1. deterministic operations research
CHAPTER 14

2. 14.6. Mathematical Programming
Definition of OR:
Set of quantitative techniques to solve decision problems, using math, statistics, and computers
Definition of Math Programming
A class of OR methods to mathematically model and solve constrained optimization problems, include:
Linear programming
Nonlinear programming
Integer programming
Dynamic programming
Stochastic programming
. . .

3. 14.6. Mathematical Programming
Math Programming Models
Used to find optimum (best) solution of the following:
Optimize f(X)
Subject to
gi(X) ≤ bi, i = 1, …, m, X = x1, …, xn
where
f(X) = objective function
Optimize = minimize or maximize
X = (x1, …, xn) = decision variables
g(X) = constraints. Can also be (=) or (≥)
Other constraints: X ≥ 0, X binary (0-1), or X general integer

4. 14.7. Unconstrained Optimization
No constraints. Can be solved by calculus. Set: f’(X) = 0
Examples: linear regression, EOQ
Stationary Point
X* is a stationary point of f(X) if:
f’(X*) = 0 necessary, not sufficient condition
Optimum Point
X* is a min (max) point of f(X) if X* is a stationary point and:
f’’(X*) > 0 min, sufficient condition
f’’(X*) < 0 max, sufficient condition

5. 14.7. Unconstrained Optimization
If f’(X*) = 0 and f’’(X*) = 0?
Higher-order derivatives must be taken
Let n = order of the first non-zero derivative
If n is odd, X* is a deflection point
If n is even and
f(n)(X*) > 0 X* is a local minimum. f(X) is convex
f(n)(X*) < 0 X* is a local maximum. f(X) is concave

6. 14.7. Unconstrained Optimization Example
Find & classify all stationary points of f(x) = x4 + x3
f’(x) = 4x3 + 3x2 = 0
= x2 (4x + 3) = 0 Stationary points: 0, - ¾
f’’(x) = 12x2 + 6x
f’’(0) = 0,
f’’(-3/4) = 12(-3/4)2 + 6(-3/4) = 9/4, - ¾ is local minimum
f’’’(x) = 24x + 6
f’’’(0) = 6, is deflection point

7. 14.7. Unconstrained Optimization for functions of Several Variables
Given f(x1, ..,xn),
Set the gradient = 0:
 f(x1, ..,xn) = 0
Solve the system to determine stationary points X*
For each stationary point, determine the Hessian Matrix H(X*) = 2 f(X*)
If H(X*) is Positive Definite, X* is a minimum point
If H(X*) is Negative Definite, X* is a maximum point
If H(X*) is InDefinite, X* is a saddle point

8. 14.7. Unconstrained Optimization for functions of Several Variables

9. 14.7. Unconstrained Optimization Example2
Find & classify all stationary points of
f(x1, x2) = 4x12 – 5x1x2 + 3x22 – 6x x2
 f(x1,x2) = 8x1 – 5x2 – 6 = 0
= – 5x1 + 6x = 0
Solving gives the stationary point
X* = (1, 0.4)

10. 14.7. Unconstrained Optimization Example2
H(f) = 2 f(x1,x2) =  – 5   – 
Leading Determinant (k = 1) = 8
Leading Determinant (k = 2) = 8*6 – (–5*–5) = 23
Since H is +DEF
(x1,x2)* = (1, 0.4) is a minimum point

11. 14.8.1. Assignment Technique Solution method: Phase I
Given n×n square matrix of distances/transport costs
Optimal solution by Hungarian Algorithm: 2 phases
Phase I
From each row, subtract its smallest number
From each column, subtract its smallest number
For each row with only 1 uncrossed zero, select the zero, cross zeros in same column
For each column with only 1 uncrossed zero, select the zero, cross zeros in same row

12. Assignment Technique Hungarian Algorithm (Optimum Solution): Phase II
If previous steps (Phase 1) produce n zeros, stop.
If m < n zeros are covered, Phase II is needed.
Phase II
Cover all zeros with m horizontal or vertical line
Subtract the minimum uncovered element from all uncovered elements
Add the minimum uncovered element to all elements covered by 2 lines (at line intersections)
Repeat Phase I

13. Assignment Technique Example
A factory has 4 identical machines (A,B,C,D) and 4 storage areas (I,II,III,IV). Given the distance matrix below, determine the best machine-storage assignment. I II
III IV A 2 6 3 5 B 1 C 4 D

14. Assignment Technique Example
Subtract the smallest element from each row. I II
III IV A 2 6 3 5 B 1 C 4 D
Sum

15. Assignment Technique Example
Subtract the smallest element from each column. I II
III IV A 4 1 3 B 2 C D
sum

16. Assignment Technique Example
Subtract the smallest element from each column. I II
III IV A 3 1 B 4 C 2 D

17. Assignment Technique ExampleII
III IV A 3 1 B 4 C 2 D

18. Assignment Technique Example
Cover all zeros with 3 lines, in rows/columns with max no. of zeros
Min uncovered element = 1 I II
III IV A 3 1 B 4 C 2 D

19. Assignment Technique Example
Subtracting & adding “1” I II
III IV A 2 1 B 5 C 3 D

20. Assignment Technique Example
Zero-cost assignment. m = 4 (stop) I II
III IV A 2 1 B 5 C 3 D

21. Assignment Example Solution
Optimum machine-storage assignment.
Total cost = = 9 I II
III IV A 2 6 3 5 B 1 C 4 D

22. End of Chapter 14
Questions?