batch 3 4 Site A Site B Site C alu wood glass +
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 assembly 4 hours 12 hours 18 hours profit 1 hour — — 2 hours 3 hours 2 hours 3.000€ 5.000€ 6 3.1 Prototype Example •Wyndor Glass Co. •Produces windows and glass doors •Plant 1 makes aluminum frames and hardware Plant 2 makes wood frames Plant 3 produces glass and assembles products • • © L.Vanhaverbeke 7 Prototype Example •Company introducing two new products •Product 1: 8 ft. glass door with aluminum frame •Product 2: 4 x 6 ft. double-hung, wood-framed window •Problem: What mix of products would be most profitable? •Assuming company could sell as much of either product as could be produced •Products produced in batches of 20 © L.Vanhaverbeke 8 OR/Analytics Cycle Grossman, Mehrotra, and Sidaoui: A Student-Centered Approach to the Business School Management Science Course 47 INFORMS Transactions on Education 16(2), pp. 42–53, © 2016 INFORMS Figure 1 The Analysis Roadmap (Business Analysis Lifecycle) REAL WORLD Downloaded from informs.org by [134.184.69.192] on 20 July 2016, at 01:44 . For personal use only, all rights reserved. Business situation “COMMUNICATE” MODEL WORLD “FORMULATE” Create a model Construct highquality spreadsheet Recommend, influence, persuade, decide, and act Managerial insights Mathematical model Perform analysis on model Make sense of and explain model insights “INTERPRET” “ENGINEER” Spreadsheet model “ANALYZE” Model insights uses his understanding of the business situation to that ultimately affect the business situation. This Grossman, Mehrotra, and Sidaou (2016) A Student-Centered Approach to the Business formulate a mathematical model that captures the School quan-Management requires leadership and Transactions managerial skills to craft a Science Course, INFORMS on Education 16(2), pp. 42–53 titative relationships inherent in the business situamessage that can be understood and will be acted tion. The act of formulation is the bridge between upon by people in the organization. the Real World and the Model World. It is here that This roadmap establishes several key ideas imme© L.Vanhaverbeke the analyst makes assumptions about what to include diately, including: and exclude from the model and which data will be • The time and effort we ask students to expend is necessary. for the ultimate purpose of improved business underThe analyst must then engineer a spreadsheet model standing, insight, and action to implement the mathematical model as a com• Any analytical shortcuts (or cool technical tricks) puter program. Because the only programming lanexist in service to the business outcome, and are guage available to business students is the spreadnot valuable in and of themselves. (It is important sheet, all models are implemented in spreadsheets. As for research-active management science instructors to discussed later in the paper, careful attention needs remember this!) to be paid to the nuances of spreadsheet design and • The importance of separating formulation of a programming. model from engineering a spreadsheet. For small The analyst next will analyze the spreadsheet model problems, especially illustrative textbook examples, using one or more structured techniques or algoone can write a spreadsheet model directly with9 rithms to generate model insights that summarize the out a separate formulation step. For larger problems, analytical results. Model insights are framed in the this approach breaks down and one must consider Model World and therefore are numeric, exact, and the conceptual representation of the business situuse technical terminology. ation (formulation) separate from the programming Next, the analyst needs to interpret the model (spreadsheet engineering). insights and articulate them as business insights that • The importance of having in hand a wellsummarize the meaning of the analytical results. Busiengineered spreadsheet model as a prerequisite to ness insights are framed in the Real World and thereconducting analysis. fore are verbal, approximate (the model is only an We keep a copy of the roadmap in a separate file and approximation to reality), and use business terminolroutinely display it throughout the course, especially ogy. Interpretation is the bridge between the Model before embarking on a technical topic that students World back into the Real World. The analyst must find challenging. In our experience, this is a very helprevisit any elements of the business situation that ful mechanism for helping students to understand were assumed away during formulation. when it is appropriate to raise a business question, Finally the analyst communicates the business which resides in “real world,” and when it is approinsights to influence real-world decisions and actions priate to raise a technical question, which resides in Prototype Example •Data needed •Number of hours of production time available per week in each plant for new products Production time used in each plant for each batch of each new product Profit per batch of each new product • • © L.Vanhaverbeke 10 Prototype Example © L.Vanhaverbeke 11 Prototype Example © L.Vanhaverbeke 12 Prototype Example © L.Vanhaverbeke 13 Prototype Example •Problem can be solved graphically •Two dimensional graph with x and x as the axes •First step: identify values of x and x permitted by 1 1 2 2 the restrictions Next step: pick a point in the feasible region that maximizes value of Z • © L.Vanhaverbeke 14 Prototype Example © L.Vanhaverbeke 15 Prototype Example © L.Vanhaverbeke 16 Prototype Example © L.Vanhaverbeke 17 3.2 The Linear Programming Model •General problem terminology and examples •Resources: money, particular types of machines, vehicles, or personnel Activities: investing in particular projects, advertising in particular media, or shipping from a particular source • •Problem involves choosing levels of activities to maximize overall measure of performance © L.Vanhaverbeke 18 The Linear Programming Model © L.Vanhaverbeke 19 The Linear Programming Model •Standard form © L.Vanhaverbeke 20 The Linear Programming Model •Other legitimate forms •Minimizing (rather than maximizing) the objective function Functional constraints with greater-than-orequal-to inequality Some functional constraints in equation form Some decision variables may be negative • • • © L.Vanhaverbeke 21 The Linear Programming Model •Feasible solution •Solution for which all constraints are satisfied •Might not exist for a given problem •Infeasible solution •Solution for which at least one constraint is violated © L.Vanhaverbeke 22 © L.Vanhaverbeke 23 The Linear Programming Model •Feasible solution •Solution for which all constraints are satisfied •Might not exist for a given problem •Infeasible solution •Solution for which at least one constraint is • violated Optimal solution •Has most favorable value of objective function •Might not exist for a given problem © L.Vanhaverbeke 24 © L.Vanhaverbeke 25 © L.Vanhaverbeke 26 The Linear Programming Model •Corner-point feasible (CPF) solution •Solution that lies at the corner of the feasible region © L.Vanhaverbeke 27 © L.Vanhaverbeke 28 The Linear Programming Model •Corner-point feasible (CPF) solution •Solution that lies at the corner of the feasible • region Linear programming problem with feasible solution and bounded feasible region •Must have CPF solutions and optimal solution(s) Best CPF solution must be an optimal solution © L.Vanhaverbeke 29 © L.Vanhaverbeke 30 3.3 Assumptions of Linear Programming •Proportionality assumption •The contribution of each activity to the value of the objective function (or lefthand side of a functional constraint) is proportional to the level of the activity If assumption does not hold, one must use nonlinear programming (Chapter 13) • © L.Vanhaverbeke 31 Assumptions of Linear Programming •Additivity •Every function in a linear programming model is the sum of the individual contributions of the activities •Divisibility •Decision variables in a linear programming model may have any values • Including noninteger values •Assumes activities can be run at fractional values © L.Vanhaverbeke 32 Assumptions of Linear Programming •Certainty •Value assigned to each parameter of a linear programming model is assumed to be a known constant Seldom satisfied precisely in real applications • •Sensitivity analysis used © L.Vanhaverbeke 33 3.4 Additional Examples •Example 1: Design of radiation therapy for Mary’s cancer treatment •Goal: select best combination of beams and their intensities to generate best possible dose distribution •Dose is measured in kilorads © L.Vanhaverbeke 34 OR/Analytics Cycle Grossman, Mehrotra, and Sidaoui: A Student-Centered Approach to the Business School Management Science Course INFORMS Transactions on Education 16(2), pp. 42–53, © 2016 INFORMS Figure 1 REAL WORLD Business situation Downloaded from informs.org by [134.184.69.192] on 20 July 2016, at 01:44 . For personal use only, all rights reserved. 47 The Analysis Roadmap (Business Analysis Lifecycle) “COMMUNICATE” MODEL WORLD “FORMULATE” Create a model Construct highquality spreadsheet Recommend, influence, persuade, decide, and act Managerial insights Mathematical model Perform analysis on model Make sense of and explain model insights “INTERPRET” “ENGINEER” Spreadsheet model “ANALYZE” Model insights uses his understanding of the business situation to that ultimately affect the business situation. This Grossman, Mehrotra, and Sidaou (2016) A Student-Centered Approach to the Business formulate a mathematical model that captures the School quan-Management requires leadership and Transactions managerial skills to craft a Science Course, INFORMS on Education 16(2), pp. 42–53 titative relationships inherent in the business situamessage that can be understood and will be acted tion. The act of formulation is the bridge between upon by people in the organization. the Real World and the Model World. It is here that This roadmap establishes several key ideas imme© L.Vanhaverbeke the analyst makes assumptions about what to include diately, including: and exclude from the model and which data will be • The time and effort we ask students to expend is necessary. for the ultimate purpose of improved business underThe analyst must then engineer a spreadsheet model standing, insight, and action to implement the mathematical model as a com• Any analytical shortcuts (or cool technical tricks) puter program. Because the only programming lanexist in service to the business outcome, and are guage available to business students is the spreadnot valuable in and of themselves. (It is important sheet, all models are implemented in spreadsheets. As for research-active management science instructors to discussed later in the paper, careful attention needs remember this!) to be paid to the nuances of spreadsheet design and • The importance of separating formulation of a programming. model from engineering a spreadsheet. For small The analyst next will analyze the spreadsheet model problems, especially illustrative textbook examples, using one or more structured techniques or algoone can write a spreadsheet model directly with35rithms to generate model insights that summarize the out a separate formulation step. For larger problems, analytical results. Model insights are framed in the this approach breaks down and one must consider Model World and therefore are numeric, exact, and the conceptual representation of the business situuse technical terminology. ation (formulation) separate from the programming Next, the analyst needs to interpret the model (spreadsheet engineering). insights and articulate them as business insights that • The importance of having in hand a wellsummarize the meaning of the analytical results. Busiengineered spreadsheet model as a prerequisite to ness insights are framed in the Real World and thereconducting analysis. fore are verbal, approximate (the model is only an We keep a copy of the roadmap in a separate file and approximation to reality), and use business terminolroutinely display it throughout the course, especially ogy. Interpretation is the bridge between the Model before embarking on a technical topic that students World back into the Real World. The analyst must find challenging. In our experience, this is a very helprevisit any elements of the business situation that ful mechanism for helping students to understand were assumed away during formulation. when it is appropriate to raise a business question, Finally the analyst communicates the business which resides in “real world,” and when it is approinsights to influence real-world decisions and actions priate to raise a technical question, which resides in Example 1: Radiation Therapy Design © L.Vanhaverbeke 36 © L.Vanhaverbeke 37 Example 1: Radiation Therapy Design •Linear programming model •Using data from Table 3.7 © L.Vanhaverbeke 38 Example 1: Radiation Therapy Design •A type of cost-benefit tradeoff problem © L.Vanhaverbeke 39 Example 2: Reclaiming Solid Wastes •SAVE-IT company collects and treats four types of solid waste materials •Materials amalgamated into salable products •Three different grades of product possible •Fixed treatment cost covered by grants •Objective: maximize the net weekly profit •Determine amount of each product grade •Determine mix of materials to be used for each grade © L.Vanhaverbeke 40 Example 2: Reclaiming Solid Wastes © L.Vanhaverbeke 41 Example 2: Reclaiming Solid Wastes © L.Vanhaverbeke 42 Example 2: Reclaiming Solid Wastes • © L.Vanhaverbeke 43 © L.Vanhaverbeke 44 3.5 Formulating and Solving Linear Programming Models on a Spreadsheet •Excel and its Solver add-in •Popular tools for solving small linear programming problems © L.Vanhaverbeke 45 Formulating and Solving Linear Programming Models on a Spreadsheet •The Wyndor example •Data entered into a spreadsheet © L.Vanhaverbeke 46 Formulating and Solving Linear Programming Models on a Spreadsheet •Changing cells •Cells containing the decisions to be made C12 and D12 in the Wyndor example below • © L.Vanhaverbeke 47 Formulating and Solving Linear Programming Models on a © L.Vanhaverbeke 48 Formulating and Solving Linear Programming Models on a Spreadsheet © L.Vanhaverbeke 49 3.6 Formulating Very Large Linear Programming Models •Actual linear programming models •Can have hundreds or thousands of functional constraints Number of decision variables may also be very large • •Modeling language •Used to formulate very large models in practice Expedites model management tasks • © L.Vanhaverbeke 50 Formulating Very Large Linear Programming Models •Modeling language examples •AMPL, MPL, OPL, GAMS, and LINGO •Solvers •CPLEX, Gurobi, … © L.Vanhaverbeke 51 3.7 Conclusions •Linear programming technique applications •Resource-allocation problems •Cost-benefit tradeoff problems •Blending problems •Not all problems can be formulated to fit a linear programming model •Alternatives: integer programming or nonlinear programming models © L.Vanhaverbeke ...
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