1. INTRODUCTION TO LINEAR PROGRAMMING
CONTENTS
Introduction to Linear Programming
Applications of Linear Programming
Reference: Chapter 1 in BJS book.

2. A Typical Linear Programming Problem
Linear Programming Formulation:
Minimize c1x1 + c2x2 + c3x3 + …. + cnxn
subject to
a11x1 + a12x2 + a13x3 + …. + a1nxn  b1
a21x1 + a22x2 + a23x3 + …. + a2nxn  b2 :
am1x1 + am2x2 + am3x3 + …. + amnxn  bm
x1, x2, x3 , …., xn  0
or,
Minimize j=1, n cjxj
j=1, n aijxj - xn+i = bi for all i = 1, …, m
xj  0 for all j =1, …, n

3. Matrix Notation
Minimize cx
subject to
Ax = b
x  0
where

4. An Example of a LP
Giapetto’s woodcarving manufactures two types of wooden toys: soldiers and trains
Constraints:
100 finishing hour per week available
80 carpentry hours per week available
produce no more than 40 soldiers per week
Objective: maximize profit

5. An Example of a LP (cont.)
Linear Programming formulation:
Maximize z = 3x1 + 2x2 (Obj. Func.)
subject to
2x1 + x2  100 (Finishing constraint)
x x2  80 (Carpentry constraint)
x  (Bound on soldiers)
x1  (Sign restriction)
x2  (Sign restriction)

6. Assumptions of Linear Programming
Proportionality Assumption
Contribution of a variable is proportional to its value.
Additivity Assumptions
Contributions of variables are independent.
Divisibility Assumption
Decision variables can take fractional values.
Certainty Assumption
Each parameter is known with certainty.

7. Linear Programming Modeling and Examples
Stages of an application:
Problem formulation
Mathematical model
Deriving a solution
Model testing and analysis
Implementation

8. Capital Budgeting Problem
Five different investment opportunities are available for investment.
Fraction of investments can be bought.
Money available for investment:
Time 0: $40 million
Time 1: $20 million
Maximize the NPV of all investments.

9. Transportation Problem
The Brazilian coffee company processes coffee beans into coffee at m plants. The production capacity at plant i is ai.
The coffee is shipped every week to n warehouses in major cities for retail, distribution, and exporting. The demand at warehouse j is bj.
The unit shipping cost from plant i to warehouse j is cij.
It is desired to find the production-shipping pattern xij from plant i to warehouse j, i = 1, .. , m, j = 1, …, n, that minimizes the overall shipping cost.

10. Static Workforce Scheduling
Number of full time employees on different days of the week are given below.
Each employee must work five consecutive days and then receive two days off.
The schedule must meet the requirements by minimizing the total number of full time employees.

11. Multi-Period Financial Models
Determine investment strategy for the next three years
Money available for investment at time 0 = $100,000
Investments available : A, B, C, D & E
No more than $75,000 in one invest
Uninvested cash earns 8% interest
Cash flow of these investments:

12. Cutting Stock Problem
A manufacturer of metal sheets produces rolls of standard fixed width w and of standard length l.
A large order is placed by a customer who needs sheets of width w and varying lengths. He needs bi sheets of length li, i = 1, …, m.
The manufacturer would like to cut standard rolls in such a way as to satisfy the order and to minimize the waste.
Since scrap pieces are useless to the manufacturer, the objective is to minimize the number of rolls needed to satisfy the order.

13. Multi-Period Workforce Scheduling
Requirement of skilled repair time (in hours) is given below.
At the beginning of the period, 50 skilled technicians are available.
Each technician is paid $2,000 and works up to 160 hrs per month.
Each month 5% of the technicians leave.
A new technician needs one month of training, is paid $1,000 per month, and requires 50 hours of supervision of a trained technician.
Meet the service requirement at minimum cost.

14. Solution: Capital Budgeting Problem
Decision Variables:
xi: fraction of investment i purchased
Formulation:
Maximize z = 13x1 + 16x2 + 16x3 + 14x4 + 39x5
subject to
11x1 + 53x2 + 5x3 + 5x4 + 29x5 £ 40
3x1 + 6x x3 + x x5 £ 20
x1 £ 1
x2 £ 1
x3 £ 1
x4 £ 1
x5 £ 1
x1, x2, x3, x4, x5 ³ 0

15. Solution: Transportation Problem
Decision Variables:
xij: amount shipped from plant i to warehouse j
Formulation:
Minimize z =
subject to
= ai, i = 1, … , m
 bj, j = 1, … , n
xij  0, i = 1, … , m, j = 1, … , n

16. Solution: Static Workforce Scheduling
LP Formulation:
Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7
subject to
x x4 + x5 + x6 + x7 ³ 17
x1+ x x5 + x6 + x7 ³ 13
x1+ x2 + x x6 + x7 ³ 15
x1+ x2 + x3 + x x7 ³ 19
x1+ x2 + x3 + x4 + x ³ 14
x2 + x3 + x4 + x5 + x ³ 16
x3 + x4 + x5 + x6 + x ³ 11
x1, x2, x3, x4, x5, x6, x7 ³ 0

17. Solution: Multiperiod Financial Model
Decision Variables:
A, B, C, D, E : Dollars invested in the investments A, B, C, D, and E
St: Dollars invested in money market fund at time t (t = 0, 1, 2)
Formulation:
Maximize z = B + 1.9D + 1.5E S2
subject to
A + C + D + S = 100,000
0.5A C S = B + S1
A B S = E + S2
A £ 75,000
B £ 75,000
C £ 75,000
D £ 75,000
E £ 75,000
A, B, C, D, E, S0, S1, S2 ³ 0

18. Solution: Multiperiod Workforce Scheduling
Decision Variables:
xt: number of technicians trained in period t
yt: number of experienced technicians in period t
Formulation:
Minimize z = 1000(x1 + x2 + x3 + x4 + x5) (y1 + y2 + y3 + y4 + y5)
subject to
160y x1 ³ y1 = 50
160y x2 ³ y1 + x1 = y2
160y x3 ³ y2 + x2 = y3
160y x4 ³ y3 + x3 = y4
160y x5 ³ y4 + x4 = y5
xt, yt ³ 0, t = 1, 2, 3, … , 5

19. Cutting Stock Problem (contd.)
Given a standard sheet of length l, there are many ways of cutting it. Each such way is called a cutting pattern.
The jth cutting pattern is characterized by the column vector aj, where the ith component, namely, aij, is a nonnegative integer denoting the number of sheets of length li in the jth pattern.
Note that the vector aj represents a cutting pattern if and only if i=1,n aijli  l and each aij is a nonnegative number.

20. Cutting Stock Problem (contd.)
Formulation:
Minimize i=1,n xi
subject to
i=1,n aij xi  bi i = 1, …, m
xi  j = 1, …, n
xi integer j = 1, …, n

21. Feasible Region
Feasible Region: Set of all points satisfying all the constraints and all the sign restrictions
Example:
Max. z = 3x1 + 2x2
subject to
2x1 + x2 £
x1 + x2 £ 80
x1 £ 40
x1 ³ 0
x2 ³ 0

22. Example 1 Maximize z = 50x1 + 100x2 subject to 7x1 + 2x2 ³ 28
Feasible region in this example is unbounded.

23. Example 2 Maximize z = 3x1+ 2x2 subject to 1/40x1 + 1/60x2 £ 1
This linear program does not have any feasible solutions.

24. Example 3
Max. z = 3x1 + 2x2
subject to
1/40 x1 + 1/60x2 £ 1
1/50 x1 + 1/50x2 £ 1
x1, x2 ³ 0
This linear program has multiple or alternative optimal solutions.