北京外国语大学：最优化方法（课件讲稿）Optimization Method（主讲：陈曦）

Optimization Method Xi Chen Department of Management Science and Engineering International Business School Beijing Foreign Studies University 4口48+4三4至，至)只0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 1/41

Optimization Method Xi Chen Department of Management Science and Engineering International Business School Beijing Foreign Studies University Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 1 / 41

General Procedure General Procedure Modeling ③Solution Approach Linear Programming o Sensitivity Analysis Duality Theory o Commercial Softwares o Integer Programming Dynamic Programming o Game Theory OLTEX 4口4+4三4至，至)只0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 2/41

General Procedure 1 General Procedure 2 Modeling 3 Solution Approach Linear Programming Sensitivity Analysis Duality Theory Commercial Softwares Integer Programming Dynamic Programming Game Theory 4 LATEX Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 2 / 41

General Procedure o Problem Description 。Modeling ●Solution Approach o Computational Experiments and Analysis 4口40+4三4至，至)只0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 3/41

General Procedure Problem Description Modeling Solution Approach Computational Experiments and Analysis Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 3 / 41

Modeling General Procedure Modeling Solution Approach Linear Programming o Sensitivity Analysis Duality Theory o Commercial Softwares o Integer Programming Dynamic Programming o Game Theory OLTEX 4口4+4三4至，至)只0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 4/41

Modeling 1 General Procedure 2 Modeling 3 Solution Approach Linear Programming Sensitivity Analysis Duality Theory Commercial Softwares Integer Programming Dynamic Programming Game Theory 4 LATEX Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 4 / 41

Modeling Example 1 Giapetto's Woodcarving,Inc.,manufactures two types of wooden toys: soldiers and trains.Demand for trains is unlimited,but at most 40 soldiers are bought each week. A soldier sells for $27 and uses $10 worth of raw materials.Each soldier that is manufactured increases Giapetto's variable labor and overhead costs by $14.A train sells for $21 and uses $9 worth of raw materials.Each train built increases Giapetto's variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor:carpentry and finishing.A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor.A train requires 1 hour of finishing and 1 hour of carpentry labor.Each week,Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. How to maximize Giapetto's weekly profit? Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 5/41

Modeling Example 1 Giapetto’s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. Demand for trains is unlimited, but at most 40 soldiers are bought each week. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. How to maximize Giapetto’s weekly profit? Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 5 / 41

Modeling o Decision Variables:The decision variables completely describe the decisions to be made.Denote by x1 the number of soldiers produced each week,and by x2 the number of trains produced each week. o Objective Function:The function to be maximized or minimized is called the objective function.Since fixed costs (such as rent and insurance)do not depend on the values of x1 and x2,Giapetto can concentrate on maximizing his weekly profit,i.e.. max3x灯+2x2. ●Constraints: Each week,no more than 100 hours of finishing time may be used. Each week,no more than 80 hours of carpentry time may be used. Because of limited demand,at most 40 soldiers should be produced each week. 2x1+2≤100，为+9≤80，x1≤40. o Sign Restrictions:x为≥0andx2≥0. 0Q0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 6/41

Modeling Decision Variables: The decision variables completely describe the decisions to be made. Denote by x1 the number of soldiers produced each week, and by x2 the number of trains produced each week. Objective Function: The function to be maximized or minimized is called the objective function. Since fixed costs (such as rent and insurance) do not depend on the values of x1 and x2, Giapetto can concentrate on maximizing his weekly profit, i.e., max 3x1 + 2x2. Constraints: 1 Each week, no more than 100 hours of finishing time may be used. 2 Each week, no more than 80 hours of carpentry time may be used. 3 Because of limited demand, at most 40 soldiers should be produced each week. 2x1 + x2 ≤ 100, x1 + x2 ≤ 80, x1 ≤ 40. Sign Restrictions: x1 ≥ 0 and x2 ≥ 0. Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 6 / 41

Solution Approach General Procedure Modeling ③Solution Approach Linear Programming o Sensitivity Analysis Duality Theory o Commercial Softwares o Integer Programming Dynamic Programming o Game Theory OLTEX 4口，4得+4艺至，三风0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 7/41

Solution Approach 1 General Procedure 2 Modeling 3 Solution Approach Linear Programming Sensitivity Analysis Duality Theory Commercial Softwares Integer Programming Dynamic Programming Game Theory 4 LATEX Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 7 / 41

Solution Approach o Linear Programming (LP) o Integer Programming (IP) o Dynamic Programming(DP) ●Game Theory 4口，4得+4之，至三双0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 8/41

Solution Approach Linear Programming (LP) Integer Programming (IP) Dynamic Programming (DP) Game Theory Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 8 / 41

Solution Approach Linear Programming General Procedure Modeling ③Solution Approach ●Linear Programming o Sensitivity Analysis Duality Theory o Commercial Softwares o Integer Programming Dynamic Programming o Game Theory OLTEX 4口，4得+4艺至，三风0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 9/41

Solution Approach Linear Programming 1 General Procedure 2 Modeling 3 Solution Approach Linear Programming Sensitivity Analysis Duality Theory Commercial Softwares Integer Programming Dynamic Programming Game Theory 4 LATEX Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 9 / 41

Solution Approach Linear Programming oSimplex Method: The Big M Method The Two-Phase Simplex Method oSensitivity Analysis ●Duality 4口，4得+4之，至三风0 Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 10/41

Solution Approach Linear Programming Simplex Method: The Big M Method The Two-Phase Simplex Method Sensitivity Analysis Duality Xi Chen (chenxi0109@bfsu.edu.cn) Optimization Method 10 / 41