Article Contents

2023, Volume 19, Issue 12: 8657-8690. Doi: 10.3934/jimo.2023056

This issue Previous Article Optimization of a linear function over an integer efficient set Next Article A redistributed proximal bundle method for nonsmooth nonconvex functions with inexact information

Pseudomonotone variational inequality in action: Case of the French dairy industrial network dynamics

Arnaud Z. Dragicevic^1,2,3,4, ,

1.
CIRANO, 1130 Sherbrooke Ouest, H3A 2M8 Montréal (QC), Canada
2.
Chulalongkorn University [Faculty of Economics], Phayathai Road Pathumwan, 10330 Bangkok, Thailand
3.
INRAE, 9 Avenue Blaise Pascal, 63170 Aubière, France
4.
AgroParisTech, INRAE, UCA, VetAgro Sup [UMR Territoires], 63170 Aubière, France

^*Corresponding author: Arnaud Z. Dragicevic
^*Corresponding author: Arnaud Z. Dragicevic

Received: November 2022

Revised: March 2023

Early access: May 2023

Published: December 2023

Abstract / Introduction Full Text(HTML) Figure(11) / Table(2) Related Papers Cited by

Abstract

A pseudomonotone operator serves to study the dynamic network equilibrium of the French dairy industry. Using a modified variational inequality model, which is based on the price variations, we reverse the typical problem formulation and claim that the absence of synchronization between the variations, which also need to be proportionate, is what causes the network disequilibrium. Provided the pseudomonotonicity of the mapping, the solution of the variational inequality problem also happens to be a fixed point. The model outputs show that the economic viability of the upstream agents is in conflict with the overall network equilibrium. The results further suggest that increasing the threshold of resale-below-cost should enlarge the asynchronicity between the upstream and instream price adjustments, which is problematic because the price variations are already asymmetric at the levels of upstream and downstream layers. The best path toward the network equilibrium would go by carrying out a further integration of the upstream layer.

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Mathematics Subject Classification: Primary: 90Bxx, 91Bxx; Secondary: 47Nxx.

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Figure 1. Model representation

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Figure 2. Model 1: Benchmark atomistic industrial network without concentration among the agents

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Figure 3. Model 2: Concentrations among manufacturers and retailers. Model 3: Grouping of producers into organizations. Model 4: Grouping of producers into organizations and concentrations among manufacturers and retailers. Model 5: Horizontal grouping of organizations of producers into associations. Model 6: Horizontal grouping of organizations of producers into associations and concentrations among manufacturers and retailers. Model 7: Vertical grouping of organizations of producers into associations and concentrations among manufacturers and retailers

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Figure 4. Dynamic evolution of the benchmark atomistic industrial network

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Figure 5. A geometric representation of the variational inequality

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Figure 6. Model 8

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Figure 7. Upstream equilibrium price variations. The $ x $-axis represents the timeline. The $ y $-axis represents the equilibrium variations of values. The red belt depicts the producer's equilibrium price variations, issued from Models 1, 3, 5, and 7, with $ \phi_{U_{i}I_{i}} = 0.50 $. The red-dotted line illustrates the producer's equilibrium price variations without revenue sharing. The blue-dotted line is the manufacturer's equilibrium price variations

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Figure 8. Upstream equilibrium price variations for $ \phi_{U_{i}I_{i}} \in [0.00,1.00] $. The $ x $-axis represents the timeline. The $ y $-axis represents the equilibrium variations of values. The belt of full curves colored in shades of red illustrates the producer's equilibrium price variations at the increasing levels of revenue sharing. The blue-dotted line is the manufacturer's equilibrium price variations

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Figure 9. Instream equilibrium price variations. The $ x $-axis represents the timeline. The $ y $-axis represents the equilibrium variations of values. The red dotted curve corresponds to the manufacturer's equilibrium price variations issued from Model 1. The red dashed curve is related to the manufacturer's equilibrium price variations issued from Models 2 and 4. The red full curve corresponds to the manufacturer's equilibrium price variations issued from Models 5, 6, and 7. The blue dotted line illustrates the retailer's equilibrium price variations

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Figure 10. Downstream equilibrium price variations. The $ x $-axis represents the timeline. The $ y $-axis represents the equilibrium variations of values. The red-dotted curve corresponds to the retailer's equilibrium price variations issued from Models 1, 3, and 5. The red-full curve is related to the retailer's equilibrium price variations issued from Models 2, 4, 6, and 7. The blue-dotted line illustrates the consumer's equilibrium price variations

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Figure 11. Smoothed price variations of the supply chain. The $ x $-axis represents the timeline. The $ y $-axis represents the equilibrium variations of values. The red-dotted curve corresponds to the producer's equilibrium price variations without revenue sharing. The red-dashed curve is related to the manufacturer's equilibrium price variations. The red-full curve illustrates the retailer's equilibrium price variations. The blue-full curve depicts the consumer's equilibrium price variations

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Table 1. Key notations

Sets	Definition
$ U $	Set of upstream agents with a representative upstream agent $ U_{i} \in U $
$ I $	Set of instream agents with a representative instream agent $ I_{i} \in I $
$ D $	Set of downstream agents with a representative downstream agent $ D_{i} \in D $
$ C $	Set of customers with a representative customer $ C_{i} \in C $
$ \mathcal{T} $	Nonempty convex set comprising the equilibrium vector with elements $ t $
Parameters	Definition
$ N $	Number of economic agents composing the network layer
$ \phi_{U_{i}I_{i}} $	Share that an instream agent receives within the revenue-sharing supply contract
$ 1.00 - \phi_{U_{i}I_{i}} $	Share that an upstream agent receives within the revenue-sharing supply contract
$ x $	Transaction cost rate
Decision variable	Definition
$ t $	Time step within a time interval going from $ 0.00 $ to $ T $
$ {\bf{t}}^{\star} $	Multidimensional equilibrium vector with elements $ t \in [0.00,T] $
$ p_{U_{i}I_{i}}(t) $	Unit price at which an upstream agent charges an instream agent at time $ t $
$ \Delta p_{U_{i}I_{i}}(t^\star) $	Upstream equilibrium price variation
$ p_{I_{i}D_{i}}(t) $	Unit price at which an instream agent charges a downstream agent at time $ t $
$ \Delta p_{I_{i}D_{i}}(t^\star) $	Instream equilibrium price variation
$ p_{D_{i}C_{i}}(t) $	Unit price at which a downstream agent charges a customer at time $ t $
$ \Delta p_{D_{i}C_{i}}(t^\star) $	Downstream equilibrium price variation
Functions	Definition
$ \Pi_{U_{i}}(t) $	Upstream agent profit function at time $ t $
$ \Pi_{I_{i}}(t) $	Instream agent profit function at time $ t $
$ \Pi_{D_{i}}(t) $	Downstream agent profit function at time $ t $
$ v_{I_{i}}(t) $	Value function of the resource at time $ t $
$ f_{U_{i}I_{i}}(t) $	Upstream production cost function at time $ t $
$ f_{I_{i}D_{i}}(t) $	Instream production cost function at time $ t $
$ f_{D_{i}C_{i}}(t) $	Downstream production cost function at time $ t $
$ c_{U_{i}I_{i}}(t) $	Transaction cost function between the upstream and instream agents at time $ t $
$ c_{I_{i}D_{i}}(t) $	Transaction cost function between the instream and downstream agents at time $ t $
$ c_{D_{i}C_{i}}(t) $	Transaction cost function between the downstream agent and the customer at time $ t $
$ g_{U_{i}I_{i}}(t) $	Foregone benefit due to the imbalance between upstream offer and instream demand at time $ t $
$ g_{I_{i}D_{i}}(t) $	Foregone benefit due to the imbalance between instream offer and downstream demand at time $ t $
$ g_{D_{i}C_{i}}(t) $	Foregone benefit due to the imbalance between downstream offer and consumers demand at time $ t $
$ T_{U_{i}I_{i}}(t) $	Transfer adjustment between the upstream and instream agents at time $ t $

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Table 2. Simulation tools and values

Expression	Definition
$ \frac{\partial{f_{U_{i}I_{i}}(t^\star)}}{\partial{t}} $	Time variation of the upstream production cost function
$ \frac{\partial{c_{U_{i}I_{i}}(t^\star)}}{\partial t} $	Time variation of the upstream transaction cost function
$ \frac{\partial{v_{I_{i}}(t^\star)}}{\partial{t}} $	Time variation of the value function of the resource
$ \frac{\partial{p_{U_{i}I_{i}}(t^\star)}}{\partial{t}} $	Time variation of the upstream price function
$ \frac{\partial{f_{{I_{i}D_{i}}}(t^\star)}}{\partial{t}} $	Time variation of the instream production cost function
$ \frac{\partial{c_{I_{i}D_{i}}(t^\star)}}{\partial{t}} $	Time variation of the instream transaction cost function
$ \frac{\partial{g_{I_{i}D_{i}}(t^\star)}}{\partial{t}} $	Time variation of the instream opportunity cost function
$ \frac{\partial{f_{{D_{i}C_{i}}}(t^\star)}}{\partial{t}} $	Time variation of the downstream production cost function
$ \frac{\partial{c_{D_{i}C_{i}}(t^\star)}}{\partial{t}} $	Time variation of the downstream transaction cost function
$ \frac{\partial{g_{D_{i}C_{i}}(t^\star)}}{\partial{t}} $	Time variation of the downstream opportunity cost function
Value	Definition
$ \phi_{U_{i}I_{i}} = [0.00,1.00] $	Revenue-sharing level
$ x=20.00\% $	Upstream transaction cost rate in the benchmark case
$ x=15.00\% $	Upstream transaction cost rate in the case of PO
$ x=10.00\% $	Upstream transaction cost rate in the case of APO
$ x=25.00\% $	Instream transaction cost rate in the case of atomicity
$ x=12.50\% $	Instream transaction cost rate in the case of oligopoly
$ x=30.00\% $	Downstream transaction cost rate in the case of atomicity
$ x=15.00\% $	Downstream transaction cost rate in the case of oligopoly
$ t=[0.00,10.00] $	Time step
$ [-30.00,30.00] $	Interval of the seasonality index

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