第一章 运筹学绪论和线性规划

Formulation as a linear programming problem (LP) 1.Define the decision variables；
x1 = number of batches of product 1 produced per week
x2 = number of batches of product 2 produced per 2.Find the objective function： Z = 3x1 + 5 x2
WinQSB OR Tutor
Introduction to Linear Programming Basic characteristic:
(1)It is used in OR widely; (2)It’s a fundamental method, Goal programming Integer programming Dynamic programming are all derived from it. (3)An very effective method of finding the optimal distribution under the scarcity, to obtain the maximum profit or minimum cost
运筹学的定义
运筹学是一门应用于管理有组织系统的科学,它为 掌握这类系统的人提供决策目标和数量分析的工 具(大英百科全书). 运筹学应用分析,试验,量化的方法,对经济管理系 , , , 统中的人,财,物等有限资源进行统筹安排,为决策 者提供有依据的最优方案,以实现最有效的管理 (中国企业管理百科全书). 运筹学是一种给出问题不坏答案的艺术，否则的 话问题的结果会更坏。
Main contents
线性规划(Linear programming) 运输问题(Transportation problem) 目标规划(Goal programming) 整数规划(Integer programming) 分配问题(Assignment problem) 动态规划(Dynamic programming)
o
x1
Conclusion: 4 probabilities for two decision variables LP model by graphical solution
Just one optimal solution, it must be a CPF solution; Multiple optimal solutions, at least two must be CPF solutions; Just has feasible solution but no optimal solution; No feasible solution.
o
1
x1
1.5 Another examples for Graphical Solution
max S = 3 x1 + x2 x1 − x2 ≤ −1 s.t x1 + x2 ≤ −1 x ≥ 0, x ≥ 0 2 1
x2
No feasible solution
It has no feasible region
Main contents
图和网络模型(Graph and network modeling) 存储论(Inventory theory) 博弈论(Game theory) 决策论(Decision theory) 排队论(Queueing theory)
Algorithms and OR courseware
运筹学的定义
运筹学是一门新兴的边缘科学,它使用数方学 法,利用计算机等现代化工具,把复杂的研究对 象当作综合系统,进行定量分析,从整体最优出 发,提出一个最优的可行方案,提供给执行机构 作为决策的参考.
早期运筹学思想
齐王和田忌赛马的故事 丁渭修皇宫的故事(丁渭修宫,一举而三役济) 丹麦工程师A.K.Erlang研究电话占线问题 哥尼斯堡七桥问题 E.Zermelo用集合论研究下棋问题 美国Thomas Edison在第一次世界大战中研 究商船航行策略,防止敌潜艇的攻击.
min S = −2 x1 + x2 x1 + x2 ≥ 1 s.t x1 − 3 x2 ≥ −3 x ≥ 0, x ≥ 0 2 1
x2 S = 1 S = −2 S = −3
⇒
1
Unbounded feasible region
It has no optimal solution
x1
We can get:
x2 = 3 x1 = 2 ⇒ is the optimal solution of the LP， x1 + 2 x2 = 8 x2 = 3 The optimal value is max S = 19
Note: (1) This problem has just one optimal solution
1.3 Another examples for Graphical Solution
max S = x1 + 2 x2 x1 ≤ 4 x ≤3 2 s.t x1 + 2 x2 ≤ 8 x1 ≥ 0, x2 ≥ 0
0
Isoline of the objective function
Max
st
3 x1 + 2 x 2 ≤ 18
constrains nonnegative
and
x1 ≥ 0, x2 ≥ 0
1.2 Graphical Solution to Example 1.1
max S = 2 x1 + 5 x2 x1 ≤ 4 x ≤3 2 s.t x1 + 2 x2 ≤ 8 x1 ≥ 0, x2 ≥ 0
-------A linear programming problem (LP) is an optimization problem for which we do the following: 1 We attempt to maximize pr minimize a linear function of the decision variables. The function that is to be maximized is called the objective function. 2 The values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or linear inequality. 3 A sign restriction is associated with each variable. For any variable x i , the sign restriction specifies either that x i must be nonnegative ( x ≥ 0 ) or that x i
x2
Isoline of the objective function
A
B
x1 + 2 x2 = 8
x1
We can get:
x1 + 2 x 2 = 8 x1 = 2 A: ⇒ x2 = 3 x2 = 3 x1 + 2 x 2 = 8 x1 = 4 B: ⇒ x1 = 4 x2 = 2 t he two values S = 8
Feasible region
x2
3
x2 =3
Optimal solution
Isoline of the objective function
2 x1 + 5 x2 = 15
x1 + 2 x2 = 8
2 x1 + 5 x2 = 6
x1 = 4
o
4
Isoline of the objective function
Operations Research
Textbook： Reference:
The origins of Operations Research
the military services early in Word War II
Operations Research (commonly referred to O.R).可直译为“运用研究”,“作业研究”. 1957年我国从“夫运筹帷幄之中,决胜于千 里之外”(见《史记•高祖本记》)这句古语 中摘取“运筹”二字,将O.R正式译成”运 筹学”,包含运用筹划,以策略取胜的意义.
Overview of the OR modeling Approach Defining the problem and gathering data Formulating a mathematical model Deriving solutions from the model Testing the model Preparing to apply the model Implementation conclusion
Besides A and B, any points on AB are optimal solutions of this problem. So it has multiple optimal solutions.
1.4 Another examples for Graphical Solution
The solving of LP is：in the feasible solutions which satisfy both (1.2)and (1.3)(x1， x2，…， xn), finding the value of the decision * * * variable (x1 ，x2 ，…，xn )（optimal solution） to make Z get the （ ） max(min) value.。 。
x1 ≤ 4 3.constrains： 2 x 2 ≤ 12 3 x + 2 x ≤ 来自百度文库8 2 1
4.Nonnegative： x1 ≥ 0, x2 ≥ 0
To summarize, we get
Z = 3 x1 + 5 x2
x1 ≤ 4 2 x 2 ≤ 12
Objective function
• The srandard Form of the Model： ：
max(min) s.t. z =c1x1 + c2x2 +…+ cnxn (1.1) … a11x1 + a12x2 +…+ a1nxn ≤ ( = , ≥) b1 … a21x1 + a22x2 +…+ a2nxn ≤ ( = , ≥) b2 … (1.2) … … (1 2) am1x1 + am2x2 +…+ amnxn ≤ ( = , ≥) bm … x1，x2，…，xn ≥ 0 (1.3)
1.1The simplification of Prototype Example: The WYNDOR GLASS CO. produces a high-quality glass products and wants to launch two new products. It has 3 plants and product 1 need plants 1 and 3, while products 2 needs plants 2 and 3.All the products (1 and 2) can be sold and table 3.1 on page 27 summarizes the data gathered by the OR team. The goal of the company is to get the maximum profit from the sold products 1 and 2.