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Linear Programming

- Transportation Networks
- CIVE 744

Objective

- Understand the basic concepts of Linear

Programming models and learn basic methods of

solving them on computer.

Index

- I. Introduction (brief summary)
- a. Overview
- b. Structure of a linear programming model

(objective function, variables and constraints). - II. Solution of a Linear Programming Problem
- a. Verbal Description of a problem and

symbolic formulation - b. Graphical solution
- III. Computer solving of a LP model
- a. Excel solving solution
- b. Lindo solving solution

I. Introduction

- OVERVIEW
- Linear programming (LP) can ease the task of

solving a particular type of planning problem.

LP is a mathematical method or set of procedures

to solve and interpret the results of a model of

Linear functions that in conjunction represents a

phenomenon, generally related to production or

industry environments. - The mathematical technique for solving LP

problems was developed by George Dantzig in 1947

to solve planning problems in the U.S. Air Force. - Practical applications of LP
- Economicss
- Engineering
- Mathematics
- Agriculture
- Business
- Transport
- Manufacturing

I. Introduction

- Structure of a LP Model
- To analyse a problem using LP, it must be moulded

into a particular structure that at least must

contain the following components - Objective to obtain the best or optimal

solution - Activities or decision variables What to do?
- Constraints or restrictions Limits on the

availability of a resource

II. Solution of a LP problemVerbal description

- Two Mines Company
- The Two Mines Company own two different

facilities that produce an ore which, after being

crushed, is graded into three classes high,

medium and low-grade. The company has contracted

to provide a smelting plant with 12 tons of

high-grade, 8 tons of medium-grade and 24 tons of

low-grade ore per week. The two mines have

different operating characteristics as detailed

below. - Mine Cost per day ()

Production (tons/day) -

High Medium Low - X 180

6 3 4 - Y 160

1 1 6 - How many days per week should each mine be

operated to fulfill the smelting plant contract?

II. Solution of a LP problemSymbolic Formulation

- Variables
- These represent the "decisions that have to be

made" or the "unknowns". - x number of days per week mine X is

operated - y number of days per week mine Y is operated
- Note here that x gt 0 and y gt 0.
- Constraints
- Ore production constraints - balance the amount

produced with the quantity required under the

smelting plant contract - OreHigh 6x y gt 12
- Medium 3x y gt 8
- Low 4x 6y gt 24
- Days per week constraint - we cannot work more

than a certain maximum number of days a week e.g.

for a 5 day week we have - x lt 5

- y lt 5

II. Solution of a LP problemSymbolic Formulation

- Objective
- Again in words our objective is to minimize cost

which is given by - Z 180x 160y
- Hence we have the complete mathematical

representation of the problem as - Minimize Z 180x 160y
- Subject to
- 6x y gt 12
- 3x y gt 8
- 4x 6y gt 24
- x lt 5
- y lt 5
- x,y gt 0

II. Solution of a LP problemSymbolic Formulation

Optimal Solution X 1.71 Y 2.86 Z 765.71

III. Computer Solving of a LP Problem Excel

Solver

III. Computer Solving of a LP Problem Lindo

Solver

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