Send it as an email attachment by responding to my email. Ism 206 lecture 4 duality and sensitivity analysis 2. Keough, 2008 for detailed analysis and discussion of the duality theory. Duality and interior point methods haitham hindi palo alto research center parc, palo alto, california 94304. The study of such changes is called sensitivity analysis.
I also remind you that lecture notes and practice quizzes are availbe on the webpage of the other section. Every commercial linearprogramming system provides this elementary. Every linear program has another linear program associated with it. Duality theory every lp is associated with another lp, called thedual in this case, the original lp is called the primal. More recently, the sensitivity analysis of multistage stochastic programs was discussed in 6 and 27. Em 605 elements of operations research duality theory, sensitivity analysis and other lp. Em605lecture 3 for pdf em 605 elements of operations. Duality theory is based directly on the fundamental insight particularly with row 0 presented in sec. Linear progammingsimplexbased sensitivity analysis and duality.
Test 2 will be on wednesday, april 8 at class time. When we study the correlation between a target response variable y and a set of. Linear progamming simplexbased sensitivity analysis and. Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system numerical or otherwise can be divided and allocated to different sources of uncertainty in its inputs. Linear programming, or lp, is a method of allocating resources in an optimal way.
This discovery revealed that every linear programming problem has associated with it another linear programming problem called the dual. Duality, warm starting and sensitivity analysis in mixed. Lecture 4 duality and sensitivity in lp slideshare. Duality theory and sensitivity analysis nc state ise. Lecture notes introduction to mathematical programming. Introduces and explains how to use duality theory and sensitivity analysis to interpret linear programming solutions.
Recently a duality theory for integer programming has been developed. Duality in linear programming is essentially a unifying theory that develops the. Sensitivity analysis example the rate of change in optimum z resulting from changing machine 1 capacity from 8 hours to 9 hours can be computed as follows. The emphasis is on sensitivity analysis, duality theory, network simplex. Discuss differences between the standard simplex method and the dual simplex method. Simplex algorithm, duality theory, sensitivity analysis. Its dual the dual complements the original linear program, the primal the theory of duality provides many insights into what is happening behind the scenes dualitysensitivity2 primal and dual. Chapter 4 duality given any linear program, there is another related linear program called the. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty. Pdf duality and sensitivity analysis for fractional programs. Sensitivity analysis sensitivity is a postoptimality analysis of a linear program in which, some components of a, b, c may change after obtaining an optimalsolution with an optimal basis and an optimal objective value. Since the problem d is a linear program, it too has a dual.
This methodology is the basis for what is called duality theory for mathematical programs. If x is a primal feasible solution to p and w is a dual feasible solution to. The solved examples section of the books website provides another example ing the primaldual table to construct the dual problem for a linear programming. Duality theory of constrained optimization robert m. You can find in this pdf a detailed example of the pivots method with practice examples. Announcements homework 2 on web due thursday 14th lecture tonight 6pm 3. Sensitivity analysis in quantitative techniques for. In the preceding chapter on sensitivity analysis, we saw that the shadowprice.
The email will be sent to math 46205620 class list. Mit opencourseware makes the materials used in the teaching of almost all of mits subjects available on the web, free of charge. Sensitivity analysis, stability, utility maximization, optimal investment, risktolerance process, arbitrage of the. Using duality and sensitivity analysis to interpret linear. In 6 the authors study the sensitivity with respect to parameters driving the considered price model. Lengthy proofs will be omitted, and they can be found in standard textbooks of analysis and linear algebra. The relation between an lp and its dual is extremely important for understanding the linear programming and nonlinear programming, indeed. Sensitivity analysis 1 is the study of the variation of the output of a model as the input of the model changes. It can be also seen as a proof of the weak duality theorem. Robert taylor this article graphically illustrates the onetoone duality mapping among the production function, the product supply equation, the derived factor demand. Sensitivity analysis in operation research pdf chap6 duality theory and sensitivity analysis. Duality is pervasive in nonlinear and linear optimization models in a wide variety of engineering and mathematical settings. You do not have access to any existing collections.
Duality theorem all possible relationships between the primal and dual problems 9 if one problem has feasible solutions and a bounded objective function and so has an optimal solution, then so does the other problem. More formally, duality theory can be thought of as the study of the socalled value function, which is a function. This paper will cover the main concepts in linear programming, including examples when appropriate. Associated with each linear program, which we shall call the primal, is another linear program called its dual. Introduction to operations research answers duality and sensitivity analysis prof. With more than 2,200 courses available, ocw is delivering on the promise of open sharing of knowledge. The study of duality is hence the study of a parameterized family of mathematical programming instances and is closely related to parametric programming and sensitivity analysis.
In this paper, sensitivity and duality analyses have been performed on. Duality theory and sensitivity analysis one of the most important discoveries in the early development of linear programming was the concept of duality and its many important ramifications. How does the solution change as the parameters change. Content introduction basic parameter in sensitivity analysis duality and sensitivity analysis example of duality and sensitivity analysis references 24172017. Sensitivity analysis, duality analysis, linear programming, optimal water treatment cost model, interiorpoint method. Duality and sensitivity analysis of multistage linear. Consider a linear programming problem in its standard form. Part of the performance excellence in the wood products industry publication series.
Here we examine some of the economic implications of this theory, in particular the necessity of using price functions in place of prices, and the possibility of carrying out sensitivity analysis of optimal solutions. Sensitivity analysis suppose that you have just completed a linear programming solution which will have a major impact on your company, such as determining how much to increase the overall production capacity, and are about to present the results to. Ch 06 sensitivity analysis and duality free download as powerpoint presentation. Duality arises in nonlinear and linear optimization models in a wide variety of settings. Introduction formally, sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system numerical or otherwise can be apportioned to different sources of uncertainty.
In this paper the authors consider algorithms, duality and sensitivity analysis for optimization problems, called fractional, whose objective function is the ratio of two realvalued functions. By creating a given set of scenarios, the analyst can determine how changes in one variable s will impact the target variable. Duality is a unifying theory that develops the relationships between a given linear program and another related linear program stated in terms of variables with this shadowprice interpretation. Linear programmingduality and sensitivity analysis. There are two types of readers that can skip this chapter. Those who have solid knowledge of analysis and linear algebra, and those who are only interested in computational aspects of optimization theory. First, these shadow prices give us directly the marginal worth of an additional unit of any of the resources. Krylov for a discussion on the subject of the paper. Sensitivity analysis is a way to predict the outcome of a decision if a situation turns out to be different compared to the key prediction s. Some elementary examples of models where duality is plays an important role are. View notes em605lecture 3 for pdf from em 605 at stevens institute of technology. Find materials for this course in the pages linked along the left. Worst case performance of the simplex method kleeminty example.
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