3205Chapter-4 - Chapter 4 Duality Theory and Sensitivity Analysis Also see Chapter 6 of the text book One of the most important discoveries in the early

3205Chapter-4 - Chapter 4 Duality Theory and Sensitivity...

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Unformatted text preview: Chapter 4 Duality Theory and Sensitivity Analysis Also see Chapter 6 of the text book. One of the most important discoveries in the early development of linear programming was the concept of duality and its many important ramifications. This discovery revealed that every linear programming problem has associated with it another linear programming problem called the dual. The relationships between the dual problem and the original problem (called the primal) prove to be extremely useful in a variety of ways. For example, you soon will see that the shadow prices described in Chapter 3 actually are provided by the optimal solution for the dual problem. We shall describe many other valuable applications of duality theory in this chapter as well. One of the key uses of duality theory lies in the interpretation and implementation of sensitivity analysis. As we already mentioned many times, sensitivity analysis is a very important part of almost every linear programming study. Because most of the parameter values used in the original model are just estimates of future conditions, the effect on the optimal solution if other conditions prevail instead needs to be investigated. Furthermore, certain parameter values (such as resource amounts) may represent managerial decisions, in which case the choice of the parameter values may be the main issue to be studied, which can be done through sensitivity analysis. For greater clarity, the first three sections discuss duality theory under the assumption that the primal linear programming problem is in our standard form (but with no restriction that the bi values need to be positive). Other forms are then discussed in Sec. 4. We begin the chapter by introducing the essence of duality theory and its applications. We then describe the economic interpretation of the dual problem and delve deeper into the relationships between the primal and dual problems (Sec. 3). Section 6.5 focuses on the role of duality theory in sensitivity analysis. The basic procedure for sensitivity analysis (which is based on the fundamental insight of Sec. 3 in last chapter) is summarized in Sec. 6 and illustrated in Sec. 7. 1 CHAPTER 6 6DUALITY ANALYSIS CHAPTER DUALITYTHEORY THEORYAND AND SENSITIVITY SENSITIVITY ANALYSIS 1 THE ESSENCE OF DUALITY THEORY ESSENCE DUALITYTHEORY THEORY ETHE ESSENCE OFOF DUALITY Given our standard form for the primal problem at the left (perhaps after conversion from Given our standard form thehas primal problem at conversion from another form), its dual problem the form shown to left the right. Given our standard form forforthe primal problem at the the left(perhaps (perhapsafter after conversion from another form), its dual problem has the form shown to the right. another form), its dual problem has the form shown to the right. Primal Problem Dual Problem Primal Problem Dual Problem m n Maximize Maximize Z n c j x j, Z  j1c j x j, j1 subject to   aij x j  bi, j1 aij x j j1 and  bi, and xj  0, i1 W  bi yi, Minimize i1 subject to subject to m subject to n n W   mbi yi, Minimize for i  1, 2, . . . , m for i  1, 2, . . . , m m aij yi  cj,  aij yi  cj, i1 for j  1, 2, . . . , n for j  1, 2, . . . , n i1 and xj  0, for j  1, 2, . . . , n. and yi  0, for i  1, 2, . . . , m. yi  0, for j  1, 2, . . . , n. for i  1, 2, . . . , m. Thus, with the primal problem in maximization form, the dual problem is in minimization Thus, with the primal problem in maximization form, the is in as minimization form instead. Furthermore, the dual problem uses exactly thedual sameproblem parameters the priThus, with the primal problem in maximization form, the dual problem is in minimization formmal instead. Furthermore, the dual problem uses exactly the same parameters as the primal problem, but in different locations, as summarized below. form instead. Furthermore, the dual problem uses exactly the same parameters as the priproblem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides mal problem, but in different locations, as summarized below. • The coefficients the objective function of the primal problem are the right-hand sides of the functionalinconstraints in the dual problem. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coefof the functional constraints in theofdual problem. ficients in the objective function the dual problem. • The right-hand sides of the functional constraints theprimal primal problem arethe the coef2. The right-hand sides the functional constraints ininthe are coef3. The coefficients of of a variable in the functional constraints of the problem primal problem are ficients in the objective function of the dual problem. the coefficients in a functional of the dual problem. ficients in the objective functionconstraint of the dual problem. 3. The coefficients ofofa avariable the functional ofofthe primal problem •ToThe coefficients variable theat functional constraints the primal problem are highlight the comparison, nowinin look these sameconstraints two problems in matrix notation (as are the coefficients in a functional constraint of the dual problem. the coefficients in a functional constraint of the dual problem. introduced at the beginning of Sec. 5.2), where c and y  [y1, y2, . . . , ym] are row vectorshighlight but b the andthe x are column now vectors. To highlight comparison, looklook at these samesame two two problems in matrix notation (as To comparison, now at these problems in matrix notation (as introduced at the beginning of Sec. 2 inwhere Chapter 3), ywhere , · · row · , ymvec] are introduced at the beginning of Sec. 5.2), c and  [y1c, and y2, .y. = . , [yy1m,]y2are Primal Problem Dual Problem row vectors butx bare and x are column tors but b and column vectors.vectors. Z  cx, Maximize PrimaltoProblem subject Ax  b Maximize Z  cx, Dual subject to Problem yA  c Minimize x  0. W  yb, subject to y  0. yA  c Ax  b and W  yb, and and subject to Minimize and To illustrate, the primal and dual problems for the Wyndor Glass Co. example of Sec. 3.1 y  0. x  0. are shown in Table 6.1 in both algebraic and matrix form. The primal-dual table for linear programming (Table 6.2) also helps to highlight the correspondence between the two problems. It shows all the linear programming paTo illustrate, the primal and dual problems for the Wyndor Glass Co. example of Sec. 3.1 rameters (the aij, bi, and cj) and how they are2 used to construct the two problems. All the are headings shown infor Table 6.1 in problem both algebraic and matrix form. the primal are horizontal, whereas the headings for the dual probTheare primal-dual tablethefor linear programming (Tableproblem, 6.2) alsoeach helps to highlight lem read by turning book sideways. For the primal column (exthe cept correspondence between the two problems. It shows all the linear programming the Right Side column) gives the coefficients of a single variable in the respective pa- rameters (the a , b , and c ) and how they are used to construct the two problems. All the 76299_ch06_195-275.qxd 11/19/08 09:50 AM Rev.Confirming Pages Page 197 6.1 primal THE ESSENCE OF DUALITY THEORY To illustrate, the and dual problems for the Wyndor Glass Co. example are197 shown in Fig. 4.1 in both algebraic and matrix form. ■ TABLE 6.1 Primal and dual problems for the Wyndor Glass Co. example Primal Problem in Algebraic Form Dual Problem in Algebraic Form Z  3x1  5x2, Maximize subject to W  4y1  12y2  18y3, Minimize subject to 3x1  2x2  4 y12y2  3y3  3 3x1  2x2  12 2y2  2y3  5 3x1  2x2  18 and x1  0, and x2  0. y1  0, Primal Problem in Matrix Form x1 x , 2 Minimize subject to 1 0 3 0 x1 2 x2  2   y3  0. Dual Problem in Matrix Form Z  [3, 5] Maximize y2  0, 4 W  [y1, y2, y3] 12 18 subject to 4 12 18 [y1, y2, y3] and 1 0 3 0 2  [3, 5] 2 and x1 0  . x2 0    [y1, y2, y3]  [0, 0, 0]. ■ TABLE 6.2 Primal-dual table for linear programming, illustrated by the Wyndor Figure 4.1: Primal and dual problems for the Wyndor Glass Co. example Glass Co. example (a) General Case y1 y2 y3 Coefficients for Objective Function (Minimize) Right Side Dual Problem Coefficient of: The primal-dual table for linear programming (Fig. 4.2) also helps to highlight the correPrimal Problem spondence between the two problems. It shows all the linear programming parameters (the Coefficient of: problems. All the headings for aij , bi , and cj ) and how they are used to construct the two Right … for the x1 the x xn dual Side the primal problem are horizontal, whereas headings problem are read by 2 turning the note sideways. For they1primal problem, each column (except the Right Side … a11 a12 a1n  b1 … y2 a21 a22in the respective a2n  b2 column) gives the coefficients of a single variable constraints and then in   the objective function, whereas eachy row (except the bottom one) gives the parameters for a … am1 am2 amn  bm m single constraint. For the dual problem, each row (except the Right Side row) gives the co… VI VI VI efficients of a single variable in the respective in the objective function, … and then c1 constraints c2 cn whereas each column (except the rightmost one)Coefficients gives theforparameters for a single constraint. In addition, the Right Side column gives the Objective righthand sides for the primal problem and Function (Maximize) the objective function coefficients for the dual problem, whereas the bottom row gives the Wyndor Glass for Co. Example objective function(b)coefficients the primal problem and the right-hand sides for the dual problem. x1 x2 1 0 3 0 2 2 VI 3 VI 5  4  12  18 3 3 2 AND SENSITIVITY ANALYSIS CHAPTER 6 DUALITY THEORY and x1 and 0 x   0. [y , y , y ]  [0, 0, 0]. 2 3 the parameters for a 1single constraint. In addition, the Right Side column gives the rig hand sides for the primal problem and the objective function coefficients for the d ■ TABLE 6.2 Primal-dual for linear illustrated by the Wyndor problem,table whereas the programming, bottom row gives the objective function coefficients for the prim Glass Co. example problem and the right-hand sides for the dual problem. (a) General Case Consequently, we now have the following general relationships between the prim and dual problems. Primal Problem Coefficient of: 1. The parameters for a (functional) constraint in either problem are the coefficients o Right variablex1in thexother problem. … xn Side 2 2. The coefficients in the…objective function of either problem are the right-hand sides y1 a11 a12 a1n  b1 the … y2 other a21problem. a22 a2n  b2   Coefficients for Objective Function (Minimize) Coefficient of: 2 … Thus, is a direct between these entities in the two problems, as su ym theream1 am2 correspondence amn  bm marized in VITable 6.3. These are a key to some of the applications of … correspondences VI VI … ality theory,c1 including sensitivity c2 cn analysis. The WorkedCoefficients Examples for section of the book’s website provides another example Objectivetable Function using the primal-dual to construct the dual problem for a linear programming mod Right Side Dual Problem 198 (Maximize) (b) Wyndor Glass Co. Example of Origin x1 x2 1 0 3 0 2 2 the Dual Problem Duality theory is based directly on the fundamental insight (particularly with regard  presented 4 row 0) in Sec. 5.3. To see why, we continue to use the notation introduced  12 Table5.9 for row 0 of the final tableau, except for replacing Z* by W* and dropping 18 asterisks from z* and y* when referring to any tableau. Thus, at any given iteration of VI VI 3 5 simplex method for the primal problem, the current numbers in row 0 are denoted shown in the (partial) tableau given in Table 6.4. For the coefficients of x1, x2, . . . , recall that z  (z1, z2, . . . , zn) denotes the vector that the simplex method added to Figure 4.2: Primal-dual table programming, by the Co. tableau. (Do vectorfor of linear initial coefficients, in the process of Wyndor reaching Glass the current c,illustrated example confuse z with the value of the objective function Z.) Similarly, since the initial coe cients of xn1, xn2, . . . , xnm in row 0 all are 0, y  (y1, y2, . . . , ym) denotes the v Consequently, we now the simplex following general the primal and [see Eq. (1) in torhave that the method hasrelationships added to thesebetween coefficients. Also recall dual problems. statement of the fundamental insight in Sec. 5.3] that the fundamental insight led to following relationships between these quantities and the parameters of the original mod • The parameters for a (functional) constraint in either problem are the coefficients of a m variable in the other problem. W  yb   bi yi , i1 • The coefficients in the objective function of either problem are the right-hand sides for m the other problem. for j  1, 2, . . . , n. z  yA, so zj   aij yi , i1 Thus, there is a direct correspondence between these entities in the two problems, as summarized in Fig. 4.3.■ These are a between key to some of the applications of TABLEcorrespondences 6.3 Correspondence entities in primal and duality theory, including sensitivity analysis. y1 y2 y3 dual problems One Problem Other Problem Constraint i ←→ Variable i Objective function ←→ Right-hand sides ■ TABLE 6.4between Notation for entries in row of a simplex Figure 4.3: Correspondence entities in primal and0 dual problemstableau Coefficient of: Iteration Basic Variable Eq. Z x1 x2 … xn xn1 xn2 … xnm Any Z (0) 1 z1  c1 z2  c2 … zn  cn y1 y2 … ym 4 Rig Si W variable in the other problem. Thus, there is aobjective direct correspondence between these in the twosides problems, 2. The coefficients in the function of either problem areentities the right-hand for as summarized in Table 6.3. These correspondences are a key to some of the applications of duthe other problem. ality theory, including sensitivity analysis. Thus, there is a direct correspondence the twoprovides problems,another as sum-example of The Worked Examplesbetween section these of theentities book’sinwebsite marized in Table 6.3. These correspondences are a key to some of the applications of duusing the primal-dual table to construct the dual problem for a linear programming model. ality theory, including sensitivity analysis. Origin of the Dual Problem The Worked Examples of the book’s website provides another example of Origin of the section Dual Problem using thetheory primal-dual tabledirectly to construct thefundamental dual probleminsight for a linear programming Duality is based on the (particularly withmodel. regard to Duality theory is based directly on the fundamental insight (particularly with regard to row 0) presented in Sec. 3 of Chapter 3. To see why, we continue to use the notation introrow presented in Sec. 5.3. To see why, we continue to ∗use the ∗notation introduced in Origin the0) Dual Problem duced in Fig.of3.11 for row 0 of the except for replacing Z by dropping Table 5.9 for row 0 offinal the tableau, final tableau, except for replacing Z*Wby and W* and dropping the ∗ ∗ the Duality asterisks from z and y when referring to any tableau. Thus, at any given iteration theory is based directly on the fundamental insight (particularly with regard to asterisks from z* and y* when referring to any tableau. Thus, at any given iteration of the of the method for the primal problem, the current numbers in row 0 are0 denoted rowsimplex 0) presented in method Sec. 5.3.for Tothe seeprimal why, we continue use thenumbers notation introduced in denoted as simplex problem, thetocurrent in row are as shown in the (partial) tableau given in Fig. 4.4. For the coefficients of x , x , · · · , xn , Table 5.9 for row 0inofthe the(partial) final tableau, replacing and dropping 1 shown tableauexcept given for in Table 6.4. Z* Forby theW* coefficients of2 xthe 1, x2, . . . , xn, recall that zfrom = (z z2 , ·y* z(z vector thatThus, the simplex method added to the asterisks z*1 ,and referringthe to any tableau. at any given iteration of the n ) , denotes recall that z· ·,when 1 z2, . . . , zn) denotes the vector that the simplex method added to the vector of initial coefficients, −c, in the process of reaching tableau. (Do simplex method for primal problem, numbers incurrent row 0the are denoted as not(Do not vector of the initial coefficients, c, incurrent the process ofthe reaching current tableau. the confuse z with the value of the objective function Z.) Similarly, since the initial coefficients shown in the (partial) tableau 6.4. For function the coefficients of x1, xsince . , xinitial 2, . . the n, confuse z with the given value in of Table the objective Z.) Similarly, coeffiof xrecall · · ,(z xn+m in row 0 all are 0, y = (y , y , · · · , y ) denotes the vector that thethe vec, z , . . . , z ) denotes the vector that the simplex method added to the n+1 , xthat n+2 , ·zcients 1 2 m 1of 2x n, . . . , x , x in row 0 all are 0, y  (y , y , . . . , y ) denotes n1 n2 nm 1 2 m simplex added to these recall [the seecurrent Eq. Also (1) in the(Do statement vectormethod of initial coefficients, c, incoefficients. thehas process oftoreaching tableau. not method torhas that the simplex addedAlso these coefficients. recall [see Eq. (1) in the of the fundamental insight in Sec. 3 of Chapter 3] that the fundamental insight led to the confuse z with the value of the objective function Z.) Similarly, since the initial coeffistatement of the fundamental insight in Sec. 5.3] that the fundamental insight led to the following between these quantities and the parameters of the original model: cients relationships of xn1 , x , . . . , x in row 0 all are 0, y  (y , y , . . . , y ) denotes the vecn2 nm 1 2 m following relationships between these quantities and the parameters of the original model: tor that the simplex method has added to these coefficients. Also recall [see Eq. (1) in the m statement of theWfundamental  yb   insight bi yi , in Sec. 5.3] that the fundamental insight led to the following relationships between i1 these quantities and the parameters of the original model: m m z byA, W  yb   i yi , zj   aij yi , so for j  1, 2, . . . , n. i1 i1 m ■ TABLE 6.3 Correspondence between   awith yi ,primal j  1, example, 2, . . . , n.the first equation gives W = z  yA, these so relationships zj entities ijin and To illustrate the for Wyndor i1 4y1 +12y2 +18y3 , which is justdual the problems objective function for the dual problem shown in the upper ■ TABLE 6.3 Correspondence between right-hand boxOne of Fig. 4.1. The second set of equations give z1 = y1 + 3y3 and z2 = 2y2 + 2y3 , Problem entities in primal andOther Problem which are the left-hand sides of the functional constraints for this dual problem. Thus, by dual problems Constraint i ←→ Variable i subtracting the right-hand sides of these ≥ constraints (c1 = 3 and c2 = 5), (z1 − c1 ) and Objective function ←→ Right-hand sides One Problem (z2 − c2 )Problem can be interpreted Other as being the surplus variables for these functional constraints. The remaining key is to express what the simplex method tries to accomplish (according Constraint i ←→ Variable i to the optimality test) in6.4 terms of thesefor symbols. Specifically, it seeks a set of basic variables, Objective function ←→ Right-hand ■ TABLE Notation sides entries in row 0 of a simplex tableau and the corresponding BF solution, such that all coefficients in row 0 are nonnegative. Coefficient of: It then stops with this optimal solution. Using the notation in Fig. 4.4: Basic Right ■ TABLE 6.4 Notation for entries tableau … … Iteration Variable Eq. in Z rowx10 of a xsimplex x x x x Side 2 n n1 n2 nm Z (0) 1 of: z  c z1  c1Coefficient z2  c2 …...
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