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1、3 Introduction to Linear Programming,3.1 Prototype Example3.2 The Linear Programming Model3.3 Assumptions of Linear Programming 3.4 Additional Examples3.5 Some Case Studies3.6 Conclusions,In this lesson: The teaching objects are : (1) Master the formulation of linear programming model (2) Understand
2、 the procedure of solving linear programming by using EXCEL SOLVER The teaching contents are : (1) Prototype Example (2) The basic steps of formulating a model of linear programming (3) Solving a linear programming by use EXCEL SOLVER,The development of linear programming has been ranked among the m
3、ost important scientific advances of the mid-20th century, and we must agree with this assessment. Its impact since just 1950 has been extraordinary. Today it is a standard tool that has saved many thousands or millions of dollars for most companies or businesses of even moderate size in the various
4、 industrialized countries of the world; and its use in other sectors of society has been spreading rapidly. Linear programming involves the general problem of allocating limited resources among competing activities in a best possible way. It use s a mathematical model to describe the problem of conc
5、ern. And the remarkably efficient solution procedure to the linear programming is called the simplex method.,3.1 Prototype example,The WKYNDOR GLASS CO. produces high-quality glass products, including windows and glass doors. It has three plants. Aluminum frames and hardware are made in Plant1, wood
6、 frames are made in Plant 2, and Plant 3 produces the glass and assembles the products. Because of declining earnings, top management has decided to revamp the companys product line. Unprofitable products are being discontinued, releasing production capacity to launch two new products having large s
7、ales potential:Product1: an 8-foot glass door with aluminum framing Product2: A 4*6 foot double-hung wood-famed window,The OR team gather the following production data:,The question is: Determine what the production rates should be for the two products in order to maximize their total profit, subjec
8、t to the restrictions imposed by the limited production capacities available in the three plants. (Each product will be produced in batches of 20, so the production rate is defined as the number of batches produced per week.),Question Analysis,The decision variable is : x1=? x2=? The objective is ma
9、ximize the profit:,The constraints are : Plant 1 Plant 2 Plant 3,Formulate the model,Another Example :,Regional Planning,THE SOUTHER CONFEDERATION KIBBUTZIM is a group of three kibbutizm (communal farming communities) in Israel. This office is planning agricultural production for the coming year. Th
10、e agriculture output of each kibbutz is limited by both the amount of available irrigable land and the quantity of water allocated for irrigation by the Water Commissioner. These data are given in Table 3.8.,The crops suited for this region include sugar beets, cotton, and sorghum, and these are the
11、 three being considered for the upcoming season. These crops differ primarily in their expected net return per acre and their consumption of water. In addition, the Ministry of Agriculture has set a maximum quota for the total acreage that can be devoted to each of these crops by the Southern Confed
12、eration of Kibbutzim, as shown in Table 3.9.,Because of the limited water available for irrigation, the Southern Confederation of Kibbutzim will not be able to use all its irrigable land for planning crops in the upcoming season. To ensure equity between the three kibbutzim, it has been agreed that
13、every kibbutz will plant the same proportion of its available irrigable land. However , any combination of the crops may be grown at any of the kibbutzim.,The question is: How many acres to devote to each crop at the respective kibbutzim while satisfying the given restrictions. The objective is to m
14、aximize the total net return to the southern Confederation of Kibbutzim as a whole.,Analyze and formulate the model,So, the decision variables are,The objective is to maximize the Net Return,The constraints are more complicated, we divided them into a few kinds,1.Usable land for each kibbutz:,2.Wate
15、r allocation for each kibbutz:,3.Total acreage for each crop:,4.Equal proportion of land planted:,5.Nonnegativity：,So, we can see, any linear programming model institute of three parts: Decision variables, Objective function, Constraints,And to formulate a model of linear programming institute three
16、 stepsDetermine decision variablesDetermine the objective functionDetermine the constraints,We can now formulate the mathematical model for this general problem of allocating resources to activities.In particular,this model is to select the values for x1,x2, ,xn so as to,We call this our standard fo
17、rm for the linear programming problem. Any situation whose mathematical formulation fits this model is a linear programming problem.,3.2 The Linear Programming Model,Common terminology for the liner programming model can now be summarized. The function being maximized, c1x1+c2x2+cnxn is called the o
18、bjective function. The restrictions normally are referred to as constraints. The first m constraints are called functional constraints.The xj0 restrictions are called nonnegativity constriants (or nonnegativity conditions).,3.3 Solving Linear Programming by using EXCEL,(1)Add-in Excel Solver,(2) Inp
19、ut the data to data cells(3)determine the location of decision variables,(4) In the objective cell, input the objective function,(5) Input the left function of each constraint,(6) Launch Solver,(7) Answer the parameters of Solver,(8) Add constrains,(9) Finish the Solver dialog box,(10) Solve the problem and get the optimal solution,The optimal solution is: produced doors=2, windows=6 maximum profit=3600,Exercise: To solve the Regional Planning,The End,
