作業研究（二） Operations Research II - 廖經芳、王敏

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 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 Time per Batch, Hours Production Time Available per Week, Hours Product 1 2 Plant 1 2 3 1 0 0 2 3 2 4 12 18 Profit per batch $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 Maximize subject to

[3] Graphical Solution (only for 2-variable cases) 10 8 6 4 Feasible region 2 0 2 4 6 8

10 8 6 4 Feasible region 2 0 2 4 6 8

Maximize: The largest value The optimal solution Slope-intercept form: 8 6 4 2 0 2 4 6 8 10

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: Minimize

3. Some functional constraints in equation form: 4. Deleting the nonnegativity constraints for some decision variables: : unrestricted in sign

[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

Multiple optimal solutions: Example Maximize Subject to and

8 Multiple optimal solutions 6 Every point on this red line segment is optimal, each with Z=18. 4 2 Feasible region 0 2 4 6 8 10

(f) An unbounded solution occurs when the constraints do not prevent improving the value of the objective function.

[7] Basic assumptions for LP models: • Additivity: c1x1+ c2x2+… • ai1x1+ ai2x2 +… • Proportionality: cixi, ai1x1 • Divisibility: xi can be any real number • Certainty: all parameters are known with certainty.

作業研究（二） Operations Research II - 廖經芳、王敏