1. Linear Programming Topics General optimization model
LP model and assumptions
Manufacturing example
Characteristics of solutions
Sensitivity analysis
Excel add-in
8/14/04
J. Bard and J. W. Barnes
Operations Research Models and Methods
Copyright All rights reserved

2. 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,” not a computer program.
Mathematical Program
Maximize / Minimize
z = f(x1, x2 ,…, xn) { }
Subject to
gi(x1, x2 , …, xn) 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 =
(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

5. Linear Programming
A linear program is a special case of a mathematical program where f and g1 ,…, gm are linear functions
Linear Program:
Maximize/Minimize z = c1x1 + c2x2 + • • • + cnxn { }
Subject to ai1x1 + ai2x2 + • • • + ainxn bi
i = 1,…,m , =
xj  uj, j = 1,…,n
xj ≥ 0, j = 1,…,n

6. LP Model Components x = decision vector = "activity levels"
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

7. Linear Programming Assumptions(
i) proportionality
(ii) additivity
linearity
(iii) divisibility
(iv) certainty

8. Explanation of LP Assumptions
(i) activity j’s contribution to obj fcn 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)
(ii)
“cross terms” such as x1x5 may not appear in the objective or constraints.

9. 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)].

10. Manufacturing Example
Machine data
Product data

11. Product Structure for Manufacturing Example

12. Data Summary: (R production fixed at 60)Q
Selling price/unit 90
100
Raw Material cost/unit 45 40
Demand (maximum)
100 40
mins/unit on A 20 10 B 12 28 C 15 6 D 10 15
Machine Availability: A  1800 min/wk; B  1440 min/wk,
C  2040 min/wk, and D  2400 min/wk
Operating Expenses = $3000/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

13. LP Formulation xP ≥ 0, xQ ≥ 0 Are we done? Are the LP assumptions
max
45 xP + 60 xQ
Objective Function
20 xP +
10 xQ
s.t.
1800
Structural
12 xP +
28 xQ
1440
constraints
15 xP +
6 xQ
2040
10 xP +
15 xQ
2400
xP  100, xQ  40
demand
xP ≥ 0, xQ ≥ 0
Are we done?
nonnegativity
Are the LP assumptions
valid for this problem? *
Optimal solution = * x x = P Q

14. 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 it’s 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.

15. Feasible Region for Manufacturing Example

16. Iso-Profit Lines and Optimal Solution for Example

17. Discussion of Results for Manufacturing Example
Optimal objective value is $4664 but when we subtract the weekly operating expenses of $3000 we obtain a weekly profit of $1664.
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 ?

19. Possible Outcomes of an LP
1. Infeasible –
feasible region is empty; e.g., if the
constraints include
x1+ x2 £ 6 and x1+ x2  7
2. Unbounded -
Max 15x1+ 7x2
(no finite optimal solution)
s.t.
x1 + x2 ³ 1
x1, x2 ³ 0
3. Multiple optimal solutions -
max 3x1 + 3x2
s.t.
x1+ x2 £ 1
x1, x2 ³ 0
4. Unique Optimal Solution
Note: multiple optimal solutions occur in many practical (real-world) LPs.

20. Example with Multiple Optimal Solutions

21. Bounded Objective Function with Unbound Feasible Region

22. Inconsistent constraint system
Constraint system allowing only nonpositive values for x1 and x2

23. Sensitivity Analysis Shadow Price on Constraint i
Amount object function changes with unit increase
in RHS, all other coefficients held constant
RHS Ranges
Allowable increase & decrease for which shadow prices remain valid
Objective Function Coefficient Ranges
Allowable increase & decrease for which
current optimal solution is valid

25. Sensitivity Analysis with Add-ins

26. 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