Stochastic Due-Date Lot-Streaming Flowshop Scheduling with Benders Decomposition and Branch-and-Bound

doi:10.16183/j.cnki.jsjtu.2023.070

Abstract

Abstract:

The lot-streaming flowshop scheduling problem with stochastic due time is addressed in this paper, with the objective of minimizing the sum of expected job delays. Closed-form expressions for the expected delays of jobs are derived under three classical distribution conditions. A mathematical model is then formulated, considering set-up times and stochastic due time. To address the highly nonlinear nature of the model, a linearization is performed. Furthermore, an optimization algorithm is designed using a Logic-based Benders decomposition (LBBD) approach combined with branch-and-bound. Two effective acceleration strategies are introduced to improve the efficiency of the algorithm. The numerical experiments demonstrate the effectiveness of the proposed algorithm, and the necessity of considering stochastic lead times is verified by comparing the results with those obtained from deterministic due time.

Key words: stochastic due time, lot-streaming, Benders decomposition, branch-and-bound

CLC Number:

TH186

SHI Yadong, LIU Ran, WANG Chengkai, WU Zerui. Stochastic Due-Date Lot-Streaming Flowshop Scheduling with Benders Decomposition and Branch-and-Bound[J]. Journal of Shanghai Jiao Tong University, 2024, 58(8): 1271-1281.

Figures/Tables 7

Fig.1

Tab.1

Tab.2

Fig.2

Tab.3

Tab.4

Numerical experiment results of exact algorithm

算例	分布类型	LBBD结合分枝定界				LBBD结合Gurobi				上界 Gap/%
算例	分布类型	上界, $B b U$	下界, $B b L$	Gap_b/%	时间/s	上界, $B b U$	下界, $B b L$	Gap_b/%	时间/s	上界 Gap/%
9	N	73.9	73.9	0	1 575	89.9	43.40	51.60	3600	17.73
10	N	282.5	282.5	0	963	306.3	38.50	87.40	3600	7.77
11	N	397.8	397.8	0	1 378	397.9	67.10	83.10	3600	0.03
12	N	338.7	338.7	0	1 621	380.2	64.00	83.20	3600	10.92
13	N	524.0	524.0	0	1 940	639.1	44.70	93.00	3600	18.01
14	E	400.3	379.2	5.20	3 600	416.5	113.30	72.80	3600	3.90
15	E	471.8	471.8	0	163	478.2	195.80	59.10	3600	1.34
16	E	424.4	420.3	1.00	1 280	434.3	150.70	65.30	3600	2.28
17	E	163.1	161.9	0.70	1 228	181.0	52.30	71.10	3600	9.87
18	E	582.5	578.5	0.70	2 847	674.6	175.40	74.00	3600	13.65
19	U	77.4	77.4	0	2 880	126.7	14.10	88.90	3600	38.91
20	U	96.2	96.2	0	422	110.5	25.70	76.70	3600	12.94
21	U	464.1	464.1	0	2 078	541.3	93.00	82.80	3600	14.26
22	U	47.4	47.4	0	152	47.8	0.06	99.90	3600	0.84
23	U	205.9	205.9	0	948	236.7	12.55	94.70	3600	13.01

Tab.4

Tab.5

References 18

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参数符号	含义
k	机器的索引号,k∈{1, 2,…, K}
j, h	工件i, g的子批索引号,j∈{1, 2,…, u_i}, h∈{1, 2,…, u_g}
i, g, n	工件的索引号,i, g, n∈{1, 2,…, N}

决策变量	含义
a_ij	辅助变量,用于顺序记录
r_ij	整数变量,工件i的第j个子批的最小批量个数,可以为0,0≤i≤N, 1≤j≤u_i
t_ijk	连续变量,工件i的第j个子批在机器k上的完工时间, 0≤i≤N, 1≤j≤u_i, 1≤k≤K
y_ij	0-1变量,r_ij>0时,y_ij=1,r_ij=0时,y_ij=0. 0≤i≤N, 1≤j≤u_i
z_ijgh	0-1变量,当工件i的第j个子批和工件g的第h个子批均不为0,且工件i的第j个子批为工件g的第h个子批的紧前子批时,z_ijgh=1;否则,z_ijgh=0. 0≤i≤N, 1≤j≤u_i, 0≤g≤N, 1≤h≤u_g
S_ij	整数变量,工件i的第j个子批的工件数量,0≤i≤N, 1≤j≤u_i

算例	分布	LBBD		企业实际方案					Gap_min/ %	Gurobi			遗传算法
算例	分布	求解结果	时间/s	策略1	策略2	策略3	策略4	时间/s	Gap_min/ %	求解结果	Gap_g/ %	时间/s	求解结果	Gap_c/ %	时间/s
24	N	1 700	3 600	3 450	3 638	5 188	4 986	<5	50.72	2 066	21.55	3 600	2 905	41.48	3 600
25	N	1 022	3 600	3 981	4 283	5 850	3 525	<5	71.01	1 109	8.51	3 600	2 484	58.86	3 600
26	N	1 608	3 600	4 256	4 390	7 181	5 437	<5	62.22	2 063	28.30	3 600	2 843	43.44	3 600
27	N	502	3 600	2 771	2 639	2 354	2 986	<5	78.68	588	17.19	3 600	2 211	77.31	3 600
28	N	2 039	3 600	5 751	5 545	7 041	7 408	<5	63.23	2 308	13.19	3 600	2 526	19.28	3 600
29	N	2 804	3 600	5 532	5 447	5 111	6 162	<5	45.14	3 045	8.59	3 600	4 412	36.45	3 600
30	E	2 019	3 600	2 752	2 884	4 027	4 813	<5	26.64	2 206	9.26	3 600	2 981	32.27	3 600
31	E	999	3 600	1 873	2 000	2 269	2 876	<5	46.66	1 259	26.03	3 600	1 617	38.22	3 600
32	E	807	3 600	1 023	1 051	1 348	2 348	<5	21.11	1 016	25.90	3 600	1 498	46.13	3 600
33	U	1 301	3 600	2 756	2 756	4 066	5 757	<5	52.79	1 865	43.35	3 600	2 686	51.56	3 600
34	U	2 148	3 600	3 825	3 943	5 119	7 178	<5	43.84	2 486	15.74	3 600	3 645	41.07	3 600
35	U	1 002	3 600	1 881	1 962	2 138	4 807	<5	46.73	1 082	7.98	3 600	2 738	63.40	3 600