Economically optimal multi-actor processing networks: material flows and price assignment of the intermediates using Lagrangian decomposition

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Abstract

Manufacturing networks are usually composed by several suppliers of intermediate products and several consumers that upgrade these to products. Coordination between the members of the network involves deciding how much intermediate product is going to be exchanged and at which price. In this work, a two-level Lagrangian approach is proposed to find the optimal transfer of intermediates (in terms of flowrates and prices) so that the profits of the actors participating in the network are balanced. The mathematical framework is first discussed for a simple network consisting of one supplier and one consumer, and then extended to multiple suppliers and consumers. A numerical example for a generic chemical process is presented to demonstrate the approach.

Introduction

Chemical processing networks commonly involve several consecutive steps which can be carried out by a single actor, that processes feedstocks to final products, or by several actors that specialize in a section of the value chain. An example of the latter, is the traditional petrochemical network where the production of intermediates (naphtha, gasoline, diesel, etc.) is decoupled from the production of final products (e.g. building block chemicals such as ethylene, propylene, aromatics, etc.).

In a generic case, a manufacturing network can be described as a set of suppliers producing several intermediate products that are uptaken by different consumers to produce other products. The decision of which intermediate product(s) is the supplier going to produce and the consumer going to buy, is made by each actor, who independently of the others, looks after its own economic benefit. Coordination among the members of the network is done in a second stage and involves (i) deciding how much intermediate product is going to be exchanged and (ii) reaching an agreement on the price for this intermediate.

Recent approaches to study interactions between suppliers and consumers include the work of Ortiz-Gutiérrez et al. (2015) who considers game theory and MILP to optimize the performance of a bio-ethanol supply chain that also guarantees a fair profit distribution between the members; and the work of Hjaila et al. (2015) that proposes a scenario-based negotiation win-win approach to negotiate the exchange price for an intermediate.

In this paper, we extend our previous work on the decomposition of biorefinery networks (see Torres et al. (2015)), to the analysis of generic processing networks that can be modeled by the suppliers-intermediates-consumers scheme. We explore how the two-level Lagrangian approach framework presented at that time, needs to be re-formulated to study different generic scenarios in order to find the values for the process design variables, amount of intermediates to be exchanged and their prices, that maximizes the profit of each actor involved in the network. The proposed framework is demonstrated for a one-supplier one-consumer case in which generic equations to estimate capital and operational costs for the different actors are considered.

Section snippets

Problem Statement

Figure 1 shows a scheme of the type of processing networks to be considered in this paper. In here, suppliers (SA, SB,) may produce one or more intermediate products (I1, I2,) that are uptaken by one or more consumers (C1, C2,). These consumers may compete for the same intermediate (e.g. C2 and C3 for I2), and their products sold to the market or back to the supplier. Consumers might also consider the use of different intermediates, from different suppliers, as feedstocks of their processes.

Solution approach

As mentioned before the goal is to find the optimal set of design variables dl¯ for each actor l and the optimal flowrates and prices for the intermediates. An outline of the procedure is as follows:

  • 1.

    After creating a MOP such as the one in Eq. 3, formulate the corresponding Lagrangian ℒ

  • 2.

    By considering the Karush-Khun-Tucker (KKT) conditions pj,k=0pi,k=0 and μlPl = 0, prove that only when the profits of all actors are equally weighed in the overall network, i.e sS1 = sS2 =  = sC1 = sC2 =  = 1, the

Illustrative numerical example

As a demonstration of the proposed framework, we considered the two actors network linked through one intermediate depicted in Fig. 3, where the process of each actor is composed by a reactor and a separator. For each case, two first order reactions in series were assumed in the reactor, and complete separation of the desired intermediate product from the reactant and byproduct were assumed in the separator. The profits for each actor are as in Section 3.1.

A cost function clml,feeddl¯=tal,tml,t

Conclusions

In this paper, we have studied manufacturing networks in which each actor is seeking after its own benefit. By proposing a multi-objective optimization problem and a two-level Lagrangian approach to solve it, the solution that offers the best balance between the profits of the participating actors has been found. In addition, a link between the value of the Lagrange multiplier and the price of the products (intermediates) that are exchanged has been established. The proposed framework has

Acknowledgements

This work was funded by the Cooperative Agreement between the Masdar Institute of Science and Technology (Masdar Institute), Abu Dhabi, UAE and the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA - Reference 02/MI/MI/CP/11/07633/GEN/G/00 for work under the Second Five Year Agreement.

References (3)

  • K. Hjaila et al.

    Scenario-based price negotiations vs. game theory in the optimization of coordinated supply chains

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