Introduction to Linear Programming Chapter 3 Introduction to Linear Programming

Introduction Linear programming An application of 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

Prototype Example Formulating the model From bottom row of Table 3.1 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 𝑍 = 3 𝑥1+5𝑥2

Prototype Example Constraints (see Table 3.1) 𝑥1≤4 2𝑥2≤12 3𝑥1+2𝑥2≤18 𝑥1≥0 𝑥2≥0 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

Prototype Example

Prototype Example

Prototype Example

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

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

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

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

Example 2: Reclaiming Solid Wastes

Example 2: Reclaiming Solid Wastes Decision variables 𝑥𝑖𝑗=𝑧𝑖𝑗𝑦𝑖 (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