线性规划和整数规划经济中的应用the models 论文

OTTO-VON-GUERICKE UNIVERSITY MAGDEBURGFACULTY OF ECONOMICS AND MANAGEMENTDEPARTMENT OF INTERNATIONAL ECONOMICSMASTER OF BUSINESS ADMINISTRATIONSome Applications of Linear Programming inEconomicMaster ThesisContents1 Introduction............................................................................................................................ - 1 -2 Linear Programming .............................................................................................................. -3 -2.1 What is a Linear Programming? ..................................................................................... - 3 -2.2 How to apply a linear programming ............................................................................... -4 -2.2.1 A linear function ...................................................................................................... - 4 -2.2.2 Graphical method..................................................................................................... - 5 -2.2.3 Simplex method ..................................................................................................... - 11 -3 The LINDO computer package ............................................................................................. - 22 -3.1Introduction LINDO software .................................................................................. - 22 -3.2 Economic applications of LINDO software .................................................................... - 24 -3.2.1How to formulate planting plans, in order to make the maximum output? ... - 25 -3.2.2 How to formulate planting plans, in order to make the greatest profits? ............ - 31 -4 An integer programming problem and relationships between integer and linear programming ................................................................................................................................................ - 35 -4.1 Formulating integer programming problems ............................................................... - 36 -4.2 Branch and bound method ........................................................................................... - 37 -4.3 The 0-1 integer programming ....................................................................................... - 42 -Summary ................................................................................................................................. - 45 -References............................................................................................................................... - 46 -List of figureF IGURE 1G RAPHICAL SOLUTION (7)F IGURE 2M ULTIPLE OPTIMAL SOLUTIONS (10)F IGURE 3LINDO OUTPUT FOR E XAMPLE 2 (22)F IGURE 4LINDO OUTPUT FOR E XAMPLE 2 (23)F IGURE 5LINDO SOLVER STATUS (23)F IGURE 6LINDO MODEL (26)F IGURE 7T HE FIRST PART OF OUTPUT (27)F IGURE 8T HE SECOND PART OF OUTPUT (29)F IGURE 9 (32)F IGURE 10 (33)F IGURE 11 (34)F IGURE 12B RANCH-AND-BOUND TREE FOR E XAMPLE 2 (37)F IGURE 13 (40)F IGURE 14B RANCH-AND-BOUND TREE FOR E XAMPLE 4 (41)F IGURE 15B RANCH-AND-BOUND TREE FOR E XAMPLE 5 (44)List of tableT ABLE 1 (6)T ABLE 2A DJACENT CPF SOLUTIONS FOR EACH CPF SOLUTION OF THE E XAMPLE 1 (9)T ABLE 3T HE INITIAL SIMPLEX TABLEAU (16)T ABLE 4T HE I TERATIVE PROCESS (17)T ABLE 5 (18)T ABLE 6S PECIFIC EXPLANATION SIMPLEX TABLEAU (19)T ABLE 7D ETERMINE THE ENTERING AND LEAVING VARIABLE (20)T ABLE 8T HE ITERATION FOR E XAMPLE 2 (21)T ABLE 9T HE YIELD OF THE DIFFERENT CORPS IN DIFFERENT FARMLAND (KG / HM^2) (24)T ABLE 10C ROPS PLANTING AREA (HM^2) (25)T ABLE 11C ROPS TOTAL OUTPUT（KG） (25)T ABLE 12 (39)T ABLE 13 (42)1 IntroductionThe linear programming method was first developed by Leonid Kantorovich in 19391. It is a fundamental branch of operations research; it is widely used in the existing science and technology. Linear programming (LP or linear optimization) is a tool to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. This mathematical model method can help decision makers to solve practical problems and choose the best policy. Through efficient management system for quantitative analysis, quantify the fiscal management system; provide solutions for decision makers to choose an optimal plan in order to achieve the best economic results.Most of the models described in this thesis will be optimization models. Generally, components of an optimization model include.1.Objective functionThere is a linear function that is to be optimized (maximized or minimized) in a linear programming model. This function is referred to as the model’s objective function. Of course, to maximize the process yield we need to find the all constraints and decision variables make our objective function as large as possible.2.Decision variablesThe variables in a given model which values are subject to manipulation by satisfying all constraints and influence the performance of the system are called decision variables. Linear programming is able to discuss of how to determine the value that maximize (or minimize) an objective function.3.Constraints1L.V. Kantorovich: A new method of solving some classes of extremal problems, Doklady Akad Sci USSR, 28, 1940, 211-214.There is a restriction on the permissible values of the decision variables of a given LP problem are called constraints2.In our life, calculation of the LP model, some of the optimal solution might be a decimal, and for some specific problems, the decision variables must be an integer. For example, it is necessary to assign people, machines, and vehicles to activities in integer quantities. There should be a special way to solve them.We will use integer programming which is a branch of LP, in order to solve the special problems.Finally this thesis studies the knowledge of how to apply linear and integer programming in economic problems, discuss the establishment of model, and explain the LINDO output.2Wayne L. Winston (1975) Operations Research Applications and Algorithms, 5 editions, 2004, 22 Linear Programming2.1 What is a Linear Programming?Linear programming is a special class of mathematical programming models in which the objective functions and the constraints can be expressed as linear functions of the decision variables. In our daily life, company or factory production, management and other economic activities will face a problem of what is the best solution and decision. So components of an economic model will be including.1.Objective functionIt represents some principal objective criterion or goal that measures the effectiveness of the system (such as maximizing profits or productivity, or minimizing cost or consumption).2.Decision variablesThey represent quantities that are, in some sense, controllable inputs to the system being modeled.3.ConstraintsThere is always some practical limitation on the availability of resources (time, materials, machines, or manpower) for the system, and such constraints are expressed as linear inequalities or equations involving the decision variables3.An economic model applies to a linear programming theory and methods to analyze and solve. There are under a set of constraints, a mathematical method for finding an optimal solution (maximum or minimum) and value.3Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 252.2 How to apply a linear programmingLinear programming is a branch of mathematical programming and is the most commonly used operations research methods. In various economic activities, a linear programming problem is generally figuring out how to best allocate limited resources, in order to best leverage the resources to obtain the best performance of the economic benefits. Linear programming prepares a program of activities or model in order to achieve optimal results. That is how to select the best decision variables solution in order to achieve the specified goals.2.2.1 A linear functionIn general, if c1,c2,…,c n are real numbers, then the function Z of the real variables x1, x2,…, x n defined bynZ=c1x1+c2x2+⋯+c n x n=∑c j x jj=1If Z is a linear function and if b is a real number, thenZ=bis called a linear equation andZ≤bZ≥bare called linear inequalities. They are both referred to as linear constraints. So, a linear programming problem is the problem of maximizing (or minimizing) a linear function subject to a finite number of linear constraints. We shall usually attach different subscripts i to different constraints and different subscripts j to different variables4. So we can get LP problems of the following form:4Vašek Chvátal (1946) Linear Programming, 1 editions, 6nMaximize Z=∑c j x j(objective function)j=1subject ton∑a ij x j≤b i (constraints)j=1x j≥0(Non−negative constraints)( j=1,2,…,n; i=1,2,…,m)When x j satisfies all constraints of an LP problem are said to constitute a feasible solution of that problem. But x j can’t satisfy all constraints of an LP problem are called infeasible solution. If a set of feasible solution can make our objective function maximize (or minimize), these solution are called an optimal solution; the corresponding value of the objective function Z is called the optimal value of the problem.After the introduction of the linear function, we have three basic components to establish a linear programming model:.1.Decision variables that we seek to determine x j,( j=1,2,…,n)2.Objective function that we aim to optimize (Z)3.Constraints that we need to satisfy52.2.2 Graphical methodThis section deals with the graphical solution of a two-variable linear programming. Though a two-variable problem hardly exists in practice, the treatment provides concrete ideas for the development of the general solution algorithm presented in Example 2.Example 1 Graphical method to solve a linear programming problem5Hamdy A. Taha, Operations Research An Introduction, seventh edition, Prentice Hall, New York (USA), 2010, 12A factory produces two types A andB of stainless steel from two raw materials – iron and carbon. The following Table 1 provides the basic data of the problem:No constraints factory productivity in our data so is ignored. The factory wants to determine the best product mix of iron and carbon that maximizes the total daily profit. How much profit the factory can get?Formulation Example 1Step1 Decision variables that we seek to determine x j,(j=1,2,…,n) We need to determine the amounts to be produced by iron and carbon. Thus, thedecision variables of the model are defined as:x1=Tons produced daily of ironx2=Tons produced daily of carbonStep2 Objective function that we aim to optimize (Z)Profit per part multiplied the quantity of production is equal to total profit, letting Z represents the total daily profit (100€). The objective function of the company is expressed asMaximize Z=2x1+3x2(1-1)Step3 Constraints that we need to satisfyRaw material restrictions are expressed as(Usage of a raw material by both goods)≤(Maximum raw material availability) From the data of the problem, the daily availabilities of raw materials iron and carbon are limited to 12 and 8 tons, so we have:{Usage of raw material IRON per day: 2x1+2x2≤12Usage of raw material CARBON per day: x1+2x2≤8(1-2)An implicit restriction is that the decision variables x1and x2cannot assume negative values. The non-negative constraints x1,x2≥0, account for this requirement.Finally, combining the non-negative constraints x1,x2≥0with objective function (1-1) and constraints (1-2) yields the following optimization model:Maximize Z=2x1+3x2 (objective function)subject to (s.t.){2x1+2x2≤12 (input Iron constraint) x1+2x2≤8 (input Carbon constraint)x1,x2≥0 (Non−negative constraints)“subject to” (s.t.) means that the values of the decision variables x1 and x2must satisfy all constraints and all sign restrictions.Graphical Example 1Step.1 Determination of the feasible solution spaceAny values of x1and x2that satisfy all the constraints of the model constitute a feasible solution. Then each inequality constraint defines a half-plane in the two dimensions, and the intersection of these half-planes comprises the feasible space for this case6, as showed by shaded area in Figure 1.Figure 1 Graphical solution6Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 25The feasible space (or feasible region) for the problem is the set of all such points that satisfies the problem constraints. The points labeled A, B, C and D are called corner-point of the feasible space.In Figure 1, we can easily find the point (x1, x2) = (2, 1) tons per day, is a feasible solution because it does not violate any of the constraints, including the non-negativity restrictions. Total daily profit is Z(2,1)=2x1+3x2=2∗2+3∗1=9. But when (x1, x2) = (0, 4) and Z(0,4)=12>Z(2,1)=9. The feasible space includes numerous solutions, which one is the best?Step.2 Draw an isoprofit lineThe feasible space consists of an infinite number of points, any point within or on the boundary ABCD of the space is feasible, in the sense that it satisfies all the constraints. The need for a systematic procedure to identify the optimal solution is obvious. When we moving the objective function (profit function) Z=2x1+3x2, the function have the same slope of -2/3 and have the same Z value, such a line is called an isoprofit line in max problems.Step.3 Move parallel the isoprofit line to determine an optimal solutionThis line is tangent to the feasible space at one of the corner point. The last point in the feasible space that contacts an isoprofit line is an optimal solution to the LP. In Figure 1, the red line represents the objective function, the optimal objective function value is Z∗=2∗4+3∗2=14at the optimal point C(x1, x2)=(4,2).Corner-point feasible solutionsHere, each constraint boundary is a line that forms the boundary of what is permitted by the corresponding constraint. The points of intersection are the corner-point solutions of the problem. The four points that lie on the corners of the feasible space A, B, C and D are the corner-point feasible solutions (CPF). However, (0, 6) (8, 0) are called corner-point infeasible solutions. In Figure 1, each corner-point solution lies at the intersection of two constraint boundaries. (Although a corner-pointsolution is defined in terms of n constraint boundaries whose intersection gives this solution, it also is possible that one or more additional constraint boundaries pass through this same point7. A to B and A to D, they share one constraint boundary. Thus, point A has two adjacent CPF solutions.Optimality test: Shift curve Z to point B in the Figure 1, point B has adjacent CPF C. Consider any linear programming problem that possesses at least one solution. If a CPF solution has no adjacent CPF solutions that are better, when we increasing the Z curve to point C (as measured by Z), then it must be an optimal solution.The graphical solution to find an optimal solution is very intuitive, if an optimal solution exists, it occurs at an extreme point of the feasible space. This fundamental property of a linear programming problem is the foundation of a general solution method called the simplex method. Because only the finitely many corner points need to be tested, an optimal solution may be found systematically by considering the objective function values at the extreme points8. It also gives concrete ideas for the development and interpretation of sensitivity analysis in a linear programming9. The sensitivity analysis is concerned with how changes in a linear programming problem’s parameters affect the linear programming problem’s optimal solution. We will be described next.When market conditions are changed, the carbon profit increase, we have to change the carbon’s parameter: i.e. the per carbon profit c2=4instead of c2=3. We get a new objective functionZ new=2x1+4x27Hillier Lieberman, Introduction to Operations Research, Seventh Edition, MacGraw-hill, United States, 2010, 1108Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 309Hamdy A. Taha, Operations Research An Introduction, seventh edition, Prentice Hall, New York (USA), 2010, 11The Figure 1 becomes Figure 2, the isoporfit line of original objective function (blue) rotation to new objective function (red).Figure 2 Multiple optimal solutions“By replacing the original data by more pessimistic or optimistic estimates of the unknown quantities, we may create a number of variations on the original theme. Since each of these new LP problems might possibly represent the actual situation, it is useful to find out how the optimal solutions vary with the changes in the data. It may be, for example, that the optimal solution is particularly sensitive to changes in only a small set of parameters; if possible, these parameters should then be estimated with greater accuracy. Other variations on the original theme arise when new variables and/or constraints are introduced. New variables may appear when new products are developed. Similarly, new constraints may result from new production policies. It may also happen that a new constraint actually had to be satisfied from the beginning but was simply forgotten in the original formulation. Investigations of such changes are referred to as sensitivity analysis.”10Market circumstance may fluctuate. Supplies of raw materials and demands for finished products are often unknown in advance, like per carbon profit from 3€ changes to 4€. Furthermore，production may be influenced by a variety of accidental events such 10Vašek Chvátal (1946) Linear Programming, 1 editions, 148as technology improvements or machine breakdowns. No matter how much raw iron used, there is always a value of carbon materials, making an optimal Z new in the feasible space. Sensitivity analysis provides efficient computational techniques to study the dynamic behavior of the optimal solution resulting from making changes in the parameters of the model. LINDO software will be used later to help us answer these questions. In Figure 2 the original optimal solution C (4, 2) becomes into an infinite number of optimal solutions, Maximize Z new=16.2.2.3 Simplex methodPreparation for the simplex methodWe are necessary to express the LP problem in standard form. Because of “A line function” section, for a linear program with n variable and m constraints, we will use the following standard form:Maximize Z=c1x1+c2x2+⋯+c n x nsubject toa11x1+a12x2+⋯+a1n x n=b1a21x1+a22x2+⋯+a2n x n=b2… … … … … … … … … … … …a m1x1+a m2x2+⋯+a mn x n=b mx1,x2,…x n,b1,b2,…,b m≥0Then the optimization problem can be expressed succinctly asnMaximize Z=∑c j x jj=1n∑a ij x j=b ij=1j=1,2,…,n; i=1,2,…,m; x j,b i≥0;Solutions of linear systemsWe now have a system of linear equations ∑a ij x j n j=1=b i which consisting of n unknowns and m equations. The n unknowns include the original decision variables and any other variables that may have been introduced in order to achieve standard form. ∑a ij x j n j=1=b i to find a basic solution: We choose a set of n −m variables (the non -basic variables ) and set each of these variables equal to zero. Then we solve for the value of the remaining n −(n −m )=m variables (the basic variables ) that satisfy ∑a ij x j n j=1=b i .Note that not all of the solutions satisfy all problem constraints and non -negativity constraints. Those that do not comply with these requirements are infeasible solutions . The ones that do meet the restrictions are called basic feasible solutions . An optimal basic feasible solution is a basic feasible solution that optimizes the objective function. The basic feasible solutions correspond precisely to the corner -points of the feasible space (as defined in our earlier discussion of graphical solutions).11In graphical method we see that in going from one canonical from to the next, we have proceeded from one basic feasible solution to a better adjacent basic feasible solution. The procedure used to go from one basic feasible solution to a better adjacent basic feasible solution is called an iteration of the simplex algorithm 12.To describe the mechanics of the algorithm, we must specify how an initialfeasible solution is obtained, how a transition is made to a better basic feasible solution, and how to recognize an optimal solution. From any basic feasible solution, we have the assurance that, if a better solution exists at all, then there is an adjacent solution(Corner -point feasible solutions) that is better than the current one. This is the principle on which the simplex method is based; thus, an optimal solution is accessible from any starting basic feasible solution.1311Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 32 12 Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 14513 Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 33How to convert linear inequalities to a standard form and special cases of standard form converting:(1)An original objective function is minimize Z, then let Z′equal to –Z, therefore, the minimize Z objective function becomes:Z=c1x1+c2x2+⋯+c n x n→min!↔Z′=–Z=−c1x1−c2x2−⋯−c n x n→max!(2) If a constraint is given by a11x1+a12x2+⋯+a1n x n≤b1.Less-than-or-equal inequalities require the introduction of variables that we will call slack variables. Then we add a non-negative slack variable x n+1, and new inequality becomes:a11x1+a12x2+⋯+a1n x n+x n+1=b1(3) If a constraint is given by a11x1+a12x2+⋯+a1n x n≤b1.Greater-than-or-equal constraints are modified by introducing surplus variables. Then we subtract a non-negative surplus variable x n+1(Also be referred to slack variables) is:a11x1+a12x2+⋯+a1n x n−x n+1=b1(4) Under special circumstances, for example x j are arbitrary value, if the variables x j unconstrained, we set x j=x j′−x j′′, x j′≥0,x j′′≥0, and replace the variables in the model. Then in any solution, the sign of the values of x j are dependent on the relative values of x j′ and x j′′.Then we get:x j′>x j′′↔x j>0x j′=x j′′↔x j=0x j′<x j′′↔x j<0In the above situation, each constraint condition adds slack variable in linear inequalities left-hand. Set x i=x1,x2…x m are basic variables, x j=x m+1,x m+2…x nnon-basic variables. a m+1,a m+2,…,a n the parameters of non- basic variables. The new formula can be expressed as:Maximize Z=c1x1+c2x2+⋯+c n x n{x1 +a1m+1x m+1+⋯+a1n x n=b1 x2 ++a2m+1x m+1+⋯+a2n x n=b2……………………x m+a nm+1x m+1+⋯+a mn x n=b1mx1,x2…x n≥0(2−1)The simplex algorithmSimplex method is an iterative algorithm that begins with an initial feasible solution, repeatedly move to a better solution, and stops when an optimal solution has been found and, therefore, no improvement can be made. In order to apply such an approach, from formula (2-1) consider a feasible canonical form with the basic variables x i=x1,x2…x m and the non-basic variables x j=x m+1,x m+2…x n:{x1=b1−a1m+1x m+1−⋯−a1k x k−⋯−a1n x n x2=b2−a2m+1x m+1−⋯−a2k x k−⋯−a2n x n……………………x l=b lm−a lm+1x m+1−⋯−a lk x k−⋯−a ln x n……………………x m=b1m−a nm+1x m+1−⋯−a mk x k−⋯−a mn x nx1,x2…x n≥0(2−2)Then the form can be expressed succinctly as:x i=b i−a i,m+1x m+1−a i,m+2x m+2−⋯−a i,n x n=b i−∑a ij x j(i=1,2,…,m)nj=m+1We remind the basic solution, which each non-basic variable has the value zero, x i0=(b1,b2,…,b m),x j=(0,0,…,0), so we can getZ0=∑c imi=1x i0=∑c i b imi=1Substituting the above expressions into the objective function:Z =c 1x 1+c 2x 2+⋯+c n x n =∑c i x i (basic variables )+∑c j x j mj=m+1m i=1(non −basic variables)=∑c i b i m i=1+∑(c j −z j )x j n j=m+1=Z 0+∑(c j −z j )x j nj=m+1The c j −z j is coefficient of non - basic variable x i . When entering a variable into the basis, if its value would increase by one unit. By means of the coefficients in the objective row we can give the following optimality criterion.(1) x 0 is an optimal solution, when c j −z j ≤0,Z ≤Z 0.(2) If c j −z j <0, we have a unique optimal solution.(3) If c j −z j ≤0, and for a non -basic variable x k we have c j −z j =0, the linear programming problem has infinitely optimal solutions.(4) If the non -basic variable x m+k we have c j −z j >0, and the coefficient corresponding the non -basic variable solution a i,m+k ≤0, so there does not exist an optimal solution to the maximum problem.The c j −z j can determine whether we have a feasible solution, c j −z j is called the optimal test . Summing up the above, we can draw a simple table 3Determination of the pivot column k and row lIf our initial feasible solution is not an optimal solution, we need to find another feasible solution, which increases the objective function.From Table 3, if c j−z j>0, a unit increases in x j (j=m+1,…,k,…,n)can increase in Z value,. So we often choose the max c j−z j, which is the corresponding value of x j is entering value.From Table 3, the feasibility of the basic solution must be maintained. Then when entering a variable into the basis, compute the ratio inRight−hand side of row coefficient of entering variable in row ↔ θ=min(b ia ik,a ik>0)=b la lk(2−3)for every constraint in which the entering variable has a positive coefficient14. If the entering variable has a non-positive coefficient in a row, the row’s basic variable will remain positive for all value of the entering variable. So we set column x k(m+1≤k≤n) to be the entering variable in non-basic solution and row x l(1≤l≤m)to be the leaving variable in basic solution.Iterative processθ=min(b i′a ik′,a ik′>0)=b l′a lk′The value b i′and a ik′are correspond to the b i and a ik after the iteration.Let x k and x l be converted, our constraint equations (2-2) l-th over a lk becomes:14Michael W. Carter, Camille C.Price, Operations research: a practical introduction, 2001, 1431 a lk x l=b lma lk−a lm+1a lkx m+1−⋯−x k−⋯−a lna lkx nThen x k column, except for x k become 1, the other is zero. And according to Gauss theorem will get new l-th:−a ika lkx l=b i−b lma lka ik−(a im+1−a lm+1a lka ik)x m+1−⋯−1−⋯(a ln−a lna lka ik)x n Thus, we can get every parameter transformation relations:a ij′={a ij−a lja lka ik (l≠k)a lja lk(l=k),b i′={b i−b la lka ik (l≠k)b la lk(l=k)x k and x l be converted can be expressed as Table 4.We will use the following simple model as an illustration for describing the simplex method.Example 2 A simple transportation problemWe often encounter the problem of transport in economic activities. Our society is no shortage of labor force, therefore, the transport constraints focused on carrying capacity and profit. Here we give an example: If a train loaded with two kinds of goods --- Machine A and B. Each machine’s size, weight, gets the profits and transports restrictions shown in the following Table 5:。