The bilevel lot-sizing problem Tamás Kis 1 joint work with András Kovács 1 Computing and Automation Research Institute Hungarian Academy of Sciences and Department of Operations Research Eötvös Loránd University, Budapest Aussois, January 9–13, 2012
Outline Uncapacitated lot-sizing with backlogging The bilevel lot-sizing problem MIP formulations Computational evaluation Conclusions
Uncapacitated lot-sizing with backlogging (ULSB) � n � � min ( p t x t + f t y t + h t s t + g t r t ) | ( 2 ) − ( 6 ) (1) t = 1 where x t + ( s t − 1 − r t − 1 ) = d t + ( s t − r t ) , t = 1 , . . . , n (2) x t ≤ My t , t = 1 , . . . , n (3) s 0 = s n = r 0 = r n = 0 , (4) x t , s t , r t , ≥ 0 , t = 1 , . . . , n (5) y t ∈ { 0 , 1 } , t = 1 , . . . , n (6) where ◮ The p t , f t , h t , g t are the cost parameters, the d t are the demands ◮ The x t , y t , s t , r t are the variables
Some related work ◮ W. I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system – A network approach. Management Science, 15(9):506–527, 1969. ◮ A. Federgruen, M. Tzur, The dynamic lot-sizing model with backlogging: A simple O ( n log n ) algorithm and minimal forecast horizon procedure. Naval Res. Logitics 40, 459–478, 1993. ◮ Y. Pochet and L. A. Wolsey. Lot-size models with backlogging: Strong reformulations and cutting planes. Mathematical Programming, 40:317–335, 1988. ◮ S. Kucukyavuz and Y. Pochet. Uncapacitated lot-sizing with backlogging: the convex hull. Mathematical Programming, Ser. A, 118:151–175, 2009.
Network representation of ULSB 0 x 1 x 2 x 3 x 4 x 5 x 5 s 1 s 2 s 3 s 4 1 2 3 4 5 r 1 r 2 r 3 r 4 d 4 d 2 d 3 d 1 d 5 *
Extreme point solutions for ULSB 0 x 1 x 2 x 3 x 4 x 5 x 5 s 1 s 2 s 3 s 4 1 2 3 4 5 r 1 r 2 r 3 r 4 d 4 d 2 d 3 d 1 d 5 *
Bilevel Optimization ◮ Two decision makers, Leader and Follower, who make decisions sequentially, in this order ◮ General form: optimistic case min f ( x , y ) (7) x , y subject to L ( x , y ) (8) y ′ ( g ( x , y ′ ) | F ( x , y ′ )) y ∈ arg min (9) ◮ General form: pessimistic case min max f ( x , y ) (10) x y subject to L ( x , y ) (11) y ′ ( g ( x , y ′ ) | F ( x , y ′ )) . y ∈ arg min (12)
Bilevel lot-sizing Rules of the game ◮ Both decision makers solve an uncapacitated lot-sizing problem with backlogging ◮ The Leader has external demand ( d 1 t ) ◮ The Leader’s production ( x 1 t ) equals the supply received from the Follower ◮ The Follower’s demand ( δ t ) is set by the Leader ◮ Both the Leader and the Follower may backlog some of its demand ◮ The Follower pays the backlogging cost to the Leader as penalty for late delivery ◮ In those periods when the Follower backlogs, there is no delivery to the Leader ( r 2 t x 1 t = 0) ◮ If the Follower does not backlog in a period, then the total delivery up to period t equals the total amount requested by the Leader up to period t
Example Optimal solution of a sample problem t 1 2 3 4 5 6 7 8 9 10 d 1 71 84 43 21 4 81 59 44 32 46 t 82 73 68 42.72 39.77 57.51 55.46 21.93 44.61 δ t x 1 82 73 68 82.49 57.51 55.46 21.93 44.61 t s 1 11 25 4 1.49 11.46 1.39 t r 1 t x 2 82 141 140 122 t s 2 68 57.51 66.54 44.61 t r 2 42.72 t f 1 = 100 p 1 = 1 h 1 = 6 g 1 = 18 f 2 = 492 p 2 = 1 h 2 = 2 g 2 = 6
Formulation n � p 1 t x 1 t + f 1 t y 1 t + h 1 t s 1 t + g 1 t r 1 t − g 2 t r 2 � � Minimize (13) t t = 1 subject to x 1 t + s 1 t − 1 − r 1 t − 1 = d 1 t + s 1 t − r 1 t = 1 , . . . , n (14) t , t = � t r 2 τ = 1 ( δ τ − x 1 τ ) , t = 1 , . . . , n (15) x 1 t ≤ My 1 t = 1 , . . . , n (16) t , x 1 t ≤ M ( 1 − β 2 t ) , t = 1 , . . . , n − 1 (17) s 1 0 = s 1 n = r 1 0 = r 1 n = 0 , (18) x 1 t , r 1 t , s 1 t , δ t ≥ 0 , t = 1 , . . . , n (19) y 1 t ∈ { 0 , 1 } , t = 1 , . . . , n (20)
Formulation (cont.d) y 2 x 2 � n � � s 2 p 2 t x 2 t + f 2 t y 2 t + h 2 t s 2 t + g 2 t r 2 � � ∈ arg min | ( 22 ) − ( 28 ) t t = 1 r 2 β 2 (21) x 2 t + ( s 2 t − 1 − r 2 t − 1 ) = δ t + ( s 2 t − r 2 t ) , t = 1 , . . . , n (22) where x 2 t ≤ My 2 t , t = 1 , . . . , n (23) s 2 0 = s 2 n = r 2 0 = r 2 n = 0 , (24) x 2 t , s 2 t , r 2 t ≥ 0 , t = 1 , . . . , n (25) y 2 t ∈ { 0 , 1 } , t = 1 , . . . , n (26) r 2 t ≤ M β 2 t , t = 1 , . . . , n − 1 (27) β 2 t ∈ { 0 , 1 } t = 1 , . . . , n − 1 . (28)
Extreme Point Solutions Definition A solution to the bilevel lot-sizing problem is an extreme point solution if the Follower’s part is an extreme point solution of ULSB with demands δ t . Assumption g 2 t + h 2 t > 0 for all t = 1 , . . . , n − 1. This assumption excludes that a solution with r 2 t s 2 t > 0 is optimal for the Follower. Lemma Under the assumption, if the bilevel optimization problem admits an optimal solution, then it admits an extreme point optimal solution. New constraints to enforce extreme point solutions: s t − 1 ≤ M ( 1 − y 2 t − β 2 t ) , t = 2 , . . . , n (29)
Formulation MIP-1 Definition Let OP 2 be the set of those (¯ r 2 , ¯ x 2 , ¯ y 2 , ¯ s 2 , ¯ δ ) vectors such that � n t = 1 ¯ δ t = � n t = 1 d 1 x 2 , ¯ y 2 , ¯ s 2 , ¯ r 2 ) is an extreme point optimal t , and (¯ solution for the ULSB of the Follower w.r.t. demand ¯ δ . Let Z ULSB ( δ ) denote the optimum value of ULSB for fixed δ > 0 Question Does OP 2 admit a compact (extended) mixed integer formulation? Answer YES! Idea: use an extended formulation for ULSB with δ in the objective function only.
Formulation MIP-1 (cont.d) Lemma (Pochet and Wolsey (1988)) The optimum value of ULSB equals the optimum value of the following mathematical program n � k − 1 n � n L SP ( δ ) = min � � � � a k ℓ v k ℓ + p k δ k z kk + b k ℓ w k ℓ + f t z tt k = 1 ℓ = 1 ℓ = k + 1 t = 1 subject to a shortest path formulation in the network below, where a k ℓ = p k δ ℓ, k − 1 + � k − 1 t = ℓ g t δ ℓ, t for 1 ≤ ℓ < k ≤ n, and b k ℓ = p k δ k + 1 ,ℓ + � ℓ − 1 t = k h t δ t + 1 ,ℓ for 1 ≤ k < ℓ ≤ n, and δ k ,ℓ = � ℓ t = k δ t for 1 ≤ k ≤ ℓ ≤ n. v 3 v 2 v 3 1 1 2 1 1' 1'' 2 2' 2'' 3 3' 3'' 4 1 v 1 z 1 w 1 v 2 z 2 w 2 v 3 z 3 w 3 1 1 1 1 2 2 2 3 3 3 w 1 w 2 2 3 w 1 3
Formulation MIP-1 (cont.d) The dual of the shortest path formulation is D SP ( δ ) = max φ 2 (30) 1 subject to φ 2 t − φ 2 k ′ ≤ a k , t , k = t , . . . , n φ 2 t ′ − φ 2 t ′′ ≤ p 2 t δ t + f 2 for all t = 1 , . . . , n . t , (31) φ 2 t ′′ − φ 2 k + 1 ≤ b t , k , k = t , . . . , n By the strong duality of linear programming Z ULSB ( δ ) = D SP ( δ ) for any fixed δ ≥ 0. Lemma δ ) ∈ OP 2 if and only if � n r 2 , ˆ t = 1 δ t = � n x 2 , ˆ y 2 , ˆ s 2 , ˆ t = 1 d 1 (ˆ t , and there φ 2 such that (ˆ exists ˆ r 2 , ˆ β, ˆ δ, ˆ x 2 , ˆ y 2 , ˆ s 2 , ˆ φ 2 ) satisfies the constraints (22)-(28), (29), (31), and the equation n � � p 2 t x 2 t + f 2 t y 2 t + h 2 t s 2 t + g 2 t r 2 � = φ 2 1 . (32) t t = 1
Formulation MIP-1 (cont.d) The complete formulation: (14)-(16), n (18)-(20), � p 1 t x 1 t + f 1 t y 1 t + h 1 t s 1 t + g 1 t r 1 t − g 2 t r 2 � � MIP-1 : min . t (22)-(28),(29), t = 1 (31),(32) Lemma There is a one-to-one correspondence between the extreme point feasible solutions of the bilevel lot-sizing problem and that of MIP-1: (i) Any feasible solution of MIP-1 can be projected onto a feasible solution of the bilevel lot-sizing problem of the same value. (ii) Conversely, any feasible extreme point solution of the bilevel lot-sizing problem can be extended to a feasible solution of MIP-1 of the same value.
Formulation MIP-2 ◮ Again, based on a shortest path formulation � 1 the requests δ i , . . . , δ k are produced in j ∈ { i , . . . , k } α ijk = 0 otherwise ◮ If α ijk = 1, then s 2 i − 1 = s 2 k = 0, and r 2 i − 1 = r 2 k = 0. ◮ Cost associated with α ijk : c ijk = a j , i + f j + p j δ j + b j , k α i, i, k i -1 i +1 i k α i, i + 1 , k α i, k , k
Formulation MIP-2 (cont.d) n � p 1 t x 1 t + f 1 t y 1 t + h 1 t s 1 t + g 1 t r 1 t − g 2 t r 2 � � MIP-2 : min t t = 1 subject to the constraints of the Leader, and r 2 t ≤ M ( 1 − β 2 t ) , t = 1 , . . . , n − 1 � β 2 t = α i , j , k , t = 1 , . . . , n − 1 i ≤ t < j ≤ k � � α i , j , k = 1 , t = 1 , . . . , n i ≤ t ≤ k i ≤ j ≤ k a j , i + f j + p j δ j + b j , k + φ i − 1 ≥ φ k , 1 ≤ i ≤ j ≤ k ≤ n a j , i + f j + p j δ j + b j , k + φ i − 1 ≤ φ k − M ′ ( 1 − α i , j , k ) , 1 ≤ i ≤ j ≤ k ≤ n φ 0 = 0 , α i , j , k ∈ { 0 , 1 } , 1 ≤ i ≤ j ≤ k ≤ n .
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