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## Introduction to Linear Programming

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**Chapter 3**Introduction to Linear Programming**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**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**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**Prototype Example**• Products produced in batches of 20 • 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**Prototype Example**• Formulating the model x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Z = total profit per week (thousands of dollars) from producing these two products • From bottom row of Table 3.1**Prototype Example**• Constraints (see Table 3.1) • Classic example of resource-allocation problem • Most common type of linear programming problem**Prototype Example**• Problem can be solved graphically • Two dimensional graph with x1 and x2 as the axes • First step: identify values of x1 and x2 permitted by the restrictions • See Figures 3.1 and Figure 3.2 • Next step: pick a point in the feasible region that maximizes value of Z • See Figure 3.3**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**The Linear Programming Model**• Standard form**The Linear Programming Model**• Other legitimate forms • Minimizing (rather than maximizing) the objective function • Functional constraints with greater-than-or-equal-to inequality • Some functional constraints in equation form • Some decision variables may be negative**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**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**3.3 Assumptions of Linear Programming**• Proportionality assumption • The contribution of each activity to the value of the objective function (or left-hand 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)**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**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**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**Example 1: Radiation Therapy Design**• Linear programming model • Using data from Table 3.7**Example 1: Radiation Therapy Design**• A type of cost-benefit tradeoff problem**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**Example 2: Reclaiming Solid Wastes**• Decision variables (for i = A, B, C; j = 1,2,3,4) number of pounds of material j allocated to product grade i per week • See Pages 56-57 in the text for solution**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**Formulating and Solving Linear Programming Models on a**Spreadsheet • The Wyndor example • Data entered into a spreadsheet**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**Formulating and Solving Linear Programming Models on a**Spreadsheet**Formulating and Solving Linear Programming Models on a**Spreadsheet**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**Formulating Very Large Linear Programming Models**• Modeling language examples • AMPL, MPL, OPL, GAMS, and LINGO • Example problem with a huge model • See Pages 73-78 in the text**3.7 Conclusions**• Linear programming technique applications • Resource-allocation problems • Cost-benefit tradeoffs • Not all problems can be formulated to fit a linear programming model • Alternatives: integer programming or nonlinear programming models