ECE 307- Techniques for Engineering Decisions Duality Concepts in Linear Programming George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DUALITY Definition: A LP is in symmetric form if all the variables are restricted to be nonnegative and all the constraints are inequalities of the type: objective type corresponding inequality type max ≤ ≥ ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DUALITY DEFINITIONS We define the primal problem as max Z = cT x s.t Ax ≤ b x ≥ (P) The dual problem is therefore W = bT y s.t AT y ≥ c y ≥ ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved (D) DUALITY DEFINITIONS The problems (P ) and (D) are called the symmetric dual LP problems ⎫ ⎪ s.t ⎪ a 11 x1 + a 12 x + + a n x n ≤ b1 ⎪ ⎪⎪ a 21 x + a 22 x + + a n x n ≤ b2 ⎬ ⎪ # ⎪ am x1 + am x2 + + amn xn ≤ bm ⎪ ⎪ x1 ≥ 0, x2 ≥ 0, , xn ≥ ⎪⎭ max Z = c1 x1 + c2 x + + cn xn ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved (P) DUALITY DEFINITIONS W = b y + b y + + b m y m ⎫ s.t a 11 y + a 21 y + + a m y m ≥ c1 a 12 y + a 22 y + + a m ym ≥ c2 # a n y + a n y + + a mn ym ≥ cn y1 ≥ 0, y2 ≥ 0, , ym ≥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭ ( D) ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved EXAMPLE 1: MANUFACTURING TRANSPORTATION PROBLEM R1 shipment cost coefficients warehouses W1 W2 retail stores R1 R2 R3 W1 R2 W2 R3 ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved EXAMPLE 1: MANUFACTURING TRANSPORTATION PROBLEM We are given that the supplies needed at Warehouse W and W are W1 ≤ 300 W2 ≤ 600 We are also specified the demands needed at retail stores R , R , and R as R1 ≤ 200 R2 ≤ 300 R3 ≤ 400 ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved EXAMPLE 1: MANUFACTURING TRANSPORTATION PROBLEM The problem is to determine the least-cost shipping schedule We define the decision variable x ij quantity shipped from W1 to R j i = 1, j = 1, 2, The shipping costs c ij may be viewed as element i,j of the transportation cost matrix ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved FORMULATION STATEMENT Z = s.t ∑∑ c i =1 j =1 ij x ij = x 11 + x 12 + x 13 + x 21 + x 22 + x 23 ≤ 300 x 11 + x 12 + x 13 x 21 + x 22 + x 23 ≤ 600 ≥ 200 + x 21 x 11 ≥ 300 + x 22 x 12 + x 23 ≥ 400 x 13 x ij ≥ i = 1, j = 1, 2, ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DUAL PROBLEM SETUP Z = ∑∑ c i =1 j =1 ij x ij s.t y1 ↔ − x 11 − x 12 − x 13 − x 21 − x 22 − x 23 ≥ − 600 y2 ↔ y3 ↔ y4 ↔ y5 ↔ ≥ − 300 + x 21 x 11 ≥ 200 ≥ 300 + x 23 ≥ 400 + x 22 x 12 x 13 xij ≥ i = 1, j = 1, 2, ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 10 DUALITY max s.t Z = T c x Ax ≤ b (P) x ≥ s.t W = bT y AT y ≥ c (D) y ≥ ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 56 DUALITY Suppose the primal problem is minimization, then, s.t Z = cTx (P) Ax ≥ b x ≥ max W = bT y s.t (D) AT y ≤ c y ≥ ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 57 INTERPRETATION The economic interpretation is Z * = max Z = c T x* = b T y * = W * = minW b i − constrained resource quantities, y *i − optimal dual variables i = 1, 2, , m Suppose, b i → b i + Δ b i ⇒ Δ Z = y *i Δ b i 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 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 58 INTERPRETATION Economists refer to this as a shadow price on the constraint resource The shadow price determines the value/worth of having an additional quantity of a resource In the previous example, the optimal dual variables indicate that the worth of another unit of resource is 1.2 and that of another unit of resource is 0.2 ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 59 GENERALIZED FORM OF THE DUAL We start out with max s.t Z = c Ax x T = ≥ x b (P ) To find ( D ), we first put ( P ) in symmetric form y+ ↔ Ax ≤ b ⎡ A ⎤ ⎡ b ⎤ symmetric x ≤ ⎢ ⎥ ⎢ ⎥ y− ↔ − Ax ≤ − b form ⎣− A ⎦ ⎣− b ⎦ x ≥ ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 60 GENERALIZED FORM OF THE DUAL Let y = y+ − y− We rewrite the problem as W = b T y s t AT y ≥ c y is unsigned The c.s conditions apply ( x *T AT y * − c ) = ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 61 EXAMPLE s.t max Z = x − x + x − x y1 ↔ x1 + x + x + x = y2 ↔ x1 ≤ y3 ↔ x2 ≤ y4 ↔ −x2 ≤ y5 ↔ x3 ≤ y6 ↔ −x3 ≤ y7 ↔ (P) x ≤ 10 x1 , x ≥ x , x unsigned ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 62 EXAMPLE W = y + y + y + y + y + y + 10 y s.t x ↔ y1 + y x ↔ y1 x ↔ y1 x ↔ y1 ≥ + y3 − y4 = −1 + y5 − y6 = ( D) + y ≥ −1 y , , y ≥ y1 unsigned ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 63 EXAMPLE We are given that x* ⎡ ⎢− = ⎢ ⎢ ⎢⎣ ⎤ ⎥ ⎥ ⎥ ⎥⎦ is optimal for (P) Then the c.s conditions obtain ( ) x *1 y *1 + y 2* − = x * = 8>0 ⇒ y + y = * * ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 64 EXAMPLE The other c.s conditions obtain ⎛ ⎞ * * y i ⎜ ∑ j x j − b i ⎟ = ⎝ j =1 ⎠ Now, x *4 = implies x *4 − 10 < and so y *7 = Also, x *3 = implies y *6 = Similarly, the c.s conditions ⎛ ⎞ * * x j ⎜ ∑ a ji y i − c j ⎟ = ⎝ i =1 ⎠ ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 65 EXAMPLE have implications on the y *i variable Since, x *2 = − then, we have y =0 * Now, with y *7 = we have y *1 > − Since, x *1 = we have y *2 = − y *1 ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 66 EXAMPLE Suppose y *1 = and so, y *2 = Furthermore, y *1 + y *3 − y *4 = − y *4 = − implies y *4 = ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 67 EXAMPLE Also y *1 + y *5 − y *6 = implies + y *5 = and so y *5 = ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 68 EXAMPLE Therefore ( ) W y * = ( )( 1) + ( )( ) + ( )( ) + ( )( ) + ( )( ) + ( )( ) + ( 10 )( ) = 16 and so W * = 16 = Z * ⇔ optimality of ( P ) and ( D ) ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 69 PRIMAL – DUAL TABLE primal (maximize) dual (minimize) A ( coefficient matrix ) A T ( transpose of the coefficient matrix ) b ( right-hand side vector ) b ( cost vector ) c ( price vector ) c ( right hand side vector ) i th constraint is = type the dual variable y i is unrestricted in sign i th constraint is ≤ type the dual variable y i ≥ i th constraint is ≥ type the dual variable y i ≤ x j is unrestricted j th dual constraint is = type xj ≥ j th dual constraint is ≥ type xj ≤ j th dual constraint is ≤ type ECE 307 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 70 ... 307 © 2 006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved EXAMPLE 1: MANUFACTURING TRANSPORTATION PROBLEM The problem is to determine the least-cost shipping... + y3 + y4 ≤ 20 ice - cream ≤ 30 150 y1 + y + y4 500 y1 + y3 + y4 ≤ 80 cheesecake soda y1 , y2 , y3 , y4 ≥ ECE 307 © 2 006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights... 307 © 2 006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 28 DUAL PROBLEMS max s.t Z = T c x Ax ≤ b (P) x ≥ s.t W = bT y AT y ≥ c (D) y ≥ ECE 307 © 2 006 – 2009