LINEAR PROGRAMMING PROBLEM

Linear : form meant a mathematical expression of the type aa11xx1 1 + a+ a22xx2 2 + ……….+ a+ ……….+ annxxnn

where awhere a11, a, a22, ….., a, ….., ann are constants, are constants,

and xand x11, x, x22, ………, x, ………, xnn are variables. are variables.

Programming : refers to the process of determining a particular program or plan of action.

Linear Programming Problem(LPP): Technique for optimizing(maximizing/minimizing) a linear function of variables called the ‘OBJECTIVE FUNCTION’ subject to a set of linear equations and/or inequalities called the ‘CONSTRAINTS’ or ‘RESTRICTIONS’.

FORMULATION OF LP PROBLEMS

LP Model FormulationObjective function

a linear relationship reflecting the objective of an operationmost frequent objective is to maximize profit or to minimize cost.

Decision variablesan unknown quantity representing a decision that needs to be made. It is the quantity the model needs to determine

Constrainta linear relationship representing a restriction on decision making

Steps in Formulating the LP Problems

1.1. Define the objective. (min or max)Define the objective. (min or max)

2.2. Define the decision variables. Define the decision variables.

3.3. Write the mathematical function for the objective. Write the mathematical function for the objective.

4.4. Write the constraints. Write the constraints.

5.5. Constraints can be in Constraints can be in <<, =, or , =, or >> form. form.

Example

Two productsTwo products: Chairs and Tables: Chairs and Tables

DecisionDecision:: How many of each to make this month? How many of each to make this month?

ObjectiveObjective: : Maximize profitMaximize profit

Data

Tables (per table)

Chairs (per chair)

Hours Available

Profit Contribution

$7 $5

carpentry 3 hrs 4 hrs 2400

Painting 2 hrs 1 hr 1000Other Limitations:• Make no more than 450 chairs• Make at least 100 tables

Decision VariablesDecision Variables::

TT = Num. of tables to make = Num. of tables to make

CC = Num. of chairs to make = Num. of chairs to make

Objective FunctionObjective Function: Maximize Profit: Maximize Profit

MaximizeMaximize $7 $7 TT + $5 + $5 CC

Solution

Constraints

Have 2400 hours of carpentry time availableHave 2400 hours of carpentry time available

3 3 TT + 4 + 4 CC << 2400 2400 (hours)(hours)

Have 1000 hours of painting time availableHave 1000 hours of painting time available

2 2 TT + 1 + 1 CC << 1000 1000 (hours)(hours)

More ConstraintsMore Constraints::Make no more than 450 chairsMake no more than 450 chairs

CC << 450 450Make at least 100 tablesMake at least 100 tables

TT >> 100 100

Non negativityNon negativity::Cannot make a negative number of chairs or tablesCannot make a negative number of chairs or tables

TT >> 0 0

CC >> 0 0

ModelMaxMax 7 7TT + 5 + 5CC

Subject to the constraintsSubject to the constraints::

33TT + 4 + 4CC << 2400 2400

22TT + 1 + 1CC << 1000 1000

C C << 450 450

T T >> 100 100

TT,, C C >> 0 0

General Formulation of LPPMax/min Max/min z = cz = c11xx11 + c + c22xx22 + ... + c + ... + cnnxxnn

subject to:subject to:aa1111xx11 + a + a1212xx22 + ... + a + ... + a1n1nxxnn (≤, =, ≥) b (≤, =, ≥) b11

aa2121xx11 + a + a2222xx22 + ... + a + ... + a2n2nxxnn (≤, =, ≥) b (≤, =, ≥) b22

:: aam1m1xx11 + a + am2m2xx22 + ... + a + ... + amnmnxxnn (≤, =, ≥) b (≤, =, ≥) bm m

xx1 1 ≥ 0, x≥ 0, x22 ≥ 0,…….x ≥ 0,…….xjj ≥ 0,……., x ≥ 0,……., xnn ≥ 0. ≥ 0.

xxjj = decision variables = decision variables

bbii = constraint levels = constraint levels

ccjj = objective function coefficients= objective function coefficients

aaijij = constraint coefficients = constraint coefficients

Example

Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional.

The number of pounds of each alloy needed per frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week?

Example

Pounds of each alloy needed per frame

Aluminum AlloyAluminum Alloy Steel AlloySteel Alloy

DeluxeDeluxe 2 3 2 3

Professional Professional 4 4 22

Solution

Define the objective

Maximize total weekly profit

Define the decision variables

x1 = number of Deluxe frames produced weekly

x2 = number of Professional frames produced weekly

Solution

Max Z = 10x1 + 15x2

Subject To

2x1 + 4x2 < 100

3x1 + 2x2 < 80

x1 , x2 >> 0

ExampleA firm manufactures 3 products A, B and C. The profits are Rs.3, Rs.2, A firm manufactures 3 products A, B and C. The profits are Rs.3, Rs.2, and Rs.4 respectively. The firm has 2 machines and below is the and Rs.4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product.required processing time in minutes for each machine on each product.

Product

A B C

Machines G 4 3 5

H 2 2 4

Machine G and H have 2000 and 2500 machine-minutes respectively. The firm must manufacture 100 A’s, 200 B’s and 50 C’s, but not more than 150 A’s. Set up an LP problem to maximize profit.

Solution

Define the objective Maximize profit

Define the decision variables x1 = number of products of type A x2 = number of products of type B x3 = number of products of type C

Solution

Max Z = 3xMax Z = 3x1 1 + 2x+ 2x22 + 4x + 4x33

Subject ToSubject To

4x4x11 + 3x + 3x22 + 5x + 5x33 ≤ 2000 ≤ 2000

2x2x11 + 2x + 2x22 + 4x + 4x33 ≤ 2500 ≤ 2500

100 ≤100 ≤ xx11 ≤ 150 ≤ 150

xx2 2 ≥≥ 200x3 ≥≥ 50

xx11, x, x22, x, x33 ≥ ≥ 0

ExampleThe Sureset Concrete Company produces concrete. Two ingredients in concrete are sand (costs $6 per ton) and gravel (costs $8 per ton). Sand and gravel together must make up exactly 75% of the weight of the concrete. Also, no more than 40% of the concrete can be sand and at least 30% of the concrete be gravel. Each day 2000 tons of concrete are produced. To minimize costs, how many tons of gravel and sand should be purchased each day?

Solution

Define the objectiveMinimize daily costs

Define the decision variablesx1 = tons of sand purchased

x2 = tons of gravel purchased

Cont…

Min Z = 6x1 + 8x2

Subject To

x1 + x2 = 1500

x1 < 800

x2 >> 600 600

x1 , x2 >> 0