Minimizing makespan in a three-stage flexible flowshop with identical machines and two batch processors

2017 12 31 4 Dec. 2017 Communication on Applied Mathematics and Computation Vol.31 No.4 DOI 10.3969/j.issn.1006-6330.2017.04.005 Æ ÛÝ ¹ÐĐÝÆßÌ ĐĐ 1, 1, 2 (1. ÆÆ ÆÆ 200444; 2. ¾Æ Æ ¾ 330031) ѺÛÏÖ Æº m Ó ÐÆÑ Ø Ù 1 Æ ³ÌÙÓ ÂÓ Ò Ø ÅÓ± Æ Öº±¾Ç±Ó Ð ¾ÛÏÖ Ó 2010 ű 90C27 Çű O223 ÈÏÚÕ» A ÈÒÙ 1006-6330(2017)04-0471-16 Minimizing makespan in a three-stage flexible flowshop with identical machines and two batch processors HUANG Huanhuan 1, HE Longmin 1, LUO Runzi 2 (1. College of Sciences, Shanghai University, Shanghai 200444, China; 2. School of Sciences, Nanchang University, Nanchang 330031, China) Abstract A three-stage flexible flowshop scheduling problem that consists of m identical machines at stage 1, one batch processor at stage 2 and the other batch processor at stage 3 is considered. The objective is to minimize the makespan. When all jobs have the same processing time at all the three stages, an algorithm for general situation and some algorithms for some special situations are provided. Key words scheduling; flexible flowshop; identical machines; batch processor 2010 Mathematics Subject Classification 90C27 Chinese Library Classification O223 «2015-03-20; 2015-08-08 ¾ «¹± Æ Æ (11632008, 11372170, 11571221, 11361043) ÅÓ Þ ³ÌÖ ÚÑ E-mail: hlm@shu.edu.cn

472 31 1 1.1 ØÙ ÞßÕÒ Ñ Ï½ÚÎÕ ÏÆß ½ÚÎÕ (flexible flow shop, FFS) (burn in, BI) ÂÊÏÑ ß FFS ÊÀ n Ð ¼Ê k Ð i m i ( 1) (i = 1, 2,, k), ¹ Á ÐÁ¹ ²ÂÒÒÅ ÉßØ Fk(m 1, m 2,, m k ) C max. FFS ß µ²ëõ F2(m 1, m 2 ) C max ßÌ k = 1 ßØ Pm C max, Garey Johnson [1] ÌÎß Pm C max Ø NP- Ì m 1 = 1 m 2 = 1 ßØ F2(1, 1) C max, ¼ß F2 C max, Johnson [2] ĐÕß F2 C max Ï ¹Ð²ÃÕÒ ÌÎ SPT-LPT ÒØÒÒß F2(m 1, m 2 ) C max ϲËÕÒÅ Âß Pm C max ß F2 C max Ï ÂÐ BI ÊÀ n Ð µá¹³ m (> 1) ¹Ò B ( 1) Ð Ï ØÛ ÒÏ Ï Å³ÛÒÍà J j ÏÍà r j (j = 1, 2,, n), Á Ä ½³Ï ²ÂÒÒÅ ßØ 1 B C max ³ m B C max. Ï 2 Ð ÒÍÃÏß 1 r j, B C max, Lee Uzsoy [3] ĐÕÛ«Ò¹Ïß 1 r j, B C max, Lee Uzsoy [3] ĐÕÐ O(n log n) ÉÒ¹ ÐÅ O(n log n) ÉÒ ÍÃÏß 1 B C max, Sung Choung [4] ÃÕ SPT-FOE ³ LPT-LOE Ï«Ò ÍÃĐÏß 1 r j, B C max ÃÕ Ï O(nc) Ò (c Ø Ï¼Ó Ò); ¹Ïß 1 r j, B C max ÃÕ O(n log n) O(n log n) max{o(n log n), O(nc)} ÏÐÉÒ 2. Ahmadi [5] F2(B 1, 1) b 1j b 1 C max F2(1, B 2 ) b 2j b 2 C max F2(B 1, B 2 ) b 1j b 1, b 2j b 2 C max ÏÐß µđõ O(n log n) O(n log n) O(n 3 ) ÏÒ Sung Yoon [6] ĐÕß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max Ï O(nB 1 ) Ò Á µ ÒÏ ß Sung [7] ĐÕ O (n m 1 ) c k ÉÒ (m Ø Ë c k Ø k ϼÓ). k=2 Ò Ï½ÚÎÕ ß F2(m, B) C max : Ø Á Ï Ò Ò He [8] µđõ max{o(n log n), O(nB)}(B Ø Ï¼Ó) ÒØ 2 Ï O(n log n) ÉÒ Ø¹ Ü [9] Ì Õ¹ O(n log n) ÃÖ Ø ()NP- NP- Ï ĐÕ O(n log n) O(n) ÏÉÒ

4 ÆÓ Ò ÐÛÏÖ Ó 473 1.2 Îļ ½ FFS BI ÑÅ Úν ÅÉ Õ Î ( À ), Õ ( ÒÏÅÉ Á±À ßÝ ±) ß ÆØÞÏ FFS-BI-BI ß Đ FFS-BI-BI ÊÀ Ϲ½ÚÎÕ n Ð ¼Ñ 1 2 3 Ï 1 m ( 1) Ò {1, 2,, m} J j Á»¹Ò Ï Ø a j (j = 1, 2,, n). 2 3 µø ÒÏÑ M 1 M 2, M 1 M 2 Ê J j Ï µø b 1j b 2j (j = 1, 2,, n), M 1 M 2 Ò ÏØÊ ÏÒÈ M 1 M 2 ¹ µò B 1 ( 1) B 2 ( 1) Ð Đ Ï Ë B 1 B 2 µø M 1 M 2 ϼӵ²¹ÒÏ Õ Ï µ C max ÃÒßØ F3(m, B 1, B 2 ) C max, F3 ÏÚÎÕ m Ï 1 m Ò {1, 2,, m} B 1 B 2 µï 1 2 M 1 M 2 Ï¼Ó ÞÐß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max, 1 2 3 Ï µ Ò 1.3 m: 1 Ï m Ò M 1 : 2 Ï M 2 : 3 Ï n: Ë N: ²Ïº N = {1, 2,, n}. J j : ²Ø j Ï (j N), j (j N), J = {J 1, J 2,, J n }. r j : J j ÏÍà (j N), r = {r 1, r 2,, r n }. a j : J j ÁÒ Ï (j N). b lj : J j Á M l Ï (j N; l = 1, 2). B l : M l Ï¼Ó (l = 1, 2). C j : J j Á M 2 Ï (j N). C max : {J j }(j N) Á M 2 Ï µ C max: {J j }(j N) Á M 2 ÏÒ µ x : x ÏÒÅÊË x : Å x ÏÒÊË arg: ±Óϲ ( argmax{r j j N} {r j }(j N) ÒÅÎϲ j). ERT (earliest release time) Ä ÍÃÍÅ

474 31 SPT (shortest processing time first) Ä ÍÅ LPT (longest processing time first) Ä ÅÍ FOE(first only empty)(n, B) Ä n Ð Ä (n ( n/b 1)B) Ð ( n/b 1) Ø LOE(last only empty)(n, B) Ä n Ð ( n/b 1) Ø «(n ( n/b 1)B) Ð Áº¹ÐßÏ ²Ë Z = f(c 1, C 2,, C n ), D º S ϹРº (D S), ¹Ã s (s S) Ϲ (C 1, C 2,, C n ). S»¹Ðà s, Á D Ç͹Ðà s, ²Î Z = f(c 1, C 2,, C n ) Z Z, Z Âà s Ï ²ËÎ D  S ϹÐÁº [10-11]. 2 Íà F3(m,B 1, B 2 ) a j a, b 1j b 1,b 2j b 2 C max Þ 2.1 ÎÄ F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max µ Ï Ò (a j a, b 1j b 1, b 2j b 2 ), ¼ 1 ¼³ J 1, J 2,, J n ÁÌÒ ÏÒ Ï 1. ³ 1 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ÂÁÉ ÏÁ Ò ÐÏ Î 1, ß Á 1 Ò ÐÏ J j Á 1 Ï j/m a ÃØ J j Íà M 1 Ï r j = j/m a(j N). É ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max. ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max, Sung Yoon [6] ¼ Ï ERT Ä ( 2) ĐÕÒà M 1 M 2 Ï M 2 ¼ FOE(n, B 2 ) Ä²Ò ( 3). M 1 Á M 2 ¼ FOE(n, B 2 ) ÄÏÒµ B 1 B 2, b 1 b 2 B 1 B 2, b 1 b 2 ÏÑÆ µ¼ FOE(n, B 1 ) Ä FOE(n, B 2 ) IJҹ ¼ FOE(n, B 2 ) Ä Ã DP(dynamic program) Ò²Ò Þ [6] ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max ÏÑ Ä ³ 2 [6] ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max, ÂÁ¹ÐÒà ERT Ä ERT ÄÏ ÓØ O(n log n). ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max, Î 1, Þ Ñ 1 Ò ÐÏ Íà 2 Ï M 1 Ï r j = j/m a (j N) 2 Ï ERT Ä

4 ÆÓ Ò ÐÛÏÖ Ó 475 2 Đ 3. ³ 3 [6] ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max, FOE(n, B 2 ) IJ M 2 Ï Ò FOE(n, B 2 ) ÄÏ ÓØ O(n). ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max, ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max, FOE(n, B 2 ) ÄØ M 2 ÏÒ (i) (v), Đß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max Á M 1 Ï DP Ò 4. (i) batching-instant-job Ï Ω Đ n Ð ºÏ¹Ðº j Á M 2 Ï»ÒÒ ¼ j Ω = {z, z + B 2,, n}, z = n ( n/b 2 1)B 2, j ¹Рbatchinginstant-job. (ii) ߹ͺ (a) j Ð {1, 2,, j} Á M 1 j ÂÁͺÏÒ ¹Ð (b) m (m=arg max{r i i Ω, i j},» 1.3 À) Ð Á M 2 FOE(n, B 2 ) Ä m ÂÁ j ÍÒÏ batching-instant-job. (iii) ÐËÏ ĐÊ (ii) Ï¹ÍºØ M 1 ÏÒ«Ø (a) φ 1 (j, b) Á M 1 «ËØ b (b min{b 1, j}) ÏÐ M 1 j Ð ÏÒ «ºØ {j b + 1, j b + 2,, j}; (b) ϕ 1 (j) M 1 j Ð ÏÒ (c) φ 2 (j, b) Á M 1 j Ð («ËØ b) ÏÐ M 2 m Ð ÏÒ (d) ϕ 2 (j) Á M 1 j Ð ÏÐ M 2 m Ð ÏÒ (iv) Õ Ê (iii) Ï (a) (b), φ 1 (j, b) = max{ϕ 1 (j b), r j } + b 1. (1) M 1 «b Ð {j b + 1, j b + 2,, j} Ï» É Ω (M 2 É ÏÍà ), Ê (iii) Ï (c) (d), φ 2 (j, b) = ϕ 2 (j b). (2) M 1 «b Ð {j b + 1, j b + 2,, j} Ϲ É Ω (M 2 ¾ Ï ÍÃÕ), Ê (iii) Ï (a) (c) (d), φ 2 (j, b) = max{φ 1 (j, b), ϕ 2 (j b)} + v(j, b)b 2, (3)

476 31 v(j, b) Á {j b + 1, j b + 2,, j} batching-instant-job ÏË j Ð ÏҹР¼ j (Û Ø jth) j / Ω j Ω ÏÑ Ð (a) j / Ω j Ð Á M 1 Þ Ë M 2 µº Ð M 2 ËÁÉ ¹Ð j Ð Ï Ò M 1 Ï M 2 Ï ¹Ð batching-instant-job Ï (»Ê (iii) Ï (a) (b)) (»Ê (iii) Ï (c) (d)) min ϕ 1 1bj φ1 (j, b), j < B 1, (j) = min φ 1 (j, b), B 1 j n, 1bB 1 ϕ 2 (j) = min{φ 2 (j, b) φ 1 (j, b) = ϕ 1 (j)}. (5) (b) j Ω M 1 M 2 j Ð Ï Ò Ã¹Ò M 2 Ï ϕ 1 (j) ϕ 2 (j) ³ φ 2 (j, b)), (»Ê (iii) Ï (c) (d)) (»Ê (iii) Ï (c) (b)) (v) Ð min ϕ 2 1bj φ2 (j, b), j < B 1, (j) = min φ 2 (j, b), B 1 j n, 1bB 1 ϕ 1 (j) = min{φ 1 (j, b) φ 2 (j, b) = ϕ 2 (j)}. (7) ϕ 1 (0) = ϕ 2 (0) = 0, (8) φ 1 (j, b) = φ 2 (j, b) = ( j > n, ³b > B 1 ). (9) Î (iii) (a) (d) ÏÐ (iv) (v) (1) (9), Đ DP Ò 4. DP Ð ϕ 1 (0) = ϕ 2 (0) = 0. φ 1 (j, b) = φ 2 (j, b) = ( j > n, ³ b > B 1 ). Õ Đ (i) φ 1 (j, b) = max{ϕ 1 (j b), r j } + b 1. (ii) M 1 «b Ð (a)» É Ω, φ 2 (j, b) = ϕ 2 (j b); (b) ÂÁ É Ω, φ 2 (j, b) = max{φ 1 (j, b), ϕ 2 (j b)} + v(j, b)b 2. (iii) jth (j Ð Ï j Ð ): (4) (6)

4 ÆÓ Ò ÐÛÏÖ Ó 477 (a) j / Ω, min φ 1 1bj φ1 (j, b), j < B 1, (j) = min φ 1 (j, b), B 1 j n, 1bB 1 ϕ 2 (j) = min{φ 2 (j, b) φ 1 (j, b) = ϕ 1 (j)}; (b) j Ω, min ϕ 2 1bj φ2 (j, b), j < B 1, (j) = min φ 2 (j, b), B 1 j n, 1bB 1 ϕ 1 (j) = min{φ 1 (j, b) φ 2 (j, b) = ϕ 2 (j)}. DP ÒÏ ÓØ O(nB 1 ). ³ 4 [6] ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max, ¼ M 2 FOE(n, B 2 ) Ä ÏÒ Ã M 1 Ï DP Ò²Ò ÊÐ Đß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ÏÒ Ò m Ò Ð M 1 M 2 Ï µ DP Ò FOE(n, B 2 ) Ä 2.2 ÎÄ F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max H1 2.1 ÀÏÐ ÃÒ H1 1. H1 ¹ 1 Á 1 ¼ÁÌÒ ÏÒ ¹ 2 Á 1 Ï ÃØ Á 2 Ï M 1 ÏÍà M 1 M 2 µ¼ DP Ò FOE(n, B 2 ) Ä Ï DP Ò FOE(n, B 2 ) ÄÏ Ó µø O(nB 1 ) O(n), Ò H1 Ï ÓØ O(nB 1 ). ³ 1 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max Ø O(nB 1 ) à 1 n = 11, m = 5, a = 7, {B 1, B 2 } = {4, 3}, {b 1, b 2 } = {5, 3}. ÎÒ H1 DP Ò FOE(n, B 2 )) Ä µî M 1 M 2 ÏÒ ¼ DP Ò² M 1 ÏÒ 11 Ð Íà M 1 ÏØ r = { 1/5 7, 2/5 7,, 11/5 7} = {7, 7, 7, 7, 7, 14, 14, 14, 14, 14, 21}(r j = j/m a (j N)), batching-instant-jobs Ϻ Ω = {2, 5, 8, 11}, φ 1 (j, b) = φ 1 (1, 1) j = 1, (1) (2) (4) (5) (8)( 1 Ï j = 1), φ 1 (1, 1) = max{ϕ 1 (0), r 1 } + b 1 = max{0, 7} + 5 = 12, φ 2 (1, 1) = φ 2 (0) = 0, ϕ 1 (1) = φ 1 (1, 1) = 12, ϕ 2 (1) = 0.

4 ÆÓ Ò ÐÛÏÖ Ó 479 1 Ï j = 11 M 1 M 2 ÏÒ µ µø ϕ 1 (11) = 26 ϕ 2 (11) = 29. (Ô 1). Ç 1 H1 1 ÜÇ 2 j = 11: ³ (j, b) = (11, 3), 11 Ð «ËØ 3, Î M 1 ÏÒ«{J 9, J 10, J 11 }, Ò µ ϕ 1 (11) = 26. j = 8: 11 Ð «3 Ð Ï 8 Ð ³ (j, b) = (8, 4), Î 8 Ð ÏÒ«{J 5, J 6, J 7, J 8 }, Ò µ ϕ 1 (8) = 19. j = 4: Òß Ï 4 Ð ³ (j, b) = (4, 4), ÎÒÄ {J 1, J 2, J 3, J 4 }, Ò µ ϕ 1 (4) = 12. 26. M 1 ÏÒ Ø {{1, 2, 3, 4)}, {5, 6, 7, 8}, {9, 10, 11}}, Ò µ ϕ 1 (11) = 3 M 2 ¼ 3 Ï FOE(n, B 2 ) =FOE(11, 3) = {{1, 2}, {3, 4, 5}, {6, 7, 8}, {9,10, 11}} Ä C max = φ 2 (11) = 29. 2.3 ÎÄ F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ز ß H2 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max, Ø (i)a m/b 1 b 1 ; (ii) a m/b 1 b 1 ; (iii) m/b 1 b 1 < a < m/b 1 b 1 Æ Ð ( 2 Ï 2),

480 31 Æ µ ÏÒ H2(1) 2 Ò H2(2) 3 Ò H2(3) 4. Á H2(1) (3) ÏÆÒ Ò H2(3)( 2 Ï 2 Ï (3)) B 1 > B 2, b 1 > b 2 (³ B 1 < B 2, b 1 < b 2 ) Ï ÓØ O(nB 1 ), ÏÒ Ó Ø O(n). Î 1, 1 Ï m Ò ÕÐ Ê ( ) (Å) ÏÆ µá M 1 M 2 Ï ÅÐ ( ) a m/b 1 b 1 H2(1) ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max a m/b 1 b 1, 1 ÏÒ m Ð («Ë m) ¼ 2 ¼ LOE(m, B 1 ) Ä À 3. b 1 B 1 /B 2 b 2, n Ð Á M 2 ¼ LOE(B 1, B 2 ) Ä b 1 B 1 /B 2 b 2, Û m Ð Á M 2 Ð (i) a m/b 2 b 2, n Ð Á M 2 ¼ LOE(m, B 2 ) Ä (ii) a m/b 2 b 2, n Ð Á M 2 ¼ LOE(n, B 2 ) ÄÐ (iii) m/b 2 b 2 < a < m/b 2 b 2, n Ð Á M 2 ¼ FOE(n, B 2 ) Ä B 1 /B 2 b 2 < b 1 < B 1 /B 2 b 2, n Ð Á M 2 ¼ FOE(n, B 2 ) Ä Ò H2(1)- Ò H2(1)- Ò H2(1)- (Ò H2(1)- Ò) Ï ÓØ O(n), Ò H2(1) Ï ÓØ O(n). ³ 2 a m/b 1 b 1 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max Á O(n) à Խ Ø a m/b 1 b 1, Ò m Ð M 1 Á a ( M 1 ÄÐ a [a, 2a] LOE(m, B 1 )) Ä Û m Ð Ð ÁÒÈ 3 Ï M 2 b 1 B 1 /B 2 b 2 M 2 ¼ LOE(B 1, B 2 ) Ä Á b 1 ( M 2 ÄÐ b 1 [a+b 1, a+2b 1 ]) B 1 Ð Ð M 2 Á ¹Ï B 1 Ð ÍÃÍ Ò ¹Ï B 1 Ð (Ô 2) n n/m m Cmax = n/m a + b 1 + B 1 n n/m m (n n/m m)/b1 B 1 Ì B 1 B 2, b 1 b 2 (n n/m m (n n/m m)/b 1 B 1 ) < B 1 Ø B 2 C max = n/m a + (n n/m m)/b 1 b 1 + b 2. b 2. b 1 B 1 /B 2 b 2 2 ½ Ï m Ð Á M 2 Ð Ëß n Ð Á M 2 ÂÚ Ð (i) a m/b 2 b 2

4 ÆÓ Ò ÐÛÏÖ Ó 481 a [ ] 2a [ ] [/ ] [ / ] Ç 2 H2(1)- M 2 ¼ LOE(m, B 2 ) Ä Á a ( M 2 ÄÐ a [a + b 1, 2a + b 1 ]) Û m Ð Ð M 2 Á ¹Ï m Ð ÍÃÍ Ò Û m Ð C max = n/m a + b 1 + n n/m m)/b 2 b 2 (Ô 3). [] [ ] [ ] [/ ] [ / ] [/ ] Ç 3 H2(1)- (i) (ii) a m/b 2 b 2 M 2 ¼ LOE(m, B 2 ) Ä Á a ( M 2 ÄÐ a [a + b 1, 2a + b 1 ]) m/b 2 B 2 Ð ¾ n Ð Á M 2 Ð ¼ LOE(n, B 2 ) Ä µ C max Ò C max = a + b 1 + n/b 2 b 2 (Ô 4). (iii) m/b 2 b 2 < a < m/b 2 b 2

482 31 Î 3, ¼ FOE(n, B 2 ) ÄÎ M 2 ÏÒ B 1 /B 2 b 2 < b 1 < B 1 /B 2 b 2 [ ] [ ] [] [] [/ ] [ / ] [/ ] Ç 4 H2(1)- (ii) Î 3, M 2 ¼ FOE(n, B 2 ) Ä ÎÒ (Å) a m/b 1 b 1 H2(2) ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max a m/b 1 b 1, 2 ¼ LOE(n, B 1 ) Ä Î M 1 ÏÒ 3: b 1 B 1 /B 2 b 2, n Ð Á M 2 ¼ LOE(B 1, B 2 ) Ä b 1 B 1 /B 2 b 2, n Ð Á M 2 ¼ LOE(n, B 2 )) ÄÐ B 1 /B 2 b 2 < b 1 < B 1 /B 2 b 2, n Ð Á M 2 ¼ FOE(n, B 2 ) Ä Ò H2(2)- Ï ÓØ O(n), Ò H2(2) Ï ÓØ O(n). ³ 3 a m/b 1 b 1 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max Á O(n) à Խ a m/b 1 b 1, Ò m Ð Û m Ð Í M 1 M 1 Á a ( M 1 ÄÐ a [a, 2a]) m/b 1 B 1 ( m) Ð LOE(n, B 1 ) ÄÎ M 1 ÏÒ ß 3: b 1 B 1 /B 2 b 2 M 2 ¼ LOE(B 1, B 2 ) Ä Á b 1 ( M 2 ÄÐ b 1 [a+b 1, a+2b 1 ]) B 1 Ð M 2 Á ¹Ï B 1 Ð ÍÃÍ Ò ¹Ï B 1 Ð C max = a + n/b 1 b 1 + (n n/b 1 B 1 )/B 2 b 2 (Ô 5). b 1 B 1 /B 2 b 2

4 ÆÓ Ò ÐÛÏÖ Ó 483 M 2 Á b 1 ( M 2 ÄÐ b 1 [a + b 1, a + 2b 1 ]) B 1 /B 2 B 2 ( B 1 ) Ð 2 ½ Ï n Ð Á M 2 ¼ LOE(n, B 2 ) Ä Ð B 1 /B 2 b 2 < b 1 < B 1 /B 2 b 2 C max = a + b 1 + n/b 2 b 2 (Ô 6). Î 3, M 2 ¼ FOE(n, B 2 ) Ä ÎÒ [] [] [ ] [ ] [/ ] [ / ] Ç 5 H2(2)- [] [] [ ] [ ] [/ ] [ / ] Ç 6 H2(2)- («) m/b 1 b 1 < a < m/b 1 b 1 2.1 ÀÏÐ Á M 2 ¼ FOE(n, B 2 ) ÄÏÒµĐ M 1 ÏÒ ²ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max ÏÒ Đ B 1 B 2, b 1 b 2

484 31 B 1 B 2, b 1 b 2 ÑÆ Á M 1 ÏÒ H2(3)-a b 5 6. H2(3)-a B 1 B 2, b 1 b 2 r j = j/m a (j N) ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max Ï M 1 FOE(n, B 1 ) Ä Ò H2(3)-a Ï ÓØ O(n). ³ 5 B 1 B 2, b 1 b 2 r j = j/m a (j N) Ò H2(3)-a ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max Î M 1 Ï O(n) Ò H2(3)-b B 1 B 2, b 1 b 2 r j = j/m a (j N) ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max Ï M 1 FOE(n, B 2 )) Ä Ò H2(3)-b Ï ÓØ O(n). ³ 6 B 1 B 2, b 1 b 2 r j = j/m a (j N) Ò H2(3)-b ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max Î M 1 Ï O(n) Ò ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ß F2(B 1, B 2 ) r j, b 1j b 1, b 2j b 2 C max. ÎÒ H2(3)-a b 2.1 ÀÏÒ H1( DP Ò FOE(n, B 2 ) Ä), Đß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max m/b 1 b 1 < a < m/b 1 b 1 ÏÒÒ H2(3) 4. H2(3) Ä ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max m/b 1 b 1 < a < m/b 1 b 1 : B 1 B 2, b 1 b 2, M 1 M 2 µ FOE(n, B 1 ) Ä FOE(n, B 2 ) Ä B 1 B 2, b 1 b 2, M 1 M 2 FOE(n, B 2 ) Ä B 1 >B 2, b 1 >b 2 (³ B 1 <B 2, b 1 <b 2 ), M 1 M 2 µ DP Ò FOE(n, B 2 ) Ò H2(3)- Ï ÓØ O(n), Ò H2(3)- Ï ÓØ O(nB 1 ), Ò H2(3) Ï ÓØ O(nB 1 ). ³ 4 m/b 1 b 1 < a < m/b 1 b 1 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max B 1 > B 2, b 1 > b 2 (³ B 1 < B 2, b 1 < b 2 ) Á O(nB 1 ) ÃÖ Ø O(n) Ã Ò H2(1) (3) 2 4, µîò H2 5. H2 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max : (i) a m/b 1 b 1, Ò H2(1); (ii) a m/b 1 b 1, Ò H2(2); (iii) m/b 1 b 1 < a < m/b 1 b 1, Ò H2(3). Ò H2(1) (2) Ï ÓØ O(n), Ò H2(3) Ï ÓØ O(nB 1 ), Ò H2 Ï ÓØ O(nB 1 ). ³ 5 ß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max Á O(nB 1 ) Ã

4 ÆÓ Ò ÐÛÏÖ Ó 485 Ò H2 ÎÍÏ 5 Ò H1 ÎÍÏ 1, ÑÒÏ ÓØ O(nB 1 ), ËÒ H2 Ò H1 µ¹ò H2 Âß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ØÆ ( 2 Ï 2: (I) a m/b 1 b 1, (II) a m/b 1 b 1, (III) m/b 1 b 1 < a < m/b 1 b 1 ) À¹Ð Æ m/b 1 b 1 < a < m/b 1 b 1 ÏÑ Ò H2(3)- Ï ÓØ O(n), Ç Ò H2(3)- Ï ÓØ O(nB 1 ). 3 º Þß F3(m, B 1, B 2 ) a j a, b 1j b 1, b 2j b 2 C max ÈÕÐ ĐÕ ¹ Ï O(nB 1 ) Ò½Æ Ï O(n) Ò ( 2). ß F3(m, B 1, B 2 ) b 1j b 1, b 2j b 2 C max F3(m, B 1, B 2 ) a j a, b 1j b 1 C max ¾ ÞÐ Û 2 ËÁ F3(m, B 1, B 2) a j a, b 1j b 1, b 2j b 2 C max Ü (Ú) ¼Í Ô Ô () 1 (») : m, B 1, B 2 ; a j = a, b 1j = b 1, b 2j = b 2 H1 O(nB 1 ) 1 b 1 B 1 /B 2 b 2 H2(1)- (i) a m/b 2 b 2 H2(1)- (i) (I) a m/b 1 b 1 b 1 B 1 (ii) a m/b 2 b 2 H2(1)- (ii) O(n) 2 (H2(1)) /B 2 b 2 (iii) m/b 2 b 2 <a< m/b 2 b 2 H2(1)- (iii) B 1 /B 2 b 2 < b 1 < B 1 /B 2 b 2 H2(1)- 2 (È) b 1 B 1 /B 2 b 2 H2(2)- (II) a m/b 1 b 1 b 1 B 1 /B 2 b 2 H2(2)- O(n) 3 (H2(2)) B 1 /B 2 b 2 < b 1 < B 1 /B 2 b 2 H2(2)- (III) m/b 1 b 1 < a B 1 B 2, b 1 b 2 H2(3)- O(n) < m/b 1 b 1 B 1 B 2, b 1 b 2 H2(3)- 4 (H2(3)) B 1 > B 2, b 1 > b 2 ( B 1 < B 2, b 1 < b 2 ) H2(3)- O(nB 1 ) (): m, B 1, B 2 ; a j = a, b 1j = b 1, b 2j = b 2 H2 O(nB 1 ) 5 Ö ÑܺÑÜ ¹ĐÕ»ÎÏÜ Á Ý ÜÉ (ÔÄ) [1] Garey M R, Johnson D S. Strong NP-completeness results: motivation, examples and implications [J]. Journal of the Association for Computing Machinery, 1978, 25: 499-508. [2] Johnson S M. Optimal two- and three-stage production schedules with setup times included [J]. Naval Research Logistics, 1954, 1: 61-68. [3] Lee C Y, Uzsoy R. Minimizing makespan on a single batch processing machine with dynamic job arrivals [J]. International Journal of Production Research, 1999, 37: 219-236. [4] Sung C S, Choung Y I. Minimizing makespan on a single burn-in oven in semiconductor manufacturing [J]. European Journal of Operational Research, 2000, 120: 559-574.

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