Lower bounds on the approximation ratios of leading heuristics for the single machine total tardiness problem
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(112) . 1|| Tj N = {1, 2, . . . , n} n j pj dj !. " S ∗ = (1, 2, . . . , n) n n ∗ T (N, S ) = i=1 Ti = i=1 max{Ci − di , 0} Ci = ni=1 pi ! 1|| Tj # $%&' ( )
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(135) $ Bn = {[1], [2], . . . , [k − 1], [k + 1], . . . , [h]} An = {[h + 1], . . . , [n]} h % & Cn (h) = j=1 p[j] n [k] h k ' . . !. Cn (h) d[h+1] h < n( Cn (h) < d[h] + p[h] h > k ( Cn (h) d[r] + p[r]
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(143) ! rEDD n " ! N 4 " SEDD N # Tmax (N, SEDD )
(144) ) SEDD # S ∗ 1|| Tj N , 4 2 1||Tmax
(145) Tmax (N, SEDD ) Tmax (N, S ∗ ) T (N, S ∗ ) > T (N, SEDD ) nTmax (N, SEDD ) nT (N, S ∗ )
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(148) ) $E1 ': N = {1, 2, . . . , n} p1 = m p2 , . . . , pn = 1 d1 = 0 d2 , . . . , dn = ε ! " S ∗ = (2, . . . , n, 1) T (N, S ∗ ) = n(n + 1)/2 + m − 1 − (n − 1)ε ! 4 " SEDD = (1, 2, . . . , n) T1 = m Ti = m + i − 1 − ε i = 2, . . . , n ! T (N, SEDD ) = nm + n(n − 1)/2 − (n − 1)ε ? m ε
(149) rEDD (N, SEDD ) ≈ n
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(154) 6 $%&': t i j max{t+pi , di } < max{t+pj , dj } max{t+pi , di } = max{t + pj , dj } pi < pj 789: 87! " ; ) .
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(157) . 7 # ': 1 dj p(S) + pj p(S ∪ E) − d j p(S) + pj < dj < p(S ∪ E) PIj = p(E) − pj 0 p(S ∪ E) dj ; .
(158) PIj /pj ! 1|| wj Tj ,D $%.&': nt i j ui > uj ui = exp[− max{di − t − pi , 0}/k p¯]/pi p¯ =. #
(159) 1 COVERT i=1 pi /n ! . 1|| wj Tj 1 . 7 # E>C: 4 1 # 7 # E > C ; %-&.
(160) !. rMDD = rPSK = rWI = rCOVERT n/2. ) # E2 : N = {1, 2, . . . , n + 1} p1 = n p2 , . . . , pn+1 = 1 d1 = n d2 , . . . , dn+1 = n + ε ! 6 t = 0 , $ ' ? 4 " S = (1, . . . , n + 1) T1 = 0 Ti = i − 1 − ε i = 2, . . . , n + 1 ! TMDD (N, S) = n(n + 1)/2 − nε ! " S ∗ = (2, . . . , n + 1, 1) T (N, S ∗ ) = n ? ε rMDD (N, S) ≈ n/2 , %/& %0& 789 ( # "
(161) 6 ) E2 i j pi = pj di = dj max{t + pi , di } = max{t + pj , dj } ( 789 ( 6 ) E2 # AB4C! 7 # 2 → 3 → · · · → n + 1
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(163) , PI1 = 1 S = ∅ E = {1, 2} ! p(S) = 0 p(E) = n + 1 PI2 = (1 − ε)/n ? # " SCOVERT = (1, 2, . . . , n + 1) . .
(164) . ! rAU nk
(165) . . k 1. ) E3 : N = {1, 2, . . . , n + 1} p1 = n p2 , . . . , pn+1 = ε d1 = n d2 , . . . , dn+1 = n − 1 E p¯ = (n + nε)/n ≈ 1 n
(166) S ∗ = (2, . . . , n, n + 1, 1) T (N, S ∗ ) = nε ,# ,D t = 0 : u1 = 1/n i = 2, . . . , n ui = (1/ε)(exp[−(n − 1 − ε)/(k p¯)]) ε = n−k n
(167) ui < u1 ! $ ' " ) ! " S = (1, 2, . . . , n) T (N, S) = n − nε
(168) rAU (N, SAU ) ≈ 1/ε = nk .
(169) . ! rNBR n/6
(170) ) # E4 : n = 2m + 2 N = {1, 2, . . . , 2m + 2} p1 = m p2 = 1 p3 = · · · = pm+1 = ε pm+2 = · · · = p2m+1 = 1 p2m+2 = 2ε d1 = m d2 = m + (ε/2) di = m + (i − 2)ε i = 3, . . . , m + 1 dm+2 = m + 1 + (m − 1)ε dj = j + (m − 1)ε j = m + 3, . . . , 2m d2m+1 = d2m+2 = 2m + (m − 1)ε ! E>C # " S¯ i1 < i2 < · · · < ik ¯ S Tik < pik pi1 > pi2 > · · · > pik , i1 , . . . , ik−1
(171) ik ) $ %-& ' E. ) E4 E>C ) 1, 2m + 1, 2m + 2
(172) (m+1)+(m+1)ε ! " # E>C SNBR = (2, 3, . . . , 2m+2, 1) T (N, SNBR ) = (m+1)+(m+1)ε ! " .
(173) S ∗ = (2, 3, . . . , m + 1, 1, m + 2, m + 3, . . . , 2m + 2) T (N, S ∗ ) = 3 + (m + 1)ε C n = 2m + 2
(174) rNBR (N, SNBR ) = ((m + 1) + (m + 1)ε)/(3 + 2ε) m/3 ≈ n/6. ! $ %0 +& > 1 1|| Tj 454: ) 7 ; k . # 4
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(179) ) # E5 : n = 2m + 1 N = {1, 2, . . . , 2m + 1} p1 = m − ε p2 = · · · = pm+1 = 1 pm+2 = · · · = p2m+1 = ε d1 = m di = m + i − 1 i = 2, . . . , m dm+1 = 2m − 1 di = 2m − 1 + ε i = m + 2, . . . , 2m + 1
(180) SEDD = (1, . . . , 2m + 1) F ! C1 (r) r m+r−1−ε r = 1, . . . , m + 1 C1 (r) = 2m + (r − m − 2)ε r = m + 2, . . . , 2m + 1. 7 r = 2, . . . , m + 1 # dr + pr > C1 (r); r = m + 2, . . . , 2m # C1 (r) > dr+1 F 2m + 1 " µ = (1, . . . , 2m + 1) θ = (2, . . . , 2m + 1, 1) T (N, µ) = (1 − ε) + εm(m − 1)/2 − (m − 2)ε + (m + 1) T (N, θ) = m + (m − 2)ε ? T (N, µ) > T (N, θ) 2m + 1 m + (m − 2)ε ! " (2, . . . , 2m) # # # 8 SDEC/EDD = θ > " S ∗ = (1, m + 2, . . . , 2m + 1, 2, . . . , m + 1) T (N, S ∗ ) = 1 + m(m − 1)ε ? ε rDEC/EDD (N, SDEC/EDD ) = (m + (m − 2)ε)/(1 + m(m − 1)ε) ≈ m ≈ n/2. ! rDEC/MDD = rDEC/PSK = rDEC/WI n/3.
(181) ) E6 : n = 3m/2 N = {1, 2, . . . , 3m/2} p1 = m2 pi = 2m i = 2, . . . , m/2 pj = 2 j = (m/2) + 1, . . . , 3m/2 d1 = m2 di = m2 + 2(i − 1)m i = 2, . . . , (m/2) − 1 dm/2 = 2m2 − 2m − ε dj = 2m2 − 2m + ε j = (m/2) + 1, . . . , 3m/2 F 3m/2 # ( + 6 " (1, 2, . . . , 3m/2)
(182) m(m + 1) − (m − 1)ε = m2 + m − (m − 1)ε ( m2 # m2 ? ε ( SDEC/MDD = (2, . . . , 3m/2, 1) T (N, SDEC/MDD ) = m2 ?
(183) " ) S ∗ = (1, 2, . . . , (m/2)−1, (m/2)+1, . . . , 3m/2, m/2)
(184) 2m+ε C n = 3m/2
(185) rDEC/MDD (N, S) = m2 /(2m + ε) ≈ m/2 = n/3 , # 7
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