linear programming formulation and graphic method
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LINEAR PROGRAMMING
FORMULATION AND GRAPHIC METHOD
DEFINITION
Mathematical technique for optimizing the use of constrained resources, where the relationship between variables is linear.
Linear – relationship between two or more variables which is directly and precisly proportional.
Programming – refers to use of certain mathematical techniques to get the best possible solution to a given problem involving limited resources.
LP
Linear Programming
Basic Requirements
Decision variables and their relationships should be linear.
Objective function should be well defined Presence of Constraints Alternative course of action Non negative restrictions Linearity: Both Objective and Constraints must
be expressed in Linear equations/inequalities.
Terminology Decision variables – variables directly under the
control of decision maker. Objective function – An expression showing
relationship between decision variables and organization's objectives.
Constraints – Limit or restriction on the availability of resources.
Non – negativity constraint – The decision variables must not assume negative values which is impossible situation.
Feasible region – The region which is common to all the constraints of given LPP.
Basic Solution – A solution of basic
variables, where the values of non-basic variables are zero.
Feasible Solution – A solution satisfying all constraints and non-negativity restriction of a LPP.
Basic Feasible Solution – A basic solution, where the values of basic variables are non-negative.
Optimal Solution – A basic feasible solution which also optimizes the objective function.
Assumptions Proportionality:
Proportionality/Linearity exists in Objective and Constraints
Additivity: Total resources used must equal sum of resources
used in individual activity Divisibility:
Solution need not be whole numbers and may take up fractional value.
Certainty: Co-efficients are known (Deterministic)
Finiteness: Finite number of alternative solutions
Applications of LPP Industry
Product Mix Production
Smoothing Blending Production
Scheduling Trim Loss
Miscellaneous Diet Problems Farm Planning Airline Routing Facility Location
Management Portfolio Selection Financial Mix Media Selection Manpower Planning Transportation Job Allocation Travelling Salesman
Problem Formulation
Problem formulation or modeling is the process of translating a verbal statement of a problem into a mathematical statement.
LP Model FormulationMax/min z = c1x1 + c2x2 + ... + cnxn
subject to:a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1
a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2
: am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bm
xj = decision variablesbi = constraint levelscj = objective function coefficientsaij = constraint coefficients
Guidelines for Model Formulation
Understand the problem thoroughly. Describe the objective. Describe each constraint. Define the decision variables. Write the objective in terms of the decision
variables. Write the constraints in terms of the
decision variables.
Linear Programming Problem A firm producing 2 products P1 and P2.
Each unit of P1 requires 1 kg material A, 2 kg of material B and 1 unit of labor. Each unit of P2 requires 3 kg of material B and 1 unit of labor. Each week firm has availability of 6 kg of material A, 19 kg of material B and 8 units of labor. One unit of P1 sold earn profit of Rs. 5 and P2 earns profit of Rs 7. How many units of each product is to be produced so as to give the maximum profit.
A Maximization Problem
LP Formulation
Max Z = 5x1 + 7x2 (Profit function)
s.t.
x1 < 6 (Material A Constraint)
2x1 + 3x2 < 19 (Material B Constraint)
x1 + x2 < 8 (Labor Constraint)
x1, x2 > 0 (Non-negativity constraint)
Graphic Method of solving LPP Formulate linear programming problem. Each inequality in the constraints may be
written as equality. Plot the constraint lines considering them as
equations. Identify the feasible solution region. Locate the corner points of the feasible region. Calculate the value of objective function on the
corner points. Choose the point where the objective function
has optimal value.
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1 2 3 4 5 6 7 8 9 10
Graphical Solution Constraint #1 Graphed
x2
x1
x1 < 6
(6, 0)
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2
1
1 2 3 4 5 6 7 8 9 10
Graphical Solution Constraint #2 Graphed
2x1 + 3x2 < 19
x2
x1
(0, 6 1/3)
(9 1/2, 0)
Graphical Solution Constraint #3 Graphed
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2
1
1 2 3 4 5 6 7 8 9 10
x2
x1
x1 + x2 < 8
(0, 8)
(8, 0)
Graphical Solution Combined-Constraint Graph
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1
1 2 3 4 5 6 7 8 9 10
2x1 + 3x2 < 19
x2
x1
x1 + x2 < 8
x1 < 6
Graphical Solution Feasible Solution Region
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1 2 3 4 5 6 7 8 9 10 x1
FeasibleRegion
x2
Example 1: Graphical Solution
The Five Extreme Points8
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1 2 3 4 5 6 7 8 9 10 x1
FeasibleRegion
EE DD
CC
BB
AA
x2
Evaluation of objective function at feasible region corner points
Corner points
Coordinates Objective function Z = 5x1 + 7x2
Value
A (0,6.33) 5(0)+7(6.33) 44.31
B (5,3) 5(5)+7(3) 46
C (6,2) 5(6)+7(2) 44
D (6,0) 5(6)+7(0) 30
E (0,0) 5(0)+7(0) 0
Since Z has maximum value at point B (5,3), so
x1 = 5, x2 = 3
Where profit is Rs 46.
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1 2 3 4 5 6 7 8 9 10
Alternate Solution by extreme points
Objective Function Line
x1
x2
(7, 0)
(0, 5)Objective Function5x1 + 7x2 = 35Objective Function5x1 + 7x2 = 35
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1 2 3 4 5 6 7 8 9 10
Alternate Graphical Solution Optimal Solution
x1
x2
Objective Function5x1 + 7x2 = 46Objective Function5x1 + 7x2 = 46
Optimal Solution(x1 = 5, x2 = 3)Optimal Solution(x1 = 5, x2 = 3)
Extreme Points and the Optimal Solution
The corners or vertices of the feasible region are referred to as the extreme points.
An optimal solution to an LP problem can be found at an extreme point of the feasible region.
When looking for the optimal solution, you do not have to evaluate all feasible solution points.
You have to consider only the extreme points of the feasible region.
Summary of the Graphical Solution Procedure for Maximization Problems
Prepare a graph of the feasible solutions for each of the constraints.
Determine the feasible region that satisfies all the constraints simultaneously.
Draw an objective function line. Move parallel objective function lines toward
larger objective function values without entirely leaving the feasible region.
Any feasible solution on the objective function line with the largest value is an optimal solution.
Minimization Problem
Min Z = 5x1 + 2x2
Subject to constraints
2x1 + 5x2 > 10
4x1 - x2 > 12
x1 + x2 > 4
x1, x2 > 0
Graphical Solution Graph the Constraints
Constraint 1: When x1 = 0, then x2 = 2; when x2 = 0, then x1 = 5. Connect (5,0) and (0,2). The ">" side is above this line.
Constraint 2: When x2 = 0, then x1 = 3. But setting x1 to 0 will yield x2 = -12, which is not on the graph. Thus, to get a second point on this line, set x1 to any number larger than 3 and solve for x2: when x1 = 5, then x2 = 8. Connect (3,0) and (5,8). The ">" side is to the right.
Constraint 3: When x1 = 0, then x2 = 4; when x2 = 0, then x1 = 4. Connect (4,0) and (0,4). The ">" side is above this line.
Graphical Solution Constraints Graphed
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1 2 3 4 5 6 1 2 3 4 5 6
x2x2
4x1 - x2 > 12
x1 + x2 > 4
4x1 - x2 > 12
x1 + x2 > 4
2x1 + 5x2 > 102x1 + 5x2 > 10
x1x1
Feasible Region
Graphical Solution Graph the Objective Function
Set the objective function equal to an arbitrary constant (say 20) and graph it. For 5x1 + 2x2 = 20, when x1 = 0, then x2 = 10; when x2= 0, then x1 = 4. Connect (4,0) and (0,10).
Move the Objective Function Line Toward Optimality
Move it in the direction which lowers its value (down), since we are minimizing, until it touches the last point of the feasible region, determined by the last two constraints.
Graphical Solution Objective Function Graphed
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5
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1 2 3 4 5 6 1 2 3 4 5 6
x2x2Min z = 5x1 + 2x2
4x1 - x2 > 12
x1 + x2 > 4
Min z = 5x1 + 2x2
4x1 - x2 > 12
x1 + x2 > 4
2x1 + 5x2 > 102x1 + 5x2 > 10
x1x1
Solve for the Extreme Point at the Intersection of the Two Binding Constraints
4x1 - x2 = 12
x1+ x2 = 4
Adding these two equations gives:
5x1 = 16 or x1 = 16/5.
Substituting this into x1 + x2 = 4 gives: x2 = 4/5
Graphical Solution
Solve for the Optimal Value of the Objective Function
Solve for z = 5x1 + 2x2 = 5(16/5) + 2(4/5) = 88/5.
Thus the optimal solution is
x1 = 16/5; x2 = 4/5; z = 88/5
Graphical Solution Optimal Solution
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2
1
5
4
3
2
1
1 2 3 4 5 6 1 2 3 4 5 6
x2x2Min z = 5x1 + 2x2
4x1 - x2 > 12
x1 + x2 > 4
Min z = 5x1 + 2x2
4x1 - x2 > 12
x1 + x2 > 4
2x1 + 5x2 > 10
Optimal: x1 = 16/5 x2 = 4/5
2x1 + 5x2 > 10
Optimal: x1 = 16/5 x2 = 4/5x1x1
Special Cases Multiple Optimal Solutions Infeasible problem Unbounded problem Redundant Constraint
Multiple optimal solutions
If objective function falls on more than one optimal point, the solution is said to be multiple optimal.
It exists when slope of the objective function and one of the constraint equation are same.
Infeasible problem
No feasible solution. It means there are no points that
simultaneously satisfy all constraints in the problem.
Common region do not develop in the first quadrant.
Objective function do not pass through any point of all constraints.
Example of Infeasible Problem
Max Z = 2x1 + 6x2
subject to constraints
4x1 + 3x2 < 12
2x1 + x2 > 8
x1, x2 > 0
Infeasible Problem There are no points that satisfy both
constraints, hence this problem has no feasible region, and no optimal solution.
x2
x1
4x1 + 3x2 < 12
2x1 + x2 > 8
3 4
4
8
Unbounded Problem
When feasible solution space formed by the constraints is not confined to the first quadrants only.
If constraints passes through 2nd, 3rd or 4th quadrant.
No limits on the constraints. Common feasible region is not bounded in
any respect.
Example of Unbounded Problem
Max Z = 3x1 + 4x2
subject to
x1 + x2 > 5
3x1 + x2 > 8
x1, x2 > 0
Unbounded Problem The feasible region is unbounded and the
objective function line can be moved parallel to itself without bound so that z can be increased infinitely. x2
x1
3x1 + x2 > 8
x1 + x2 > 5
Max 3x1 + 4x2
5
5
8
2.67
Redundant Constraint
When one constraint is less restrictive than other constraint, it is said to be redundant constraint.
Advantages & Drawbacks
S. No. Advantages Drawbacks
1 Problem Insight Fraction Values
2 Considers all possible Solution Co-efficients must be Deterministic
3 Better Decisions Function must be Linear
4 Better Tools Time & Un-certainty effect not considered
5 Highlights Bottlenecks Approximation of Complex problems
6 Clear specification of Management problems