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ひろみきせんばる
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1 Holt Modigliani Muth Simon 1960 HMMS HMMS 2 recovery reuse reverse Thierry 1995 direct reuse repair recycling remanufacturing 4

2 container Minner Kistner Dobos Minner Kleber 2001 Dobos 2003 Kistner Dobos Dobos

3 N 2 Two-store Reverse Logistics Store1) α Store2 Store2 Store1 1 Store1 Store2 3 αˆ i 1 (k +1)=i 1 (k)+p m (k)+p r (k) d(k) (1a) i 2 (k +1)=i 2 (k) p r (k) s(k)+r(k) (1b) Dobos 2003

4 i 1 (k) k Store1 p m (k) k p r (k) Store2 k d(k) k i 2 (k) k Store2 s(k) k r(k) k d(k τ) d(k) α(k) αˆ r(k) = α(k)d(k) (2) i 1 (k +1) i 2 (k +1) = i 1 (k) i 2 (k) p m (k) p r (k) s(k) + 1 α(k) d(k) (3a) i(k + 1) = Ai(k) + Bp(k) + c(k)d(k) (3b) i(k) T = [i 1 (k) i 2 (k)] T 2 p(k) T = [p m (k) p r (k) s(k)] T 3 ( ) T A B c(k) A = B = c(k) = 1 α(k) (4) 1 (c m ) 1 (c u ) 1 (h 1 ) 1 (h 2 ) 1 (c d ) 1 (c r ) TC

5 N 1 TC = 1 [h 1 (i 1 (k) ī 1 (k)) 2 + h 2 (i 2 (k) ī 2 (k)) 2 ] 2 k=0 N [c m (p m (k) p m (k)) 2 + c u (p r (k) p r (k)) 2 + c d (s(k) s(k)) 2 + c r α(k)d(k)] 2 k= [h 1(i 1 (N) ī 1 (N)) 2 + h 2 (i 2 (N) ī 2 (N)) 2 ] (5) N 1 TC = 1 [(i(k) 2 ī(k))t Q(i(k) ī(k)) + (p(k) p(k))t R(p(k) p (k)) + c r α(k)d(k)] k= (i(n) ī(n))t Q(i(N) ī(n)) (6) ī (k) 2 p(k) 3 Q R ī(k) = īi(k) ī 2 (k), p(k) = p m (k) p r (k) s(k), Q = h h 2, R = c m c u 0 (7) 0 0 c d sδ α l = αˆ sδ α(k) α u = αˆ + sδ (8)

6 d(k) c(k) α(k) α * (k) N 1 N 1 V (N 1) V (N 1) = 1 2 [(i(n 1) ī(n 1))T Q(i(N 1) ī(n 1)) +( p(n 1) p(n 1)) T R(p(N 1) p(n 1))] [{Ai(N 1) + Bp(N 1) + c(n 1)d(N 1) ī(n)}t P (N) {Ai(N 1) + Bp(N 1) + c(n 1)d(N 1) ī(n)}] (9) V (N 1) p(n 1) V (N 1) p(n 1) = R(p(N 1) p(n 1)) + [B T P (N){Ai(N 1) + Bp(N 1) +c(n 1)d(N 1) ī(n)}] =0 (10) N 1 p (N 1) = {B T P (N)B + R} 1 [B T P (N)Ai(N 1) + B T P (N)c(N 1)d(N 1) R p(n 1) B T P (N)ī(N)] K(N)i(N 1) + L(N)d(N 1) + M(N) p(n 1) + N(N)P (N)ī(N) (11) p (N 1) V (N 1) N 1 V (N 1) = 1 2 i(n 1)T P (N 1)i(N 1) + g(n 1) T i(n 1) + η(n) (12) P P (N 1) = Q + K T RK +(A + BK) T P (N)(A + BK), P (N) =Q (13) P (N) =Q g(n 1) = Qī (N 1) + {K(N ) T RL(N ) + (A+BK(N )) T P(N )(BL(N ) + c(k ))}d(n 1) + {K(N ) T R(M(N ) I) + (A+BK(N )) T P(N )(BM(N )} p (N 1) + {K(N ) T RN(N ) + (A+BK(N )) T P(N )(BN(N )}g (N ), (14) g (N ) = P(N )ī (N )

7 η (N 1) = 1 2 [ ī (N 1)T Qī (N 1) + {L(N )d(n 1) + (M(N ) I) p (N 1) + N(N )g(n )} T R {L(N )d(n 1) + (M(N ) I) p (N 1) + N(N )g(n )} + {(BL(N) + c(n 1))d(N 1) + BM(N) p(n 1) + BN(N)g(N)} T P(N) {(BL(N) + c(n 1))d(N 1) + BM(N) p(n 1) + BN(N)g(N)}] + g(n ) T {(BL(N ) + c(n 1))d(N 1)+BM(N ) p(n 1)+BN(N )g(n )} + η (N ), (15) η (N ) = ī (N )T Qī (N ) / 2 N 1 V (N 1) α(n 1) ν (α, N 1) ν (α, N 1) = d(n 1){L(N ) T RK(N)+(BL(N)+c(N 1)) T P(N)(A+BK(N ))}i(n 1) [{L(N )T RL(N)+(BL(N)+c(N 1)) T P(N)(BL(N)+c(N 1))}d(N 1) 2 + 2ī (N )T P(N){N(N ) T RL(N )+(BN(N) I) T P(N)(BL(N)+c(N 1))} d(n 1) + 2d(N 1){L(N ) T R(M(N) I)+(BL(N) + c(n 1)) T P(N)BM(N)} p(n 1) = d(n 1){c(N 1) T P(N)(I B{B T P(N )B + R} 1 B T P(N ))A}i(N 1) [{c(n 1)T (P(N) 1 +BR 1 B T )c(n 1)}d(N 1) 2 + 2ī (N )T {P(N )(B{B T P(N)B + R} 1 P(N ) I )c(n 1))}d(N 1) + 2d(N 1){c(N 1) T P(N)B{B T P(N)B + R} 1 R} p(n 1)] (16) d(n 1) 2 {L(N) T RL(N) + (BL(N) + c(n 1)) T P(N)(BL(N) + c(n 1))} = c(n 1) T P(N) B{B T P(N )B + R} 1 R{B T P(N )B + R} 1 B T P(N )c(n 1) + c(n 1) T P(N) B{B T P(N )B + R} 1 B T P(N )(B{B T P(k + 1)B + R} 1 B T P(N )c(n 1) c(n 1) T P(N) B{B T P(N )B + R} 1 P(N )c(n 1)) c(n 1) T P(N) B{B T P(N )B + R} 1 P(N )c(n 1)) + c(n 1) T P(N )c(n 1) = c(n 1) T [P(N ) 1 + BR 1 B T ]c(n 1) (17) P(N) R α(k) 2 P(N) 1 +BR 1 B T (3, 3) p r r 33 1 ν (α, N 1) α 2

8 V(N 1) α l = αˆ sδ α(k) α u = αˆ + sδ α = α l α = α u ν (α, N 1) c(n 1) dν(α, N 1) dc(n 1) =0 (18) α α 0 (N 1) α 0 (N 1) αˆ ν (α, N 1) α * (N 1) = α l α 0 (N 1) αˆ α * (N 1) = α u ν (α, N 1) α 0 (N 1) N 1 α 0 (N 1) i(n 1) α 0 (N 1) i(n 1) α 0 (N 1) α * (N 1) i(n 1) ν (α u, N 1) ν (α l, N 1) ν(α, N 1) α * (N 1) V(N 1) α * (N 1) c(n 1) c * (N 1) c * (N 1) L(N) L * (N) L * (N) P(N 1) P * (N 1) N 1 p * (N 1) p * (N 1) = K * (N)i(N 1) + L * (N)d(N 1) + M * (N) p(n 1) + N * (N)P(N )ī(n ) (19) K * (N ) = {B T P * (N)B + R} 1 B T P * (N)A (20a) L * (N ) = {B T P * (N)B + R} 1 B T P * (N)c * (N 1) (20b) M * (N ) = {B T P * (N)B + R} 1 R N * (N ) = {B T P * (N)B + R} 1 B T P * (N) = Q, g(n) = P * (N)ī(N ) (20c) (20d) N 1 0 k V(k) { α * (k); k = N 1,, 0 } c(k) (2, 1) α(k) α * (k) c * (k) c * (k) L(k + 1) L * (k + 1) L * (k + 1) P(k) P * (k) k p * (k ) p * (k ) = K * (k + 1)i(k) + L * (k + 1)d(k) +M * (k + 1) p(k) + N * (k + 1)g * (k + 1) (21)

9 K * (k + 1) = {B T P * (k + 1)B + R} 1 B T P * (k + 1)A (22a) L * (k + 1) = {B T P * (k + 1)B + R} 1 B T P * (k + 1)c * (k) (22b) M * (k + 1) = {B T P * (k + 1)B + R} 1 R (22c) N * (k + 1) = {B T P * (k + 1)B + R} 1 B T (22d) P * (k ) = Q + K * (k + 1) T RK * (k + 1)+(A + BK * (k + 1)) T P * (k + 1)(A + BK * (k + 1)) (23) P * (N) = P(N) = Q g * (k) = Qī(k) + {K * (k + 1) T RL * (k + 1)+(A + BK * (k + 1)) T P * (k + 1)BL * (k + 1)+c * (k ))}d(k) + {K * (k + 1) T R(M * (k + 1) I)+(A + BK * (k + 1)) T P * (k + 1)BM * (k + 1)} p(k) + {K * (k + 1) T RN * (k + 1)+(A + BK * (k + 1)) T P * (k + 1)BN * (k + 1)}g * (k + 1) + (A + BK * (k + 1)) T g * (k + 1), (24) g * (N) = g(n) = P(N)ī(N ) αˆ sδ α l = αˆ sδ α(k) α u = αˆ + sδ s δ The Proposed Manufacturing/Reuse/Disposal Policy; PMRDP { α(k); k = 0,, N 1} The Optimal Manufacturing/Reuse/Disposal Policy; OMRDP αˆ The Sub-optimal Manufacturing/Reuse/Disposal Policy; SMRDP Store1 i 1 (0) = 0.7 Store2 i 2 (0) = 0.5 {d(k) = 0.4; k = 0,, 9} {ī1(k) = 0.4; k = 0,, 9} {ī2(k) = 0.3; k = 0,, 9} { p m (k) = 0.3; k = 0,, 9} { p r (k) = 0.3; k = 0,, 9} { s(k) = 0.2; k = 0,, 9} 1 h 1 = h 2 = c m = c u = c d = c r = 2.0 αˆ δ

10 1 2

11 αˆ δ s TC of OMRDP TC of SMRDP TC of PMRDP

12 [1] Dobos, I Optimal Production-inventory Strategies for a Reverse Logistics System, Int. J. Production Economics 81 82, [2] Holt, C.C., Modigliani, F., Muth, J.F., and Simon, H.A Planning Production, Inventories, and Work Force, Prentice-Hall, Inc., Englewood Clffs, N.J. [3] Kistner, K-P. and Dobos, I Optimal Production-Inventory Strategies for a Reverse Logistics System, Optimization, Dynamics, and Economic Analysis. Essays in Honor of Gustav Feichtinger, Dockner, E.J., Hartl, R.F., Luptacik, M., and Sorger, G.(eds), Physica-Verlag [4] Minner, S. and Kleber, R Optimal Control of Production and remanufacturing in a Simple Recovery Model with Linear Cost Functions, OR Spektrum 23, [5] Minner, S Multiple-supplier Inventory Models in Supply Chain Management: A Review, Int. J. Production Economics 81 82,

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy