Duality & The Dual Simplex Method & Sensitivity Analysis for Linear Programming. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1

Transcription

1 Dualit & The Dual Simple Method & Sensitivit Analsis for Linear Programming Metodos Cuantitativos M. En C. Eduardo Bustos Farias

2 Dualit EverLP problem has a twin problem associated with it. One problem is called primal, while the other is called dual. These problems possess ver closel related properties so that the optimal solution of one problem can ield complete information about the optimal solution of the other problem! Metodos Cuantitativos M. En C. Eduardo Bustos Farias

3 Dualit In certain cases, these relationships ma prove useful in reducing the computational effort associated with solving LP problems Metodos Cuantitativos M. En C. Eduardo Bustos Farias 3

4 Dualit Ma Z n Primal c j j j= n st.. a b, i =,,..,m j= = ij j j i 0, j=,,...,n MinZ st.. m i= = a ij m i= i Dual b c i i i j, 0,i j =,,..,n =,,..., m Metodos Cuantitativos M. En C. Eduardo Bustos Farias 4

5 Construction of the Dual The dual problem is constructed from the primal problem as follows: Each constraint in one problem corresponds to a variable in the other problem The elements of the r.h.s. of the constraints in one problem are equal to the respective coefficients of the objective function in the other problem Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5

6 Construction of the Dual The maimization problem has (<) constraints, and the minimization problem has (>) constraints The variables in both problems are nonnegative (> 0) From smmetr of the two problems, the dual of the dual is the primal Metodos Cuantitativos M. En C. Eduardo Bustos Farias 6

7 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 7 Primal Eample 0, : 30 : 0 5 : 45 3 : = s t Z Ma 0,,, : : Dual Variables Dual = s t Z Min Primal Variables

8 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 8 Eample : 30 : 0 5 : 45 3 : s s s s s t Z Ma = = = = = Dual Variables Primal in Standard Form Dual in Standard Form 0,,,,, 4 3 s s s s : : Z ' A s A s t s MA MA Ma = = = 0,,,,,,, 4 3 A s A s Primal Variables

9 Observations Notice in the preceding eample, the dual problem has fewer constraints than the primal, and ma be computationall more efficient to solve than the primal Computational difficult in LP s is mainl associated with the number of constraints rather than the number of variables Metodos Cuantitativos M. En C. Eduardo Bustos Farias 9

10 Primal-Dual Relationships The optimal solution of the primal (dual) gives, directl the optimal solution of the dual (primal) The following two rules are used to determine the optimal solution of one problem from the optimal (simple) solution of the other problem Metodos Cuantitativos M. En C. Eduardo Bustos Farias 0

11 Rule If the dual variable corresponds to a slack starting variable in the primal problem, its optimum value is given directl b the coefficient of this slack variable in the optimal Z-equation row Metodos Cuantitativos M. En C. Eduardo Bustos Farias

12 Eample Primal in Standard Form Ma Z = 5 6 st.. 9 s = 60 : 3 s = 45 : 5 s = 0 : 3 3 s = 30 : 4 4,, s, s, s, s 3 4 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 0 Dual Variables

13 Simple Solution of the Primal Z s s 4 Dual variables 3 4 Z s s s s b Metodos Cuantitativos M. En C. Eduardo Bustos Farias 3

14 Rule If the dual variable corresponds to an artificial starting variable in the primal problem, its optimal value is given b the coefficient of this artificial variable in the optimal Z-equation row of the simple tableau, ignoring an Big M term Metodos Cuantitativos M. En C. Eduardo Bustos Farias 4

15 Eample ' Ma Z = MA MA Dual in Standard Form 3 4 st.. 5 s A = 5 : s A = 6 : 3 4,,,, s, A, s, A Primal Variables Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5

16 Simple Solution of the Dual Primal Variables Z' s A s A b Z' M M Metodos Cuantitativos M. En C. Eduardo Bustos Farias 6

17 Sensitivit Analsis Coefficients in an LP formulation are assumed to be known with absolute certaint! Sensitivit analsis provides an approach for studing the effect upon the optimal solution of variations in the cost/profit coefficients in the objective function, or in the resource availabilities (r.h.s. s) of the constraints. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 7

18 General Form of a Linear Programming (LP) Problem MAX (or MIN): c X c X c n X n Subject to: a X a X a n X n <= b : a k X a k X a kn X n <= b k : a m X a m X a mn X n = b m How sensitive is a solution to changes in the c i, a ij, and b i? Metodos Cuantitativos M. En C. Eduardo Bustos Farias 8

19 Sensitivit Problem A Change in objective function coefficients The goal of this sensitivit analsis is to determine the range of variation in each of the objective function coefficients that will keep the current optimum corner point of the feasible solution space unchanged. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 9

20 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 0 Blending Problem Eample 0 0, s.t. 4 3 Ma = Z Consider the following eample to illustrate sensitivit problem

21 Eample - Blending Problem Optimal Solution Ma Z = 38.5 =30.77, =.54 Ma Z s.t = , Metodos Cuantitativos M. En C. Eduardo Bustos Farias

22 Eample In the previous eample, if p is held constant at 4, we can determine the permissible range values on p so that the current solution remains optimal as follows: 3 p 5 4 p 5 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5 4 5

23 Eample Similarl, if p is held constant at 3, then or so p 3 3 p p Metodos Cuantitativos M. En C. Eduardo Bustos Farias 3

24 Sensitivit Problem How much is a resource unit worth? The goal of this analsis is to stud the sensitivit of the optimum solution to changes in the right hand sides of the constraints, i.e., the b i s Metodos Cuantitativos M. En C. Eduardo Bustos Farias 4

25 Sensitivit Problem This amounts to changing the availabilit of a resource. The results are given as predetermined ranges of the r.h.s. within which the optimum objective value will increase or decrease at a given, constant rate. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5

26 Eample - Blending Problem Suppose the amount of Crude B is varied from the 50 units available. This is reflected b constraint inequalit Increasing or decreasing the availabilit of Crude B has the effect of moving the constraint line parallel to itself We wish to determine the range of r.h.s. values so that the optimal solution is determined b the intersection of constraints and Metodos Cuantitativos M. En C. Eduardo Bustos Farias 6

27 Eample - Blending Problem Thus, the acceptable range on Crude B is: 0 Crude B 50 If the availabilit of Crude B were increased b 50 units (to a total of 00 units), then profit would increase b: 50(. 0654) = 3077., or a total of Z = = These values are called dual prices or shadow prices and are defined as the worth per unit of resource values. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 7

28 Shadow Prices Goods (resources) having positive shadow prices are called scarce goods Goods (resources) having zero shadow prices are called free goods Metodos Cuantitativos M. En C. Eduardo Bustos Farias 8

29 Blending Eample Primal in Standard Form Ma Z = 3 4 MA MA st s = 00 : 3 5 s = 50 : 5 4 s A = 00 : s A = 80 : 4 4 4,, s, s, s, A, s, A Metodos Cuantitativos M. En C. Eduardo Bustos Farias 9 0 Dual Variables

30 Simple Solution of the Primal 3 4 Z s s s A s A b Z M 0 M s s Shadow Prices (ignore M s) Metodos Cuantitativos M. En C. Eduardo Bustos Farias 30

31 Blending Eample ' Ma Z = MA MA Dual in Standard Form 3 4 st s A = 3 : s A = 4 : 3 4,,,, s, A, s, A Primal Variables Metodos Cuantitativos M. En C. Eduardo Bustos Farias 3

32 Simple Solution of the Dual Z' s A s A b Z' M M Metodos Cuantitativos M. En C. Eduardo Bustos Farias 3

33 Sensitivit Analsis in Simple MAXIMUM CHANGE IN RESOURCE AVAILABILITY A change in the resource availabilities (b i s) will onl change the r.h.s. of the tableau (b column), which means that such a change can onl affect the feasibilit of the solution. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 33

34 Sensitivit Analsis in Simple CHANGES IN MARGINAL COST/PROFIT Changes in the coefficients of the objective function will affect onl the objective function equation (Zequation row) in the optimal tableau. Such changes can affect onl the optimalit of the current solution. Goal of sensitivit analsis is to determine the range of variation for the objective coefficients (one at a time) for which the current optimum solution remains unchanged! Metodos Cuantitativos M. En C. Eduardo Bustos Farias 34

35 Approaches to Sensitivit Analsis Change the data and re-solve the model! Sometimes this is the onl practical approach. Solver also produces sensitivit reports that can answer various questions When solving LP problems, be sure to select the Assume Linear Model option in the Solver Options dialog bo. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 35

36 The Wndor Glass Problem Ma Z = 3 5 st.. 3 8, Metodos Cuantitativos M. En C. Eduardo Bustos Farias 36

37 Wndor Glass EXCEL Solution Metodos Cuantitativos M. En C. Eduardo Bustos Farias 37

38 Solver creates Worksheet tabs when these options are clicked Metodos Cuantitativos M. En C. Eduardo Bustos Farias 38

39 Wndor Glass Answer Report Metodos Cuantitativos M. En C. Eduardo Bustos Farias 39

40 Wndor Glass Sensitivit Analsis Metodos Cuantitativos M. En C. Eduardo Bustos Farias 40

41 Solver s Sensitivit Report Answers questions about: Amounts b which objective function coefficients can change without changing the optimal solution. The impact on the optimal objective function value of changes in constrained resources. The impact on the optimal objective function value of forced changes in decision variables. The impact changes in constraint coefficients will have on the optimal solution. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 4

42 Blue Ridge Hot Tubs Eample... MAX: 350X 300X S.T.: X X <= 00 9X 6X <= 566 X 6X <= 880 X, X >= 0 } profit } pumps } labor } tubing } non-negativit Metodos Cuantitativos M. En C. Eduardo Bustos Farias 4

43 Answer Report Microsoft Ecel 9.0 Answer Report Worksheet: [Solver Class Eamples Solved.ls]Blue Ridge Report Created: 7/7/005 :56:0 PM Target Cell (Ma) Cell Name Original Value Final Value $D$6 Unit Profits Total Profit $66,00 $66,00 Adjustable Cells Cell Name Original Value Final Value $B$5 Number to Make Aqua-Spas $C$5 Number to Make Hdro-Lues Constraints Cell Name Cell Value Formula Status Slack $D$0 - Labor Req'd Used 566 $D$0<=$E$0 Binding 0 $D$ - Tubing Req'd Used 7 $D$<=$E$ Not Binding 68 $D$9 - Pumps Req'd Used 00 $D$9<=$E$9 Binding 0 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 43

44 Sensitivit Report Microsoft Ecel 9.0 Sensitivit Report Worksheet: [Solver Class Eamples Solved.ls]Blue Ridge Report Created: 7/7/005 :56:0 PM Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Number to Make Aqua-Spas $C$5 Number to Make Hdro-Lues Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $D$0 - Labor Req'd Used $D$ - Tubing Req'd Used E30 68 $D$9 - Pumps Req'd Used Metodos Cuantitativos M. En C. Eduardo Bustos Farias 44

45 Changes in Objective Function Coefficients Values in the Allowable Increase & Allowable Decrease columns for the Changing Cells indicate the amounts b which an objective function coefficient can change without changing the optimal solution, assuming all other coefficients remain constant. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 45

46 Alternate Optimal Solutions Values of zero (0) in the Allowable Increase or Allowable Decrease columns for the Changing Cells indicate that an alternate optimal solution eists. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 46

47 Changes in Constraint RHS Values The shadow price of a constraint indicates the amount b which the objective function value changes given a unit increase in the RHS value of the constraint, assuming all other coefficients remain constant. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 47

48 Changes in Constraint RHS Values Shadow prices hold onl within RHS changes falling within the values in Allowable Increase and Allowable Decrease columns. Shadow prices for nonbinding constraints are alwas zero. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 48

49 Comments About Changes in Constraint RHS Values Shadow prices onl indicate the changes that occur in the objective function value as RHS values change. Changing a RHS value for a binding constraint also changes the feasible region and the optimal solution (see graph on following slide). To find the optimal solution after changing a binding RHS value, ou must re-solve the problem. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 49

50 Other Uses of Shadow Prices Suppose a new Hot Tub (the Tphoon-Lagoon) is being considered. It generates a marginal profit of $30 and requires: pump (shadow price = $00) 8 hours of labor (shadow price = $6.67) 3 feet of tubing (shadow price = $0) Q: Would it be profitable to produce an? A: $30 - $00* - $6.67*8 - $0*3 = -$3.33 = No! Metodos Cuantitativos M. En C. Eduardo Bustos Farias 50

51 Analzing Changes in Constraint Coefficients Q: Suppose a Tphoon-Lagoon required onl 7 labor hours rather than 8. Is it now profitable to produce an? A: $30 - $00* - $6.67*7 - $0*3 = $3.3 = Yes! Q: What is the maimum amount of labor Tphoon- Lagoons could require and still be profitable? A: We need $30 - $00* - $6.67*L 3 - $0*3 >=0 The above is true if L 3 <= $0/$6.67 = $7.0 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5

52 Ke Points - I The shadow prices of resources equate the marginal value of the resources consumed with the marginal benefit of the goods being produced. Resources in ecess suppl have a shadow price (or marginal value) of zero. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5

53 Ke Points-II The reduced cost of a product is the difference between its marginal profit and the marginal value of the resources it consumes. Products whose marginal profits are less than the marginal value of the goods required for their production will NOT be produced in an optimal solution. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 53

54 Formal definitions Metodos Cuantitativos M. En C. Eduardo Bustos Farias 54

55 Dualit via the Lagrangian Primal Problem P minimize z = c subject to A = b Lagrangian Problem D(π) 0 minimize v(π)= c - π(a b) subject to 0 optimum value is z* optimum value is v*(π) Note: if * is feasible for P, then it is also feasible for D(π). Thus v*(π) z*, and D(π) provides a lower bound for P. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 55

56 Computing the Highest Lower Bound Lagrangian Problem D(π) minimize subject to 0 v(π)= c - π(a b) = (c-πa) πb v(π)= - unless (c-πa) 0. v(π)= πb if (c-πa) 0 The highest lower bound is found b solving maimize v = πb subject to c-πa 0 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 56

57 Primal Problem P minimize z = c subject to A = b 0 Dual Problem D maimize v = πb subject to πa c optimum value is z* optimum value is v* Theorem. (Strong Dualit) If both P and D are feasible, then z* = v*. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 57

58 PRIMAL PROBLEM: maimize z = subject to 3 4 = = 3,, 3, 4 0 DUAL PROBLEM: minimize 3 Subject to Observation. The constraint matri in the primal is the transpose of the constraint matri in the dual. Observation. The RHS coefficients in the primal become the cost coefficients in the dual. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 58

59 PRIMAL PROBLEM: maimize z = subject to 3 4 = = 3,, 3, 4 0 DUAL PROBLEM: minimize 3 Subject to Observation 3. The cost coefficients in the primal become the RHS coefficients in the dual. Observation 4. The primal (in this case) is a ma problem with equalit constraints and non-negative variables The dual (in this case) is a minimization problem with constraints and variables unconstrained in sign. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 59

60 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 60 LP Model: Standard (Inequalit) Form s.t. ma,,, b a a a b a a a b a a a c c c n m n mn m m n n n n n n

61 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 6 Dualit Theor Standard (Inequalit) Primal Form: Dual Form: 0, 0, 0,.. min 0, 0, 0,.. ma c a a a c a a a c a a a b b b b a a a b a a a b a a a c c c m n m mn n n m m m m m n n m n mn m m n n n n n n t s t s

62 LP Model: Standard (Inequalit) Matri Form A = a a... a m a a a... m a a... a n n mn, b = b b... b m, c T = c c... c n Matri Form : ma s.t. b, 0. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 6 c A

63 Primal-Dual in Matri Form Standard (Inequalit) Primal Form: Ma c s.t. A b 0 Dual Form: Min b s.t. A c 0 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 63

64 Primal-Dual in Matri Form: Equalit Standard (Equalit) Primal Form: Min c s.t. A = b 0 Dual Form: Ma b s.t. A c Metodos Cuantitativos M. En C. Eduardo Bustos Farias 64

65 Relations Between Primal and Dual. The dual of the dual problem is again the primal problem.. Either of the two problems has an optimal solution if and onl if the other does; if one problem is feasible but unbounded, then the other is infeasible; if one is infeasible, then the other is either infeasible or feasible/unbounded. 3. Weak Dualit Theorem: The objective function value of the primal (dual) to be maimized evaluated at an primal (dual) feasible solution cannot eceed the dual (primal) objective function value evaluated at a dual (primal) feasible solution. c T >= b T (in the standard equalit form) Metodos Cuantitativos M. En C. Eduardo Bustos Farias 65

66 Relations between Primal and Dual (continued) 4. Strong Dualit Theorem: When there is an optimal solution, the optimal objective value of the primal is the same as the optimal objective value of the dual. c T * = b T * 5. Complementar Slackness Theorem: Consider an inequalit constraint in an LP problem. If that constraint is inactive for an optimal solution to the problem, the corresponding dual variable will be zero in an optimal solution to the dual of that problem. * j (c-a T *) j = 0, j=,,n. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 66

67 Optimalit Conditions Primal Feasibilit: A = b 0 Dual Feasibilit: A T c Strong Dualit: c T = b T or Complementar Slackness: (c-a ),...,n T Metodos Cuantitativos j j M. En C. Eduardo Bustos Farias 67 = 0, j =

68 Optimalit Conditions Primal Feasibilit: A = b 0 Dual Feasibilit: A T r = c r 0 Strong Dualit: c T = b T or Complementar Slackness: r =,...,n j j Metodos Cuantitativos M. En C. Eduardo Bustos Farias 68 0, j =

69 Metodos Cuantitativos M. En C. Eduardo Bustos Farias 69 Complementar Slackness Conditions in the Primal Simple Method The simple method maintain the complementar slackness condition, and moving toward B c c A c r N j B, j r B B c c B c r B c b B B N N N N j j B B B B B N B 0 or 0 Thus, 0, 0, = = = = = = = = =

70 Matri form of the initial tableau for the inequalit standard form Basic Variable s RHS Z -c 0 0 s A I b Matri form of the tableau with a selected basis B: Basic Variable s RHS Z c B B - A -c c B B - c B B - b B B - A B - B - b Metodos Cuantitativos M. En C. Eduardo Bustos Farias 70

71 The Primal Simple Method in Tableau. Initialization: A:=(A, I) and c:=(c, 0); B =B - b 0.. Calculate =c B B - and r = A - c 0. If r 0, then optimal and stop; otherwise, goto net step. Determine the entering basic variable: sa select the basic variable with the lowest value in r ; determine the leaving basic variable: whose coefficient in B reaches zero first as the entering variable increases (use the ratio test of the entering column of B - A against B - b). If the increase is unlimited (the column contains all non-positive numbers), then stop, the primal problem is unbounded. Otherwise, using the pivoting procedure to update B, B - A and B - b and Metodos return Cuantitativos to Step. M. En C. Eduardo Bustos Farias 7

72 The Dual Simple Method in Tableau. Initialization: A:=(A, I) and c:=(c, 0); =c B B - such that r = A - c 0. Calculate B =B - b. If B 0, then optimal and stop; otherwise, goto net step. Determine the leaving basic variable: sa select the basic variable with the lowest value in B ; determine the entering basic variable: whose coefficient in r reaches zero first as the dual variable of the leaving row increases (use the ratio test of the leaving row of B - A against r). If the increase is unlimited (the row contains all non-negative numbers), then stop, the primal problem is infeasible or dual is unbounded. Otherwise, using the pivoting procedure to update B, B - A, =c B B - and r = A c, and goto Step. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 7

73 Reduced Cost and Objective Coefficient Range All positive variables have zero reduced cost In general, the reduced cost of an zero variable is the amount the objective coefficient of that variable would have to change, with all other data held fied, in order for it to become a positive variable in the OS. If the OS is degenerate, the objective coefficient of a zero variable would have to change b at least, and possibl more that, the reduced cost in order to become a positive variable in the OS. The objective coefficient ranges give the ranges of the objective function over which no change in the OS will occur. If the OS is degenerate, an objective coefficient must be changed b at least, and possibl more than, the indicated allowable amounts in order to produce a new optimal solution. One of the allowable increase and decrease for a zero variable is infinite and the other is the reduced cost. If a zero variable has zero reduced cost, then there eist an alternative optimal solution. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 73

74 Dual (Shadow) Price and Constraint RHS Ranges All inactive constraint have zero dual price In general, the dual price on a given active constraint is the rate of improvement in the OV as the RHS of the constraint increases with all other data held fied. If the RHS is decreased, it is the rate at which the OV is impaired. The constraint RHS ranges give the ranges of the constraint RHS over which no change in the dual price will occur. If the solution is degenerate, the dual price ma be valid for one-sided changes in the RHS. One of the allowable increase and decrease for an inactive constraint is infinite and the other equals to the slack or surplus. In general, when the RHS of an active constraint changes, both the OV and OS will change Metodos Cuantitativos M. En C. Eduardo Bustos Farias 74