Linear Programming

1
Linear Programming

• Topics
• General optimization model
• LP model and assumptions
• Manufacturing example
• Characteristics of solutions
• Sensitivity analysis
• Excel add-ins

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Deterministic OR Models
Most of the deterministic OR models can be
formulated as mathematical programs. "Program"
in this context, has to do with a plan and not
a computer program.
Mathematical Program
Maximize / Minimize
z f (x1,x2,,xn)



gi(x1,x2,,xn)
Subject to
³
bi , i 1,,m



xj 0, j 1,,n
3
Model Components


xj are called decision variables. These are
things that you control



gi(x1,x2,,xn)
bi

are called structural
³


(or functional or technological) constraints
xj 0 are nonnegativity constraints
f (x1,x2,,xn) is the objective function
4
Feasibility and Optimality
(
)
x
1
.
.
A feasible solution x
satisfies all the
.
x
n
constraints (both structural and nonnegativity)
The objective function ranks the feasible
solutions call them x1, x2, . . . , x?. The
optimal solution is the best among these. For a
minimization objective, we have z min f (x1),
f (x2), . . . , f (x?) .
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Linear Programming
A linear program is a special case of a
mathematical program where f(x) and g1(x) ,,
gm(x) are linear functions
Linear Program
Maximize/Minimize z c1x1 c2x2
cnxn



Subject to ai1x1 ai2x2 ainxn
i 1,,m
³
bi
,

xj ? uj, j 1,,n xj ³ 0, j 1,,n
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LP Model Components
xj ? uj are called simple bound constraints

x decision vector "activity levels"
aij , cj , bi , uj are all known data ? goal is
to find x (x1,x2,,xn)T (the symbol T
means)
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Linear Programming Assumptions
(
i) proportionality
(ii) additivity
linearity
(iii) divisibility
(iv) certainty
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Explanation of LP Assumptions
(i) activity js contribution to objective
function is cjxj
and usage in constraint i is aijxj
both are proportional to the level of activity j
(volume discounts, set-up charges, and nonlinear
efficiencies are potential sources of violation)
1
no cross terms such as x1x5 may not appear
in the objective or constraints.
(ii)
2
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Explanation of LP Assumptions
(iii) Fractional values for decision variables
are permitted
(iv) Data elements aij , cj , bi , uj are known
with certainty

• Nonlinear or integer programming models should
  be used when some subset of assumptions (i), (ii)
  and (iii) are not satisfied.
• Stochastic models should be used when a problem
  has significant uncertainties in the data that
  must be explicitly taken into account a
  relaxation of assumption (iv).

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Product Structure for Manufacturing Example
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Data for Manufacturing Example
Machine data
Product data
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Data Summary
P
Q
R
Selling price/unit
90
100
70
45
40
20
Raw Material cost/unit
Maximum sales
100
40
60
Minutes/unit on A
20
10
10
12
28
16
B
C
15
6
16
10
15
0
D
Machine Availability 2400 min/wk
Structural coefficients
Operating Expenses 6,000/wk (fixed cost)
Decision Variables


xP of units of product P to produce per
week xQ of units of product Q to produce per
week xR of units of product R to produce per
week


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LP Formulation
Objective Function
xQ 50 xR 6000

60

max z 45 xP





10 xQ 10 xR
20 xP

2400
s.t.

Structural



constraints
12 xP

28 xQ 16 xR
2400




15 xP

6 xQ 16 xR
2400





10 xP

15 xQ 0 xR
2400



demand
xP ? 100, xQ ? 40, xR ? 60
xP ? 0, xQ ? 0, xR ? 0
nonnegativity
Are we done?
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Discussion of Results for Manufacturing Example

• Optimal objective value is 7,664 but when we
  subtract the weekly operating expenses of 6,000
  we obtain a weekly profit of 1,664.

• Machines A B are being used at maximum level
  and are bottlenecks.
• There is slack production capacity in Machines C
  D.

How would we solve model using Excel Add-ins ?
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Solution to Manufacturing Example
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Characteristics of Solutions to LPs
A Graphical Solution Procedure (LPs with 2
decision variables

can be solved/viewed this way.)
1.
Plot each constraint as an equation and then
decide which

side of the line is feasible (if its an
inequality).
2.

Find the feasible region.
3.

Plot two iso-profit (or iso-cost) lines.
4.

Imagine sliding the iso-profit line in the
improving direction. The last point touched as
the iso-profit line leaves the feasible region
region is optimal.
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Two-Dimensional Machine Scheduling Problem -- let
xR 60
Objective Function
xQ 3000
max z 45 xP

60






10 xQ
20 xP

1800
s.t.

Structural



constraints
12 xP

28 xQ
1440




15 xP

6 xQ
2040





10 xP

15 xQ
2400



demand
xP ? 100, xQ ? 40
xP ? 0, xQ ? 0
nonnegativity
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Feasible Region for Manufacturing Example
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Iso-Profit Lines and Optimal Solution for Example
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Possible Outcomes of an LP
1. Unique Optimal Solution
2. Multiple optimal solutions
Max 3x1 3x2
s.t.
x1 x2 1 x1, x2 ³ 0
feasible region is empty e.g., if the
3. Infeasible
constraints include
x1 x2 6 and x1 x2 ? 7
4. Unbounded
Max 15x1 7x2
(no finite optimal solution)
s.t.
x1 x2 ³ 1 x1, x2 ³ 0
Note multiple optimal solutions occur in many
practical (real-world) LPs.

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Example with Multiple Optimal Solutions
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Bounded Objective Function with Unbound Feasible
Region
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Inconsistent constraint system

Constraint system allowing only nonpositive
values for x1 and x2
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Sensitivity Analysis
Shadow Price on Constraint i Amount object
function changes with unit increase in RHS,
all other coefficients held constant Objective
Function Coefficient Ranging Allowable increase
decrease for which current optimal solution
is valid RHS Ranging Allowable increase
decrease for which shadow prices remain valid
25
Solution to Manufacturing Example
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Sensitivity Analysis with Add-ins
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What You Should Know About Linear Programming

• What the components of a problem are.
• How to formulate a problem.
• What the assumptions are underlying an LP.
• How to find a solution to a 2-dimensional problem
  graphically.
• Possible solutions.
• How to solve an LP with the Excel add-in.