ECE 307- Techniques for Engineering Decisions

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1 ECE 307- echniques for Engineering Decisions Dualit Concepts in Linear Programming George Gross Department of Electrical and Computer Engineering Universit of Illinois at Urbana-Champaign ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 1

2 DUALIY Definition: A LP is in smmetric form if all the variables are restricted to be nonnegative and all the constraints are inequalities of the tpe: objective tpe ma corresponding inequalit tpe min ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 2

3 DUALIY DEFINIIONS We define the primal problem as ma Z = c s.t. A b (P) 0 he dual problem is therefore min W = b s.t. A c (D) 0 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 3

4 DUALIY DEFINIIONS he problems (P) and (D) are called the smmetric dual LP problems s.t. ma Z = c + c c a + a a b n n 1 a + a a b n n 2 a + a a b m1 1 m2 2 mn n m 0, 0,..., n n n ( P ) ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 4

5 DUALIY DEFINIIONS s.t. min W = b + b b a + a a c m 1 m 1 a + a a c m2 m 2 a + a a c 1n 1 2n 2 mn m n 0, 0,..., m m m D ( ) ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 5

6 EXAMPLE 1: MANUFACURING RANSPORAION PROBLEM shipment cost coefficients R 1 retail stores warehouses W 1 R 1 R 2 R 3 W R 2 W W 2 R 3 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 6

7 EXAMPLE 1: MANUFACURING RANSPORAION PROBLEM We are given that the supplies needed at Warehouse. W and W are 1 2 W W We are also specified the demands needed at retail stores R, R, and R as R R R ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 7

8 EXAMPLE 1: MANUFACURING RANSPORAION PROBLEM he problem is to determine the least-cost shipping schedule We define the decision variable ij quantit shipped from W1 to Rj i = 12, j =1,2,3 c ij he shipping costs ma be viewed as element i,j of the transportation cost matri ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 8

12 INERPREAION OF HE DUAL PROBLEM he moving compan proposes to the manufacturer to: bu all the 300 units at W1 at 1 / unit bu all the 600 units at W2 at 2 / unit sell all the 200 units at R1 at 3 / unit sell all the 300 units at R2 at 4 / unit sell all the 400 units at R at / unit 3 5 o convince the manufacturer to get the business, the mover ensures that the deliver is for less than the transportation costs the manufacturer would incur (the dual constraints) ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 12

13 INERPREAION OF HE DUAL PROBLEM + c = c = c = c = c = c = he mover wishes to maimize profits, i.e., revenues costs dual cost objective function maw = ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 13

14 EXAMPLE 2: FURNIURE PRODUCS Resource requirements item desks tables chairs sales price ($) lumber board requirements finishing labor carpentr ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 14

15 EXAMPLE 2: FURNIURE PRODUCS he Dakota Furniture Compan manufacturing: resource desk table chair available lumber board (ft) finishing (h) carpentr (h) We assume that the demand for desks, tables and chairs is unlimited and the available resources are alread purchased he decision problem is to maimize total revenues ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 15

16 PRIMAL AND DUAL PROBLEM FORMULAION We define decision variables 1 = number of desks produced 2 = number of tables produced 3 = number of chairs produced he Dakota problem is ma Z = s.t lumber finishing carpentr,, ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 16

19 INERPREAION OF HE DUAL PROBLEM An entrepreneur wishes to purchase all of Dakota s resources He thus needs to determine the price to pa for each unit of each resource = price paid for 1 lumber board ft 1 = price paid for 1 h of finishing 2 = price paid for 1 h of carpentr 3 We solve the Dakota dual problem to determine 1, 2, and 3 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 19

20 o induce Dakota to sell the raw resources, the resource prices must be set sufficientl high For eample, the entrepreneur must offer Dakota at least $ 60 for a combination of resources that includes 8 ft of lumber board, 4 h of finishing and 2 h of carpentr since Dakota could use this combination to sell a desk for $ 60: this consideration implies the construction of the dual constraint INERPREAION OF HE DUAL PROBLEM ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 20

21 INERPREAION OF DUAL PROBLEM In the same wa we obtain the two additional constraints for a table and for a chair he i th primal variable corresponds to the i th constraint in the dual problem statement he j th dual variable corresponds to the j th constraint in the primal problem statement ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 21

22 EXAMPLE 3: DIE PROBLEM A new diet requires that all food eaten come from one of the four basic food groups : chocolate cake, ice cream, soda and cheesecake he four foods available for consumption are as given in the table Minimum requirements for each da are: 500 cal 6 oz chocolate 10 oz sugar 8 oz fat ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 22

23 EXAMPLE 3: DIE PROBLEM food calories chocolate (oz) sugar (oz) fat (oz) costs (cents) brownie chocolate ice cream (scoop) cola (bottle) pineapple cheesecake (piece) ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 23

24 PROBLEM FORMULAION Objective of the problem is to minimize the costs of the diet Decision variables are defined for each da s purchases = number of brownies = number of chocolate ice cream scoops = number of bottles of soda = number of pineapple cheesecake pieces ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 24

27 INERPREAION OF HE DUAL We consider a sales person of nutrients who is interested in assuming that each dieter meets dail requirements b purchasing calories, sugar, fat and chocolate he ke decision is to determine the prices i = price per unit to sell to dieters Objective of sales person is to set the prices so i as to maimize revenues from selling to the dieter the dail ration of required nutrients ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 27

28 INERPREAION OF DUAL Now, the dieter can purchase a brownie for 50 and have 400 cal, 30 oz of chocolate, 2 oz of sugar and 2 oz of fat Salesperson must set i sufficientl low to entice the buer to get the required nutrients from the brownie: brownie constraint We derive similar constraints for the ice cream, the soda and the cheesecake ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 28

31 COROLLARY 1 OF HE WEAK DUALIY HEOREM feasible for P c b ( ) for an feasible for D ( ) * c b = minw for an feasible for P, ( ) c minw ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 31

32 COROLLARY 2 OF HE WEAK DUALIY HEOREM feasible for D c b ( ) for ever feasible for P ( ) * * ma Z = ma c = c < b for an feasible of D, ( ) b ma Z ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 32

33 COROLLARIES 3 AND 4 OF HE WEAK DUALIY HEOREM If ( P ) is feasible and ma Z is unbounded, i.e., Z + ; then, ( D ) has no feasible solution If ( D ) is feasible and min Z is unbounded, i.e., Z ; then, ( P ) is infeasible ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 33

35 DUALIY HEOREM APPLICAION he corresponding dual is given b min W = b s.t. A c (D) 0 With the appropriate substitutions, we have ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 35

37 GENERALIZED FORM OF HE DUAL Consider the primal decision = 1, i = 1, 2, 3,4 ; decision is feasible for (P) with i Z = c = 10 he dual decision i = 1, i = 1, 2 is feasible for (D) with W = b = ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved

38 DUALIY HEOREM APPLICAION Clearl, ( ) ( ) < 40 = 1 2 Z,,, = W, and so clearl, the feasible decision for (P) and (D) satisf the Weak Dualit heorem Moreover, we have corollar corollar ( * *) min W = W, ( * * * *) maz=z,,, b = 40 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 38

39 CORROLARIES 5 AND 6 (P) is feasible and (D) is infeasible, then, (P) is unbounded (D) is feasible and (P) is infeasible, then, (D) is unbounded ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 39

40 EXAMPLE Consider the primal dual problems: ma Z = + st.. Now ,, 0 min W = 2 + st ( P ) ( D) = 0 is feasible for ( P) ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved , 0

41 EXAMPLE but is impossible for (D) since it is inconsistent with Since (D) infeasible, it follows from Corollar 5 that , 2 0 You should be able to show this result b solving (P) using the simple scheme Z ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 41

42 OPIMALIY CRIERION HEOREM We consider the primal-dual problems (P) and (D) with 0 is feasible for ( P) 0 is optimal for ( P) 0 is feasible for ( D) and c = b is optimal for ( D) We net provide the proof: 0 0 is feasible for is feasible for ( P) ( D) Weak Dualit heorem c b 0 0 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 42

43 OPIMALIY CRIERION HEOREM but we are given that and so it follows that 0 and therefore is optimal ; similarl so it follows that 0 0 c = b feasible, feasible c b = c feasible, feasible b c = b is optimal ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 43

44 MAIN DUALIY HEOREM (P) is feasible and (D) is feasible; then, ( ) * feasible for P which is optimal and ( ) * feasible for D which is optimal such that c = b * * ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 44

45 COMPLEMENARY SLACKNESS CONDIIONS and * * if and onl if are optimal for (P) and (D) respectivel, ( ) * * * ( *) 0 = A c + b A = b c * * We prove this equivalence result b defining the m n slack variables u and v such that and are feasible; at the optimum, + =, * * * * A u b u 0 =, * * * * A v c v 0 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 45

46 COMPLEMENARY SLACKNESS CONDIIONS where the optimal values of the slack variables u and v * * correspond to the optimal values Now, and * * A + u = b = b * * * * * * A v = c = c * A* * * * * * * ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 46

47 COMPLEMENARY SLACKNESS CONDIIONS his implies that * * * * * * u + v = b c We need to prove optimalit which is true if and onl if * * * * u + v = 0 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 47

48 COMPLEMENARY SLACKNESS CONDIIONS However,, * * are optimal Main Dualit heorem * * * * * * c = b u + v = 0 Also, u + v = 0 b = c * * * * * * Optimalit Criterion heorem ( ) ( ) * * is optimal for P and is optimal for D ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 48

49 COMPLEMENARY SLACKNESS CONDIIONS Note that * * * *,, u, v > 0 component - wise each element 0 * * * * u + v = 0 u = 0 i = 1,..., m and * * i i * * v j j = 0 j = 1,..., n At the optimum, n * * i bi ai j j = 0 i = 1,..., m j = 1 and m * * j aji i cj = 0 j = 1,..., n i = 1 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 49

50 COMPLEMENARY SLACKNESS CONDIIONS Hence, for i = 1, 2,, m and Similarl for j = 1, 2,, n and n * * i > i = ij j j = 1 0 b a n * * i ij j i j = 1 b a > 0 = 0 m i = 1 m * * j > ji i = j i = 1 0 a c a c > 0 = 0 * * ji i j j ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 50

53 EXAMPLE, * * optimal ( ) = 0 * * * * * ( ) = 0 * * * * * * 1.2 = 0.2 is given as an optimal solution with min W = 28 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 53

54 EXAMPLE + 2 = > 1 = 0 * * * = > 2 = 0 * * * = = 3 * * = = 4 * * 1 2 so that = 20 * * * * = 20 * * * * = 20 = 4 * * * = 20 = 4 * * * ma Z = 28 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 54

55 USES OF HE COMPLEMENARY SLACKNESS CONDIION Ke applications use finding optimal (P) solution given optimal (D) solution and vice versa verification of optimalit of solution (whether a feasible solution is optimal) We can start with a feasible solution and attempt to construct an optimal dual solution; if we succeed, then the feasible primal solution is optimal ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 55

58 INERPREAION he economic interpretation is * * * * Z = maz = c = b = W = minw bi * i constrained resource quantities, optimal dual variables Suppose, * bi bi + Δbi Δ Z = i Δbi In words, the optimal dual variable for each primal constraint gives the net change in the optimal value of the objective function Z for a one unit change in the constraint on resources ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 58 i = 1,2,..., m

59 INERPREAION Economists refer to this as a shadow price on the constraint resource he shadow price determines the value/worth of having an additional quantit of a resource In the previous eample, the optimal dual variables indicate that the worth of another unit of resource 1 is 1.2 and that of another unit of resource 2 is 0.2 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 59

60 GENERALIZED FORM OF HE DUAL We start out with ma Z = c s.t. A = b (P ) 0 o find (D), we first put (P) in smmetric form + A b A b smmetric A b A b form 0 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 60

61 GENERALIZED FORM OF HE DUAL Let = + We rewrite the problem as min W st.. A c b is unsigned he c.s. conditions appl = ( ) * * A c = 0 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 61

64 We are given that EXAMPLE 5 * 8 4 = 4 0 is optimal for (P) hen the c.s. conditions obtain ( ) + = 0 * * * = 8 > 0 + = 1 * * * ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 64

65 EXAMPLE 5 he other c.s. conditions obtain 4 * * i ai j j bi = 0 j = 1 * * Now, implies and so 4 = < 0 * 7 = 0 * Also, implies 3 = 4 * 6 = 0 Similarl, the c.s. conditions 7 * * j aji i cj = 0 i= 1 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 65

66 EXAMPLE 5 * have implications on the variable * Since, = then, we have 2 4 * Now, with = 0 we have * Since, = we have * 3 = 0 = 1 * * 2 1 ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 66 i * 1 > 1

69 EXAMPLE 5 herefore ( *) ( 8)( 1) ( 8)( ) ( 4)( ) ( 4)( 2) W = ( 4)( 0) + ( 2)( 0) + ( 10)( 0) = 16 and so ( ) ( ) * * W = 16 = Z optimalit of P and D ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 69

70 PRIMAL DUAL ABLE primal (maimize) A ( coefficient matri ) b ( right-hand side vector ) c ( price vector ) i th constraint is = tpe i th constraint is tpe i th constraint is tpe j is unrestricted j 0 j 0 dual (minimize) A ( transpose of the coefficient matri ) b ( cost vector ) c ( right hand side vector ) the dual variable i is unrestricted in sign the dual variable i 0 the dual variable i 0 j th dual constraint is = tpe j th dual constraint is tpe j th dual constraint is tpe ECE George Gross, Universit of Illinois at Urbana-Champaign, All Rights Reserved. 70

ISE 203 OR I. Chapter 3 Introduction to Linear Programming. Asst. Prof. Dr. Nergiz Kasımbeyli

ISE 203 OR I. Chapter 3 Introduction to Linear Programming. Asst. Prof. Dr. Nergiz Kasımbeyli ISE 203 OR I Chapter 3 Introduction to Linear Programming Asst. Prof. Dr. Nergiz Kasımbeyli 1 Linear Programming 2 An Example 3 The Data Gathered 4 Definition of the Problem Determine what the production