Decisions for a three-echelon supply chain with logistics outsourcing under a dominant retailer

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1 JOURNAL OF SYSTEMS ENGINEERING Vol.31 No.2 Apr ,2, 1,2, 1,2, 1,2, 1,2 (1., ; 2., ) :,.,,., 3, ;, ;. : ; 3 ; ; : F224 : A : (2016) doi: /j.cnki.jse Decisions for a three-echelon supply chain with logistics outsourcing under a dominant retailer Chen Xiaoxu 1,2, Wang Yong 1,2, Sun Hailei 1,2, Chen Zhangyue 1,2, Wang Yili 1,2 (1. School of Economics and Business Administration, Chongqing University, Chongqing , China; 2. Chongqing Key Laboratory of Logistics, Chognqing University, Chognqing , China) Abstract: When the retailer s order quantity is sensitive to the manufacture s wholesale price and the 3PL s service price simultaneously, a supply chain model dominated by retailer with logistics outsourcing is built. The optimal prices and order quantity are acquired by using game theory. Furthermore, the impact of the two sensitivity coefficients, the demand range and the allocation ratio of logistics cost on the decision variables are analyzed. Results show that the retailer can inhibit the manufacturer and 3PL to raise their prices and increase his profit, by controlling the two sensitivity coefficients. As the demand changes, all supply chain members decisions and profits increase or decrease simultaneously. Moreover, the one who bears the ratio of the logistics fee more gets more profit. Key words: retailer dominanting; the third party logistics provider; supply chain; game equilibrium 1, 3 (3PL) [1]., 42% 3PL,, 70% [2]. : ; : : ( ; ); (12YJA630135); (CQKLL12002).

2 2 : % 3PL [3],, 3PL [4 6]., [7],, , ,,,, 3PL., ; 3PL,.,. 21,,, 3, 3PL 3.,, 3. Cai [8] 3PL 3,, WMC WDC ; Woo [9] 3, 3PL,,, ; Chen [10] 3PL,, 3PL,, 3PL, 3PL ; [11] 3,, 3 ; [12] 3,,, ; Lei [13], EOQ,, 3.,,. Wang [14], ; Pan [15],,, ; [16], ; [17],,,, Prateo, ; [18] 3 (, 3 ),.,.,.,, 3PL ( ), 3PL, 3PL 3., 3PL.,

3 PL 3,,, 3PL,. [19], [20, 21], 3PL q = q 0 αw βs, q 0, α > 0,β > 0 w 3PL s., 3PL,. ( 3PL), 3PL,, 3PL. : 1),,, ; 2) 3PL, ; 3) 3PL, 3PL, 3PL ; 4) [I 1,I 2 ](0 < I 1 < I 2 ), µ,σ, I 1 = µ 3σ,I2 = µ + 3σ; 5) 3PL ; 6). : d( ), d > 0 f( ),F( ); v 3PL ; c ; p ; θ 1 (0 θ 1 1) ; θ 2 (0 θ 2 1) ( θ 1 + θ 2 = 1 ); k r ; k m ; k l 3PL ; : q 0 ; q ; w ; s 3PL ;

4 2 : 257,, pd (w + θ 1 s)(q 0 αw βs) k r, d q π r = (p w θ 1 s)(q 0 αw βs) k r, d > q., d q,,,,, ( ), ;, d > q,,. ( ),. E[π r ] = (p w θ 1 s)(q 0 αw βs) k r p q 0 (q 0 αw βs x)f(x) dx. (1), w,, 3PL., π m = (w c θ 2 s)(q 0 αw βs) k m. (2),, 3PL. s., 3PL, π l = (s v)(q 0 αw βs) k l. (3) q 0 w, 3PL, q 0 w 3PL s, q = q 0 αw βs., (, 3PL ). 1, q 0 ; 2, w ; 3, 3PL s 3PL ;, q = q 0 αw βs., Max E[π r ] = (p w θ 1 s)(q 0 αw βs) k r p q (q q 0 0 αw βs x)f(x) dx 0 s.t. Max π m = (w c θ 2 s)(q 0 αw βs) k m w Max π l = (s v)(q 0 αw βs) k l. s 2.2 3PL 3PL, 3, 3PL (3),, (3) s 1 2 π l s = q 0 αw 2βs + βv, (4)

5 π l = 2β. (5) s2 β > 0, 2 π l < 0,, (4), 3PL s2 s = q 0 αw + βv. (6) 2β 2.3, (2), (6) (2), w 1 2 π m w = β2 v + (q 0 + αc 2αw)β θ 2 α(αw q 0 ), (7) 2β 2 π m w = 2αβ + θ 2α 2. (8) 2 2β 2 π m w 2 < 0,, (7), w = β(q 0 + αc βv) + θ 2 αq 0. (9) α(2β + θ 2 α) 2.4 (1) A1 0 = 6σ(2c + pθ 2 + 2vθ2 2 )α 2 + (p + 2vθ 2 vθ 1 )βα + 2vβ 2 3pαβ + 12σ((θ1 + 2θ 2 )α + 2β) q + ( 3pcαβ + 3pvαβθ 2 )α + vβ 2 + 4µβ. (10) 3pαβ + 12σ((θ1 + 2θ 2 )α + 2β) 2.5 3, θ 1 = 1 θ 2. (10) q0,, q0 w., 3PL q0 w s., q0, w s. A2 3p((c + w θ2 v)β + 2µθ 2 )α + 2µβ + 6σα(2θ2 2 v + θ 2 p + 2c) = 7 + 3αβp + 12((1 + θ2 )α + 2β)σ 6σβ(2c + 4θ 2 v + p 2v) 3αβp + 12((1 + θ2 )α + 2β)σ, (11) s = 3pα(vβ + µ) + 3σ((p + 2(2θ2 + 1)v 2c)α + 4vβ), (12) 3αβp + 12((1 + θ2 )α + 2β)σ q = αβ( 3pµ 3(2v + 2c p)σ) 3αβp + 12((1 + θ2 )α + 2β)σ. (13) q > 0,, 3pµ 3(2v + 2c p)σ > 0. (14) (14),

6 2 : 259 p > 2(v + c), (14),,,,,. p < 2(v + c),, (14), µ 3(2v + 2c p) σ >. p,,, ( ),,,,.,,, µ > 3(2v + 2c p)σ/p,., α 3PL β, σ θ 2, 1. 1 α,, 3PL,. α. A3. 1,,,, 3PL,. α,,,,, α,., 3PL,, 3PL, 3PL, 3PL, ;. 3PL,, 3PL. 2 3PL β, 3PL,, σ > pαθ 2 4,, σ < pαθ 2 3θ 1 4, 3θ 1. β 3PL,. A4. 2 3PL,,, 3PL,,,, ;,.,. β, 3PL,,, 3PL, β 3PL, 3PL,.,,,,, ;,,., β. 3 σ,, 3PL. 1 θ 1 = 1 θ 2,, θ 2, θ 2.

7 A5. 3,, 3PL. p < 2(v + c), 2,, µ > 0 (µ, ), µ > (p 2v 2c)βα/(4α(1 + θ 2 ) + 8β).,,,,,,, 3PL,,,. p > 2(v + c), 2,,,,. (p 2v 2c)βα µ > 4α(1 + θ 2 ) + 8β,,,,,, 3PL,. (p 2v 2c)βα µ < 4α(1 + θ 2 ) + 8β,,,,,,,,,,,. 3PL,, 3PL,,.,. ( ) ( ), 3PL ; ( ), 3PL. 4 θ 2, 3PL,, p + (2θ θ 2 )v 2c > 0,. A6. 4, 3PL,. p + (2θ θ 2 )v 2c > 0, p > 2c (2θ θ 2 )v,. p > 2c (2θ θ 2)v,,,., θ 2,,. 3PL,, θ 2.,,. 3, q = q 0 αw βs, α β,. σ θ 2.

8 i 2 : 261 α β σ θ 2,. c = 15,v = 5,p = 50,k m = 100,k l = 50,k r = 100, U[100,300], I 1 = 100,I 2 = 300,µ = (I 1 + I 2 )/2 = 200,σ = (I 2 I 1 ) 2 /12 = 100/ 3,θ 1 = 0.5,θ 2 = 0.5,α = 50,β = 50. π. 2, 1,, , α. 1, α,,,. 1.,. α,,,,,. 1, α,, 3PL. 3PL,, α 50 60, = 0.67, = 0.10,,,.,,., α,. Table 1 1 α The effects of the sensitivity coefficient α to the supply chain members decisions π πl π m π r α α Fig. 1 The effects of the sensitivity coefficient α to the supply chain members profits 3.2 3PL, β. σ = 100 < pαθ = 625, 3θ 1 3 2, β,,,. 2. 2

9 i ,. β,,,. 2, β,, 3PL.,,,, ;,,,, ;, 3PL., β,,, β 3PL. Table 2 2 β The effects of the sensitivity coefficient β to the supply chain members decisions π πl π m π r β β Fig. 2 The effects of the sensitivity coefficient β to the supply chain members profits 3.3 Table 3 3 σ The effects of the risk parameter σ to the supply chain members decisions 100/ / / / / / , σ. σ,,. 3, σ,. (p 2v 2c)βα 3, p = 50 > 2(v+c) = 40, ( ), µ = 200 > 4α(1 + θ 2 ) + 8β = 250 7,,,. 3PL. 3,, 3PL

10 i i : 263.,, 3PL, 3PL,,, πl π m π r π σ σ Fig. 3 The effects of the risk parameter σ to the supply chain members profits 3.4, θ Table 4 4 θ 2 The effects of the ratio of logistics fee he manufacture bears θ 2 to the supply chain members decisions π πl π m π r θ2 4 θ 2 Fig. 4 The effects of the ratio of logistics fee he manufacture bears θ 2 to the supply chain members profits

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13 (14), s α = 24( 3pµ 3σ(2v + 2c p))σβ ( 3pα β + 12σ((θ 2 + 1)α + 2 β)) 2 > 0, q α = 24( 3pµ 3σ(2v + 2c p))σβ 2 ( 3pαβ + 12σ((θ 2 + 1)α + 2 β)) 2 > 0, w α = (( 6β p + 12(v + c)β + 24 µ(θ 2 1)) 3pσ 72(2v p + 2c)(θ2 1)σ 2 6 p 2 µβ)β ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2 G = 1 β ( 3pα β + 12 σ(αθ 2 + α + 2 β)) 2. G 1 = ( 6β p + 12(v + c)β + 24 µ(θ 2 1)) 3pσ 72(2v p + 2c)(θ 2 1)σ 2 6 p 2 µβ = 24(θ 2 1)σ( 3pµ 3(2v p + 2c)σ) 2 3βp( 3pµ 3(2v p + 2c)σ) = (24(θ 2 1)σ 2 3βp)( 3pµ 3(2v + 2c p)σ) = (24(1 θ 2 )σ + 2 3βp)( 3pµ 3(2v + 2c p)σ) < 0. w α < 0.. A4 2 (14) 3pµ 3(2v + 2c p)σ > 0, s 3pα + 24σ)( 3pµ 3(2v + 2c p)σ)α β = ( ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2 < 0, q β = 12(1 + θ 2)( 3pµ 3(2v + 2c p)σ)σα 2 ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2 > 0, w β = ((( 6α p + 12(v + c)α)θ 2 24 µ(θ 2 1)) 3pσ + 72(2v + 2c p)(θ2 1)σ 2 6 p 2 µαθ 2 )α ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2 = G 2 α ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2, G 2 = (( 6α p + 12(v + c)α)θ 2 24 µ(θ 2 1)) 3pσ + 72(2v + 2c p)(θ 2 1)σ 2 6 p 2 µαθ 2 = 24(θ 2 1)σ( 3pµ 3(2v p + 2c)σ) 2 3αpθ 2 ( 3pµ 3(2v p + 2c)σ) = ( 24(θ 2 1)σ 2 3αpθ 2 )( 3pµ 3(2v + 2c p)σ) = 2 3(4 3(1 θ 2 )σ αpθ 2 )( 3pµ 3(2v + 2c p)σ). 4 αpθ 3(1 θ 2 )σ αpθ 2 > 0, σ > 2 4 3(1 θ 2 ) = αpθ 2 4, G 2 > 0, w 3θ 1 β > 0; 4 3(1 θ 2 )σ αpθ 2 < 0, σ < αpθ 2 4 3(1 θ 2 ) = αpθ 2 4, G 2 < 0, w 3θ 1 β < 0.. A5 3 G 3 = ((2v p + 2c)β + 4 µ(θ 2 + 1))α + 8 µβ, q0 σ = 6 3(2 β + α θ2 )α pg 3 ( 3pαβ + 12 σ(α θ 2 + α + 2 β)) 2, w σ = 6 3(β + α θ2 ) pg 3 ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2, s σ = 3 3αpG3 ( 3pαβ + 12 σ(αθ 2 + α + 2 β)) 2,

14 2 : 267 q σ = 3 3αβpG3 ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2. Q σ A6 G 3 > 0, µ > > 0, w σ > 0, s σ 4 (p 2v 2c)βα 4α(1 + θ 2 ) + 8β > 0, q σ > 0., q 0 σ < 0, w σ (14), s = 12( 3pµ 3(2v + 2c p)σ)σα 2 θ 2 ( 3pα β + 12 σ(αθ 2 + α + 2 β)) 2 < 0, < 0, s σ q < 0, σ < 0; G (p 2v 2c)βα 3 < 0, µ < 4α(1 + θ 2 ) + 8β, q = 12( 3pµ 3(2v + 2c p)σ)σβα 2 θ 2 ( 3pα β + 12 σ(α θ 2 + α + 2 β)) 2 < 0, q0 = 6 3pσα 2 ((p + (2 + 4θ 2 )v 2c)αβ + 8vβ 2 + 4µα) + 72σ 2 α(p + (2θ θ 2 )v 2c) θ 2 ( 3pαβ + 12 σ(αθ 2 + α + 2 β)) 2 + 3p 2 α 3 β(βv + 2µ) + 576σ 2 αβv((1 + θ 2 )α + β) ( 3pαβ + 12 σ(α θ 2 + α + 2 β)) 2, w = 6 3pσα((p + (2 + 4θ2 )v 2c)αβ + 8vβ 2 + 4µ(β + α 2 )) + 72σ 2 α 2 (p + (2θ θ 2 )v 2c) θ 2 ( 3pαβ + 12 σ(αθ 2 + α + 2 β)) αβσ 2 (p + (6 + 8θ 2 )v 2c) + 3p 2 α 2 β(βv + 2µ) + 576σ 2 β 2 v ( 3pα β + 12 σ(αθ 2 + α + 2 β)) 2., p + (2θ θ 2)v 2c > 0, p + (2θ θ 2)v 2c (p + (2 + 4θ 2 )v 2c) = 2(1 θ2 2 )v 0, p + (2 + 4θ 2 )v 2c (p + (6 + 8θ 2 )v 2c) = (4 + 4θ 2 )v < 0, p + (2 + 4θ 2 )v 2c > 0, p + (6 + 8θ 2 )v 2c > 0, q 0 > 0, w > 0.. θ 2 θ 2.