作業研究（二） Operations

Research II

- 廖經芳、王敏

Topics- Revised Simplex Method

- Duality Theory

- Sensitivity Analysis and

Parametric Linear Programming

- Integer Programming

- Markov Chains

- Queueing Theory

- …..

Grading:- 廖經芳老師 (65%)

- 2 exams, 50%

- Homework and Attendance, 15%

- 王敏老師 (35%)

- …..

Reference:Introduction to Operations Research, Hillier & Lieberman, 8th ed., McGraw Hill, 2005 （滄海）

Linear Programming (LP)

- George Dantzig, 1947

[1] LP Formulation

(a) Decision Variables :

All the decision variables are non-negative.

(b) Objective Function : Minimize or Maximize

(c) Constraints

nxxx ,,, 21

21 32 xxZMinimize

0,0

414

343..

21

21

21

xx

xx

xxts

s.t. : subject to

[2] Example

A company has three plants, Plant 1, Plant 2, Plant 3. Because of declining earnings, top management has decided to revamp the company’s product line.

Product 1: It requires some of production capacity

in Plants 1 and 3.

Product 2: It needs Plants 2 and 3.

The marketing division has concluded that the

company could sell as much as could be

produced by these plants.

However, because both products would be

competing for the same production capacity in

Plant 3, it is not clear which mix of the two

products would be most profitable.

The data needed to be gathered:

1. Number of hours of production time available per week in each plant for these new products. (The available capacity for the new products is quite limited.)

2. Production time used in each plant for each batch to yield each new product.

3. There is a profit per batch from a new product.

Production Timeper Batch, Hours

Production TimeAvailable

per Week, HoursPlant

Product

Profit per batch

1

2

3

4

12

18

1 2

1 0

0 2

3 2

$3,000 $5,000

: # of batches of product 1 produced per week : # of batches of product 2 produced per week : the total profit per week

Maximizesubject to

1x2x

Z

1 2

1 2

1 2

1 2

1 2

3 5

1 0 4

0 2 12

3 2 18

0, 0

Z x x

x x

x x

x x

x x

1x0 2 4 6 8

2x

2

4

6

8

10

[3] Graphical Solution (only for 2-variable cases)

0,0 21 xx

Feasibleregion

1x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

41 x

Feasibleregion

1x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

122 2 x41 x

Feasibleregion

1x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

122 2 x41 x

1823 21 xx

Feasibleregion

1x0 2 4 6 8 10

2x

2

4

6

8

21 5310 xxZ

21 5320 xxZ

Maximize:

21 5336 xxZ

)6,2(

The optimal solution

The largest value

Slope-intercept form:

21 53 xxZ

Zxx

5

1

5

312

1 1 2 2 n nZ c x c x c x

22222121

11212111

bxaxaxa

bxaxaxa

nn

nn

0,,0,0 21

2211

n

mnmnmm

xxx

bxaxaxa

Max

s.t.

[4] Standard Form of LP Model

[5] Other Forms

The other LP forms are the following:

1. Minimizing the objective function:

2. Greater-than-or-equal-to constraints:

.2211 nn xcxcxcZ

1 1 2 2i i in n ia x a x a x b

Minimize

3. Some functional constraints in equation form:

4. Deleting the nonnegativity constraints for

some decision variables:

ininii bxaxaxa 2211

jx : unrestricted in sign

where 0, 0j j j j jx x x x x

[6] Key Terminology

(a) A feasible solution is a solution

for which all constraints are satisfied

(b) An infeasible solution is a solution

for which at least one constraint is violated

(c) A feasible region is a collection

of all feasible solutions

(d) An optimal solution is a feasible solution

that has the most favorable value of

the objective function

(e) Multiple optimal solutions have an infinite

number of solutions with the same

optimal objective value

,23 21 xxZ

1x

0,0

1823

21

21

xx

xx

122 2 x4

and

Maximize

Subject to

Example

Multiple optimal solutions:

21 2318 xxZ

1x0 2 4 6 8 10

2x

2

4

6

8

Feasibleregion

Every point on this red line

segment is optimal,

each with Z=18.

Multiple optimal solutions

(f) An unbounded solution occurs when

the constraints do not prevent improving

the value of the objective function.

2x

1x

[7] Basic assumptions for LP models:

1. Additivity: c1x1+ c2x2+…

ai1x1+ ai2x2 +…

2. Proportionality: cixi, ai1x1

3. Divisibility: xi can be any real number

4. Certainty: all parameters are known with certainty.