A Lagrange relaxation algorithm for capacitated lot-size problem(clsp) with minimum lot-size constraint

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1 Control Theory & Applications Vol. 26 No. 2 Feb : (2009) ,,, (, ) :,., W-W(Wagner-Whitin). O(T 3 ).,,.,,.,. : ; ; ; : TP273 : A A Lagrange relaxation algorithm for capacitated lot-size problem(clsp) with minimum lot-size constraint PAN Chang-chun, YANG Gen-ke, SUN Kai, LU Heng-yun (Department of Automation, Shanghai Jiaotong Unviversity, Shanghai China) Abstract: A Lagrange relaxation heuristic-based procedure is presented to solve the capacitated lot-size problem(clsp) with minimum lot-size constraint. The problem is first decomposed into a series of sub-problems W-W(Wagner-Whitin) with dynamic economic minimum lot-size constraint. To deal with the sub-problems, an optimal forward iterative algorithm with runtime complexity of O(T 3 ) is proposed. The Lagrange dual problem is then handled by the sub-gradient optimization algorithm to obtain a tight lower bound. If the solution to the Lagrange dual problem is infeasible, the setup variables are fixed and the remaining problem is reformulated as a linear programming problem which can be solved efficiently by any off-the-shelf solver. Finally, the computational experiments demonstrate the algorithm s efficiency. Key words: CLSP; minimum lot-size; Lagrange relaxation; sub-gradient optimization 1 (Introduction) CLSP(capacitated lot-size problem),,,, [1] W-W,, [2,3]. [4 7],. [8] Lagrangean CLSP,. [9] NP-hard., CLSP NP-Hard.,, [10,11] (LP). [12] (GA) LP. [13]. [14] Lagrangean., [15] (ACO). [16],.,,.,,, : ; : : ( ).

2 ,,.,. CLSP,,, W-WO(T 3 ). 2 (Problem formulation) : t:, t = 1,, T ; j:, j = 1,, J. : d jt : t, j; m jt : t, j ; a j : j ; h jt : jt t + 1 ; s jt : jt ; p jt : jt ; r t : t. : X jt : t j; Y jt :, j t 1, 0; I jt : tj. (P): min J T +1 h j,t 1 I jt + T j=1 t=2 t=1 j=1 J (p jt X jt + s jt Y jt ) (1) s.t. I jt + X jt I j,t+1 = d jt, j, t, (2) J X jt a j r t, t, (3) j=1 X jt ( rt / aj ) Y jt, j, t, (4) X jt Y jt m jt, j, t, (5) X jt 0, I jt 0, Y jt {0, 1}, j, t, (6) I j1 = 0, j, t. (7) : (1), ; (2); (3) ; (4) ; (5) ; (6)(7),. (P) CLSP : i) (5) ; ii) I j,t Lagrangean (Lagrangean relaxation-based strategy) 3.1 Lagrangran (Lagrangean relaxation) Lagrangran β = [β 1, β 2,, β T ], β t 0, (3) Lagrangean (1), Lagrangean L(β)=min { J T +1 h j,t 1 I jt + j=1 t=2 J j=1 t=1 J j=1 t=1 s.t. (2)(4) (6). T X jt (p jt + β t a j ) + T s jt Y jt T } r t β t, (8) t=1 J X jt a j r t, β 0, L(β) j=1 (P). (P), Lagrangean : LD = max (L(β), β 0). (9) Lagrangean LD, Lagrangean β, Lagrangean L(β), Lagrangean,. L(β). L(β),, L(β)J, j : L j (β) = min{ T +1 h jt 1 I jt + T {(p jt + β t a j ) X jt + t=2 t=1 s jt δ(x jt ) } }, (10) s.t. I jt + X jt I j,t+1 = d jt, t, (11) X jt δ (X jt ) m jt, t, (12) X jt 0, I jt 0, t, I j1 = 0. (13) { 1, x > 0, δ(x) = 0,.. (11), L j (β) W-W., (9) (12) W-W (E-WW). 3.2 E-WW (The optimal forward plan policy) W-W (S-WW), :, I t X t = 0, t.

3 2 : 135, 0, I t + X t < k d τ, X t X [17] t. E-WW S-, I t + X t = WW, E-WW k d τ, X t = X t X t, k,, S-WW ( X [18,19]). t, : f = p k X t ;, t X t, I t, X t, : f + = t t. E-WW, : (p t + k 1 h τ ) X t. f > f +, 1 t d t m t, E-WWS-WW. S-WW E-WW, [1], S-WW X, t Xt = 0 k, t k T, Xt = k d i., i=t t, d t m t, X E-WW, E-WW. E-WW S-WW (E-WWS-WW, ). S-WW E-WW.. 2 E-WW, t, X t m t k, k t I t + X t = k d j. j=t. E-WW,, t X t, k, k > t, : k 1 d τ < I t + X t < k d τ, : p t + k 1 h r p k :, I t + X t > k 1 d τ, X t X t, I t + X t = max(m t + I t, k 1 d τ ). X t = X t X t,, X t k. t,, : f = X t (p t + k 1 h τ ); k, : f + = X t p k. f f +, f = f +,, I t + X t = max(m t + I t, k 1 d τ ), f > f +, ; p t + k 1 h r < p k :,,.. 1 E-WW, t, I t X t > 0, l, X l = m l, l < t, t. 2, : 1) X l 0; 2) k l I l +X l = k d j. I t X t > 0, X l + I l > t 1 X l + I l = X l + I l = k k j=l d τ, k {l,, t 1}, d τ, k {t,, T }, d τ,, 2,. 1 I t = 0,, (1,, t 1)(t,, T ). [1,17], E-WW. E-WW, : F (t): t ; ˆF (t): It+1 = 0 t, I t+1 = 0, ˆF (t) = ; F (t, l) : t, l l t, I l = 0, F (t) = min {F (t, l)}. (14) l {1,,t} l, 1,, {1,, l 1} {l,, t}. : F (t) = ˆF (t) = min {C(t, l) + ˆF (l 1)}, (15) l {1,,t} min {Ĉ(t, l) + ˆF (l 1)}. (16) l {1,,t} : C(t, l) {l,, t} ; Ĉ(t, l) I t+1 = 0 {l,, t}, I t+1 = 0

4 136 26, Ĉ(t, l) =. C(1, 1) = { s 1 + m 1 p 1 + h 1 (m 1 d 1 ), m 1 > d 1, s 1 + d 1 p 1, m 1 d 1, { +, m 1 > d 1, Ĉ(1, 1) = s 1 + d 1 p 1, m 1 d 1, ˆF (0) = (Algorithmic complexity analysis ) (15)(16), ˆF (t), l {1,, t} Ĉ(t, l), Ĉ(t, l), l, l,, (t l + 1),,,. τ, t X τ = max(m τ, d τ I τ ). Ĉ(t, l) τ= τ O(t), T ˆF (T ) O(T 3 ), F (T ) ˆF (T ). 3.4 Lagrangean (Subgradient optimization for the Lagrangean dual problem ), Lagrangean. Lagreagean [20]. : 1) Lagrangean ; 2)., Lagrangean β k+1 = max { 0, β k + λ k g k}. 3.5 (Procedure for feasibility ) Lagrangean,,. Lagrangean Y it,,. 4 (Computational experiments), E- WW, [1], Table 1 E-WW Test data for E-WW Table 2 2 Mode of generating the test data g k (T ) L(β) β k, : g k = [gt k ], gt k = J a j Xjt k r t. j=1 λ k k., : λ k = σ k ( Z u L(β k ) ) g k 2. : σ k (0, 2], Z u LD(β)., σ 1 = 1.8, Z u = 1.08 Z, Z., 20, σ k+1 = 0.8σ k, 100 ( [8]). : J = 500, T = 10 : J = 1000, T = 20 : J = 2000, T = 30 d jt U[100, 1000] h jt U[0, 1] p jt U[0, 1] s j U[50, 100] a j U[1, 5] m jt U[50, 500] P r t k t a j d jt j 1, ; 2,. 1, : X = [98, 0, 97, 0, 121, 0, 0, 112,

5 2 : 137 0, 67, 135, 0], : 864[1] ; 2, : X = [134, 0, 0, 80, 102, 0, 0, 112, 0, 80, 122, 0], : 906., E-WW S-WW. [21], 3,, 2. VC++, PentiumIV 2.8GHz, 1G PC, Lingo8.0. 3, 10, 3 4. Lagrangean,, Lagrangean β,, β.,,. 3 4, SOL LB Gap = 100% LB LB., 1%,,.,. 1 : k t = U[1, 1.5]; : k t = U[1.5, 2]. Table 3 3 Computational results with tight capacity J T Gap/% CPU /s Table 4 4 Computational results with loose capacity J T Gap/% CPU /s Lagrangean,,,,.. 1 Lagrangean Fig. 1 Convergent curves of the Lagrangean dual problem under two circumstances of different capacity mode 4.1 (Conclusion),., CLSP. Lagrangean,..,,, CLSP,,,. (References): [1] WANGER H M, WHITIN T M. Dynamic version of the economic lot sizing model[j]. Management Science, 1958, 5(1): [2] DREXL A, KIMMS A. Lot sizing and scheduling survey and extensions[j]. European Journal of Operational Research, 1997, 99(2): [3] JENS R, DEGRAEVE Z. Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches[j]. European Journal of Operational Research, 2007, 177(5): [4] BARANY I, VAN ROY T J, WOLSEY L A. Strong formulations for multi-item capacitated lotsizing[j]. Management Science, 1984, 30(12): [5] EPPEN G D, MARTIN R K. Solving multi-item capacitated lotsizing problems using variable redefinition[j]. Operations Research, 1987, 35(3): [6] WOLSEY A A. Solving multi-item lot-sizing problems with an MIP solver using classification and reformulation[j]. Management Science, 2002, 48(12): [7],. CLSP[J]. ( ), 2003, 24(1): (GAO Zhen, TANG Lixin. Branch-and-Price algorithm for CLSP[J]. Journal of Northeastern University(Nat & Sci), 2003, 24(1): ) [8] DIABY M, BAHL H C, KARWAN M H, et al. A lagrangean relaxation approach for very-large-scale capacitated lot-sizing[j]. Management Science, 1992, 38(2):

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