LP: THE GRAPHICAL METHOD
Today’s Topic:
1. A Simple LP Example
2. Graphical solution of LP
3. Special Cases
4. Graphical Sensitivity Analysis
 Change of an objective function coefficient
 Change of a right-hand-side
 Simultaneous changes of several coefficients
1. A Simple LP Example
RMC, Inc. is a firm that produces chemical-based products. Three raw materials
are used to produce two products. The material requirements per ton are shown below.
Product
Material 1
Material 2
Material 3
Fuel additive
2/5
0
3/5
Solvent base
1/2
1/5
3/10
For the current production period, RMC has available the following quantities of each
raw material. Because of spoilage, any materials not used for current production must be
discarded.
Material
Material 1
Material 2
Material 3
Available
20
5
21
If the contribution to profit is $40 for each ton of fuel additive and $30 for each ton of
solvent base, how many tons of each product should be produced in order to maximize
the total contribution to profit?
-- LP formulation
 Decision variables:
 Objective function:
 Constraints:
-- Interpretation
 feasible vs. infeasible
 feasible solution vs. optimal solution
 slack vs. surplus
0
+
-
slack = rhs - lhs
≤ constraints
all resources used
extra resource
not enough resource
(infeasible)
surplus = lhs - rhs
≥ constraints
requirement just satisfied
over performed
not satisfied
(infeasible)
2. Graphical Solution of LP -- for LP with only 2 variables
Step 1. Draw the solution space (describe all feasible solutions)
-- For each constraint
1. draw the equation line
2. decide the direction
Step 2. Draw the objective function line
1. Set obj. fun. Z = convenient value
2. Draw the obj. equation line
3. Determine the direction of improvement for obj. fun. Z
Step 3. Shift the obj. line along the improvement direction and identify an optimal
solution.
Step 4. Calculate the optimal solution and the corresponding objective value
-- Observations from graphical solution:
 One of the optimal solutions is at a corner point of the solution space
 Binding and non-binding constraints
 An optimal solution is determined by binding constraints
 Optimal solution is not always an integer
 Minimization problem?
3. Special Cases of LP
-- Possible outcomes from solving an LP
Solve LP
Infeasible
Feasible
Unbounded
Solution
Optimal Solution
Unique
Multiple
Degenerate
-- LP infeasible:
conflict requirements ==> no common solution space
algebra example:
graphical example:
happens frequently in practice, due to:
 mistakes
 mis-coordination
 unrealistic requirements
-- Unbounded solution:
if the solution space is unbounded along the improvement direction of the obj.
fun.
graphical example:
caution: unbounded solution space does not necessarily imply unbounded solution
graphical example:
due to: mistakes in formulation
-- Multiple optimal solution:
any linear combination of two optimal solutions is also an optimal solution,
all optimal solutions have the same objective value
graphical example:
advantage: more flexibility for managerial decision
-- Degenerate solution:
optimal solution overly determined
graphical example:
two variable situation:
 one of the binding constraints is redundant
 same solution space with or without the redundant constraint
more than two variables:
 all constraints may be useful, dropping any may change the solution
space (but not the solution).
may cause problems for simplex method
4. Graphical Sensitivity Analysis
I. Change of One Objective Function Coefficient
-- Question: What will happen to the current production plan if the price of a product
changes?
-- Observation:
 Obj. coeff. change => slope of obj. line change => obj. line rotates
 Small change of an obj. coeff. does not change the current optimal solution.
-- Slope of the obj. line:
-- Slopes of the two binding constraints:
-- Conditions to maintain the current optimal solution:
-- Ranges of the obj. coeff. (to maintain the current optimal solution):
II. Global View of An Obj. Coefficient Change (Large Change)
-- Within the obj. coeff range, the obj. value changes proportionally to the coeff. change.

Maximization LP:

Minimization LP:
III. Change of A Right-Hand-Side
-- Questions:
 What will happen to the current production if more resources are available?
 By how much the profit will change (with the additional resources)?
-- Observations:
 RHS change => constraint shifts => potential solution space change
 RHS change for binding constraints => the optimal solution changes
 RHS change (small) for non-binding constraints => no effect on the opt. solution
 If binding constraints remains binding => obj. value changes proportionally
-- Dual Price: the improvement rate of the obj. function value as the RHS increases.
 Dual price > 0: obj. improve
 Dual price < 0: obj. deteriorate
-- Ranges of a RHS (to maintain the same dual price):
-- Determine the boundary points the opt. solution can move to
-- Determine the associated RHS’s:
-- Range of the RHS:
-- Calculate the dual price:
Constraint
Boundary Points
Coordinates
RHS
Range of RHS
Dual Price
Material 1
2
6
(35, 0)
(22.5, 25)
14
21.5
(14, 21.5)
33.33
Material 2
3
Infinity
(25, 20)
Infinity
4
Infinity
(4, Infinity)
0
Material 3
4
7
(18.75,25)
(50, 0)
18.75
30
(18.75, 30)
44.44
IV. Global View of One RHS Change (Large Change)
-- Within the RHS range, obj. value changes proportionally to the RHS change.

Maximization LP:
* < constraint:
* > constraint:

Minimization LP:
* < constraint:
* > constraint:
V. Simultaneous Change of Coefficients
-- 100% rule:
 For objective coefficients: if the total percentage changes w.r.t the allowable increases
or decreases is within 100%, the current optimal solution does not change.
 For RHS’s: if the total percentage changes w.r.t the allowable RHS increases or
decreases is within 100%, all dual prices do not change.
 Only a sufficient condition.
Concepts on Linear Program and Graphical Solution:
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Decision variable
Objective function and its coefficients
Constraints (RHS and LHS)
Solution space
Feasible solution and optimal solution
Infeasible linear program
Unique, multiple, and unbounded optimal solution
Slack and surplus
Binding and non-binding constraints
Range of objective function coefficient (range of optimality) and its interpretation
Range of the RHS of a constraint and its interpretation
Dual price
How does objective function value change as an objective function coefficient
changes (within and outside the range)?
How does objective function value change as a RHS of a constraint changes (within
and outside the range)?
Procedure of graphical solution of linear program
Calculation of range of optimality, range of RHS, and dual price via graphical method