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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% - PowerPoint PPT Presentation
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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.
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