Supply Chain Network Competition in Price and Quality with Multiple Manufacturers and Freight Service Providers

Transcription

1 Supply Chain Network Competition in Price and Quality with Multiple Manufacturers and Freight Service Providers Anna Nagurney Sara Saberi Department of Operations and Information Management University of Massachusetts, Amherst Jonas Floden School of Business, Economics and Law University of Gothenburg, Sweden 2015 INFORMS Annual Meeting 1-4 November 2015

2 This presentation is based on the paper: Nagurney, A., Saberi, S., Shukla, S., Floden, J Supply chain network competition in price and quality with multiple manufacturers and freight service providers. Transportation Research Part E: Logistics and Transportation Review 77,

3 Outline 1 Introduction 2 Contributions 3 Supply Chain Network Model with Price and Quality Competition 4 Variational Inequality Formulations 5 Dynamics 6 Algorithm 7 Numerical Results 8 Summary and Conclusions

4 Introduction Background Manufacturers and freight service providers are fundamental decision-makers in today s globalized supply chain networks. The decisions that the firms make affect the prices and quality of products as well as that of the freight services provided, which, in turn, impact their own profitability. Quality and price have been identified empirically as critical factors in transport mode selection for product/goods delivery (cf. Floden, Barthel, and Sorkina (2010), Saxin, Lammgard, and Floden (2005)). Quality has also become one of the most essential factors in the success of supply chains of various products.

5 Introduction Motivation Increasingly, tough customer demands are also putting the transport system under pressure. The providers may offer flexibility to meet customer needs of safety, and/or traceability and, furthermore, differentiate themselves from the rest of the competition. Quality of freight encompasses factors such as on-time deliveries, reliability, and frequency.

6 Introduction Relevant Literature Akerlof, G.A., The market for lemons: Quality uncertainty and the market mechanism. Quarterly Journal of Economics, 84(3), Spence, M., Monopoly, quality, and regulation. The Bell Journal of Economics, 6(2), Sheshinski, E., Price quality and quantity regulation in monopoly situation. Economica, 43, Mussa, M., Rosen, S., Monopoly and product quality. Journal of Economic Theory, 18,

7 Introduction Relevant Literature Dixit, A., Quality and quantity competition. Review of Economic Studies, 46(4), Gal-or, E., Quality and quantity competition. Bell Journal of Economics, 14, Brekke, K.R., Siciliani, L., Straume, O.R., Price and quality in spatial competition. Regional Science and Urban Economics, 40, Nagurney, A., Li, D., A dynamic network oligopoly model with transportation costs, product differentiation, and quality competition. Computational Economics, 44(2), Nagurney, A., Li, D., Nagurney, L.S., Spatial price equilibrium with information asymmetry in quality and minimum quality standards. International Journal of Production Economics, 158,

8 Contributions Contributions We model explicit competition among manufacturing firms and freight service providers (carriers) in terms of prices and quality.

9 Contributions Contributions We model explicit competition among manufacturing firms and freight service providers (carriers) in terms of prices and quality. The transportation costs differ by mode, leading to an evaluation of quality vs. costs for the freight service providers and the modes of transportation that they offer to the customers.

10 Contributions Contributions We model explicit competition among manufacturing firms and freight service providers (carriers) in terms of prices and quality. The transportation costs differ by mode, leading to an evaluation of quality vs. costs for the freight service providers and the modes of transportation that they offer to the customers. We handle heterogeneity in the providers cost functions and in the consumers demands and do not limit ourselves to specific functional forms.

11 Contributions Contributions We model explicit competition among manufacturing firms and freight service providers (carriers) in terms of prices and quality. The transportation costs differ by mode, leading to an evaluation of quality vs. costs for the freight service providers and the modes of transportation that they offer to the customers. We handle heterogeneity in the providers cost functions and in the consumers demands and do not limit ourselves to specific functional forms. Utilities of each manufacturing firm and freight service provider considers price and quality for not just his own products, but that of other firms or providers as well.

12 Supply Chain Network Model with Price and Quality Competition The Supply Chain Network Model with Price and Quality Competition Manufacturing Firms F 1 F i F N Freight Service Providers C 1 C j C O 1 M Transportation Modes O 1 1 M M M 1 M O j 1 M O 1 M j 1 M1 1 M j 1 k Q Demand Markets Figure : The Supply Chain Network Structure of the Game Theory Model

13 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition As in Nagurney and Li (2014), we define and quantify quality as the quality conformance level, that is, the degree to which a specific product conforms to a design or specification (Gilmore (1974), Juran and Gryna (1988)).

14 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition As in Nagurney and Li (2014), we define and quantify quality as the quality conformance level, that is, the degree to which a specific product conforms to a design or specification (Gilmore (1974), Juran and Gryna (1988)). Firm F i manufactures a product of quality q i at the price p i.

15 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition As in Nagurney and Li (2014), we define and quantify quality as the quality conformance level, that is, the degree to which a specific product conforms to a design or specification (Gilmore (1974), Juran and Gryna (1988)). Firm F i manufactures a product of quality q i at the price p i. The quality and price associated with freight service provider C j retrieving the product from firm F i and delivering it to demand market k via mode m are denoted, respectively, by qijk m, and pm ijk ; i, j, k, m.

16 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition As in Nagurney and Li (2014), we define and quantify quality as the quality conformance level, that is, the degree to which a specific product conforms to a design or specification (Gilmore (1974), Juran and Gryna (1988)). Firm F i manufactures a product of quality q i at the price p i. The quality and price associated with freight service provider C j retrieving the product from firm F i and delivering it to demand market k via mode m are denoted, respectively, by qijk m, and pm ijk ; i, j, k, m. Demand is denoted by dijk m for consumer market k, mode m coming from firm i through provider j.

17 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition Demand Function: d m ijk = d m ijk(p F, q F, p C, q C ); i, j, k, m. Demand depends on firm s price and quality, its competitors, and freight service providers. The Firms Behavior: Supply of Firm: s i (p F, q F, p C, q C ) = O M Q j j=1 k=1 m=1 d m ijk(p F, q F, p C, q C ); i. The Production Cost: PC i = PC i (s F (p F, q F, p C, q C ), q F ), i

18 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition The Utility of Firm: Bounds on Quality: U Fi (p F, q F, p C, q C ) = p i [s i (p F, q F, p C, q C )] PC i, i. q i q i q i, i. q i = 100 corresponds to perfect quality conformance level. Positive lower bound corresponds q i to a minimum quality standard. Bounds on Price: 0 p i p i, i. Let Ki 1 denote the feasible set for firm F i and the bounds on price and quality hold. K 1 N i=1 K i 1. Functions are continuous and continuously differentiable.

19 Supply Chain Network Model with Price and Quality Competition Supply Chain Network with Price and Quality Competition The Freight Service Providers Behavior: The Transportation Cost: TC m ijk = TC m ijk(d(p F, q F, p C, q C ), q C ), i, j, k, m. The Utility of Freight Service Provider: U Cj = N M O j i=1 k=1 m=1 [p m ijkd m ijk TC m ijk], j. Bounds on Quality: Bounds on Price: q m ijk qm ijk q m ijk, i, j, k, m. 0 p m ijk p m ijk, 1, j, k, m. Feasible set, K 2 j ; K 2 O j=1 K 2 j.

20 Supply Chain Network Model with Price and Quality Competition The Equilibrium Conditions Definition 1: Nash Equilibrium in Prices and Quality Levels A price and quality level pattern (pf, q F, p C, q C ) K 3 N i=1 K i 1 O said to constitute a Nash equilibrium if for each firm F i ; i = 1,..., N: where j=1 K 2 U Fi (p i, ˆp i, q i, ˆ q i, p C, q C ) U F i (p i, ˆp i, q i, ˆ q i, p C, q C ), (p i, q i ) K 1 i, j, is ˆp i (p 1,..., p i 1, p i+1,..., p N) and ˆ q i (q 1,..., q i 1, q i+1,..., q N), and if for each freight service provider C j ; j = 1,..., O: U Cj (p F, q F, p C j, ˆ p C j, q C j, ˆ q C j ) U Cj (p F, q F, p Cj, ˆ p C j, q Cj, ˆ q C j ), where ˆ p C j (p C 1,..., p C j 1, p C j+1,..., p C O ) qˆ C j (qc 1,..., qc j 1, qc j+1,..., qc O ).

21 Variational Inequality Formulations Variational Inequality Formulation Theorem 1: Variational Inequality Formulations of Nash Equilibrium in Prices and Quality (p F, q F, p C, q C ) K3 is a Nash equilibrium according to Definition 1 if and only if it satisfies the variational inequality: N i=1 O U Fi (p F, q F, p C, q C ) p i N j=1 i=1 k=1 m=1 (p i p i ) M Q j U Cj (pf, q F, p C, q C ) O N j=1 i=1 k=1 m=1 p m ijk N i=1 U Fi (p F, q F, p C, q C ) q i (q i q i ) (p m ijk p m ijk ) M Q j U Cj (pf, q F, p C, q C ) q m ijk (p F, q F, p C, q C ) K 3, (q m ijk q m ijk ) 0,

22 Variational Inequality Formulations Variational Inequality Formulation Standard Form Determine X K where X is a vector in R n, F(X) is a continuous function such that F (X) : X K R n, and F (X ), X X 0, X K. We define the vector X (p F, q F, p C, q C ) and F(X) (F pf, F qf, F pc, F qc ) with the i-th component of F pf and F qf given, respectively, by: F pi = U F i p i ; F qi = U F i q i, and the (i, j, k, m)-th component of F pc and F qc, respectively, given by: F p m ijk = U C j pijk m ; F q m ijk = U C j qijk m.

23 Variational Inequality Formulations Existence of the Solution Theorem 2: A Solution to the Variational Inequality Existence of a solution to the variational inequalities is guaranteed since the feasible set K is compact and the function F (X) in our model is continuous, under the assumptions made on the underlying functions.

24 Dynamics Dynamics We now propose dynamic adjustment processes for the evolution of the firms product prices and quality levels and those of the freight service providers (carriers). Rate of change of p i : ṗ i = Rate of change of q i : q i = { UFi (p F,q F,p C,q C ) p i, if 0 < p i < p i max { 0, min{ U F i (p F,q F,p C,q C ) p i, p i } }, if p i = 0 or p i = p i. { UFi (p F,q F,p C,q C ) q i, if q i < q i < q i max { q i, min{ U F i (p F,q F,p C,q C ) q i, q i } }, if q i = q i or q i = q i.

25 Dynamics Dynamics Rate of change of p m ijk : ṗijk m = Rate of change of q m ijk : q ijk m = U Cj (p F,q F,p C,q C ) p, if 0 < p m ijk m ijk < pm ijk max { 0, min{ U C j (p F,q F,p C,q C ) p, p m ijk m ijk }}, if pijk m = 0 or pm ijk. U Cj (p F,q F,p C,q C ) q, if q m < ijk m ijk qm ijk < qm ijk max { q m, min{ U C j (p F,q F,p C,q C ) ijk q, q m ijk m ijk }}, if qijk m = qm or ijk qm ijk.

26 Dynamics Dynamics Ordinary Differential Equation (ODE) for the adjustment processes of the prices and quality levels of firms and freight service providers, in vector form: The projection operator: Ẋ = Π K (X, F(X)), X(0) = X 0. P K (X δf(x)) X Π K (X, F(X)) = lim, δ 0 δ with P K denoting the projection map: P K (X) = argmin z K X z.

27 Dynamics Dynamics Theorem 3 According to Dupuis and Nagurney (1993) X solves the variational inequality problem if and only if it is a stationary point of the ODE, that is, Ẋ = 0 = Π K (X, F (X )). This theorem demonstrates that the necessary and sufficient condition for a product and freight service price and quality level pattern X = (pf, q F, p C, q C ) to be a Nash equilibrium, according to Definition 1, is that X = (pf, q F, p C, q C ) is a stationary point of the adjustment processes defined by ODE, that is, X is the point at which Ẋ = 0.

28 Algorithm Explicit Formulae for the Euler Method Applied to the Multitiered Supply Chain Network Problem Closed form expressions of price and quality of firms: p τ+1 i { { = max 0, min p i, pi τ [ O M Q j + a τ j=1 k=1 m=1 d m ijk (pτ F, qτ F, pτ C, qτ C ) +p τ i O M Q j j=1 k=1 m=1 d m ijk (pτ F, qτ F, pτ C, qτ C ) p i q τ+1 i N PC i (s F (pf τ, qτ F, pτ C, qτ C ), qτ F ) s l (pf τ, qτ F, pτ C, qτ C ) } ] }, s l p i l=1 { { = max q i, min q i, qi τ [ O M Q j + a τ p τ i N l=1 j=1 k=1 m=1 d m ijk (pτ F, qτ F, pτ C, qτ C ) PC i (s F (pf τ, qτ F, pτ C, qτ C ), qτ F ) s l (pf τ, qτ F, pτ C, qτ C ) PC i (sf τ, qτ F ) } ] }. s l q i q i q i

29 Algorithm Explicit Formulae for the Euler Method Applied to the Multitiered Supply Chain Network Problem Closed form expressions of price and quality of freight service providers: { { p m(τ+1) ijk = max 0, min p ijk m, [ pmτ ijk + a τ d m ijk (pf τ, qτ F, pτ C, qτ C ) + N M Q j l=1 s=1 t=1 d t ljs (pτ F, qτ F, pτ C, qτ C ) p m ijk M N Q j ( N O Q M v TCljs t (d(pτ F, qτ F, pτ C, qτ C ), qτ C ) d z l=1 s=1 t=1 r=1 v=1 w=1 z=1 rvw { q m(τ+1) ijk = max {q mijk, min q ijk m, [ N M Q j qmτ ijk + a τ l=1 s=1 t=1 M N Q j ( N O Q M v TCljs t (d(pτ F, qτ F, pτ C, qτ C ), qτ C ) d z l=1 s=1 t=1 r=1 v=1 w=1 z=1 rvw p tτ ljs d rvw z (pτ F, qτ F, pτ C, qτ C ) )] }} pijk m, d t ljs (pτ F, qτ F, pτ C, qτ C ) q m ijk p tτ ljs d rvw z (pf τ, qτ F, pτ C, qτ C ) ) qijk m M N Q j TCljs t (d τ, qc τ ) ] }} q m. l=1 s=1 t=1 ijk

30 Algorithm Convergence Theorem 4 In our multitiered supply chain network game theory model, assume that F (X)= U(p F, q F, p C, q C ) is strictly monotone. Also, assume that F is uniformly Lipschitz continuous. Then, there exists a unique equilibrium price and quality pattern (pf, q F, p C, q C ) K and any sequence generated by the Euler method as given by the closed form expressions, where {a τ } satisfies τ=0 a τ =, a τ > 0, a τ 0, as τ converges to (pf, q F, p C, q C ).

31 Example 1 Numerical Results Manufacturing Firm F 1 Freight Service Provider Demand Market C 1 1 Figure : The Supply Chain Network Topology for Example 1 The demand function for demand market 1 is: The supply of F 1 is: d = p q p q 1. s 1 = d

32 Numerical Results Example 1 The production cost and utility of manufacturing firm F 1 is: PC 1 = 1.55(s q1), 2 U F1 = p 1 s 1 PC 1. The quality and price of the firm are bounded as per the following constraints: 0 p 1 80, 10 q The transportation cost of freight service provider C 1 is: TC =.5d (q111) 1 2. The utility of freight service provider C 1 is: U C1 = p111d TC 1 111, with the following limitations on his price and quality: 0 p , 9 q

33 Numerical Results Example 1 The equilibrium result, after 60 iterations, is: p = 16.63, p 1 = 19.57, q = 12.90, q 1 = Figure : Prices and Quality Levels for the Product and Freight of Example 1

34 Numerical Results Example 2 The supply chain network topology is depicted as here: Manufacturing Firm F 1 Freight Service Provider Demand Market C The demand functions are: Figure : The Supply Chain Network Topology d = p q p q p q 2 111, d = p q p q p q

35 Numerical Results Example 2 The supply of manufacturing firm F 1 is : s 1 = d d The transportation costs of the freight service provider C 1 for modes 1 and 2 are: TC =.5d (q 1 111) 2, TC =.45d (q 2 111) d 2 111q The utility of freight service provider C 1 is: U C1 = p 1 111d p 2 111d TC TC 2 111, 0 p , 9 q

36 Numerical Results Example 2 The equilibrium solution, after 166 iterations, is: p = 21.68, p = 24.16, p 1 = 27.18, q = 14.58, q = 22.43, q 1 =

37 Numerical Results Example 3 and Variant Manufacturing Firm Freight Service Providers Demand Market F 1 C 1 C Figure : The Supply Chain Network Topology for Example 3 and Variant

38 Example 3 The demand functions are: Numerical Results d = p q p q p q p q1 121, d = p q p q p q p q1 121, d = p q p q p q p q s 1 = d d d The transportation costs of freight service provider C 1 are: TC =.5d (q111) d121, 1 TC =.45d (q111) d111q 2 111, 2 and that of freight service provider C 2 is: TC =.64d (q121) 1 2. The utility of C 2 is: U C2 = p121d TC p , 12 q

39 Numerical Results Example 3 The new equilibrium solution, computed after 218 iterations, is: p = 45.69, p = 45.32, p = 44.82, p 1 = 53.91, q = 31.69, q = 41.32, q = 41.24, q 1 = Add trajectories.

40 Variant of Example 3 Numerical Results d = p q p q p q p q1 121, d = p q p q p q p q1 121, d = p q p q p q p q The equilibrium solution, computed after 553 iterations, is: p111 1 = 8.71, p2 111 = 63.17, p1 121 = 16.22, p 1 = 24.80, q111 1 = 9.00, q2 111 = 93.15, q1 121 = 16.92, q 1 =

41 Numerical Results Example 4 Manufacturing Firms Freight Service Providers Demand Markets F 1 F 2 C 1 C Figure : The Supply Chain Network Topology for Example 4 and Variant

42 Example 4: Result Numerical Results The equilibrium solution, after 254 iterations, is: p = 56.79, p = 55.45, p = 72.96, p = 36.93, p = 55.19, p = 53.55, p = 62.77, p = 53.28, p = 72.94, p = 65.91, p = 76.15, p = 83.73, p 1 = 63.76, p 2 = 64.90, q 1 = , q 2 = , q = 39.53, q = 51.20, q = 74.61, q = 23.54, q = 50.93, q = 51.05, q = 46.25, q = 36.72, q = 76.89, q = 69.56, q = 61.18, q = The price and quality levels have gone up as well as utilities for both manufacturers and carriers as compared to Example 6.4 since there are two demand markets to be satisfied now as opposed to one.

43 Summary and Conclusions Summary We developed a game theory supply chain network model in both static and dynamic versions with multiple manufacturers and freight service providers competing on price and quality.

44 Summary and Conclusions Summary We developed a game theory supply chain network model in both static and dynamic versions with multiple manufacturers and freight service providers competing on price and quality. Variational inequality theory was employed in the formulation of the equilibrium governing the behaviors with respect to price and quality.

45 Summary and Conclusions Summary We developed a game theory supply chain network model in both static and dynamic versions with multiple manufacturers and freight service providers competing on price and quality. Variational inequality theory was employed in the formulation of the equilibrium governing the behaviors with respect to price and quality. The computational procedure utilized was the Euler method.

46 Summary and Conclusions Summary We developed a game theory supply chain network model in both static and dynamic versions with multiple manufacturers and freight service providers competing on price and quality. Variational inequality theory was employed in the formulation of the equilibrium governing the behaviors with respect to price and quality. The computational procedure utilized was the Euler method. The discrete-time algorithm, also serving as an approximation to the continuous time trajectories, yields an equilibrium price and quality patterns.

47 Summary and Conclusions Summary We developed a game theory supply chain network model in both static and dynamic versions with multiple manufacturers and freight service providers competing on price and quality. Variational inequality theory was employed in the formulation of the equilibrium governing the behaviors with respect to price and quality. The computational procedure utilized was the Euler method. The discrete-time algorithm, also serving as an approximation to the continuous time trajectories, yields an equilibrium price and quality patterns. we then provided solutions to a series of numerical examples - small to large scenarios and their variants.

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