LINEAR PROGRAMMING PROBLEM
Jan 28, 2015
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