Modelling complex market interactions using PDP systems
Abstract
Last decade has witnessed an increasing effort on modelling and simulation of phenomena within a wide range of areas such as Biochemistry, Ecology, Robotics or Engineering by using membrane computing, providing solutions for relevant problems (signalling pathways, population dynamics, robot control or fault diagnosis, among others). However, for no apparent reasons, other areas have not been investigated to such extent. This is the case of computational economics, where Gh. and R. Păun explored the socalled producer–retailer problem and, in a foundational paper, proposed an initial model making use of membrane computing modelling tools. In the present paper, we design a solution based on population dynamics P systems for an enriched version of that problem. This enhanced model, closer to reality, takes into account several economic issues not considered in the initial model, including: depreciation of production capacity, decision mechanism to increase manufacturing capability, dividends payment and costs associated to production factors. Additionally, the model has been simulated making use of the framework provided by PLingua and MeCoSim, and delivering a custom application based on them to reproduce the virtual experiments. Finally, several scenarios have been analysed focusing on different elements included in the model.
Keywords
Membrane computing Economy Producer–retailer problem Computational modelling Population dynamics P systems1 Introduction
The main goal of this paper is to continue extending the success obtained by membrane computing as a modelling tool in different fields (see [2, 3], among others) to a less explored one, namely computational economics. Soon after an initial paper opening the topic [13], few others continued the initial exploring, with approaches as numerical P systems [14] or membrane systems with attributes [8]. However, no further significant attention was paid to this line along the last decade, so this paper aims to regain that focus, deepening into its study.
In the foundational paper [13], generic multiset rewriting rules were proposed for modelling some economic processes associated to the socalled producer–retailer problem. More specifically, it involved the reception of orders from consumers, the production of goods from raw material and the purchase of transactions.
Taking advantage of the latest developments on population dynamics P systems (PDP systems [6], for short), and in order to reinforce the interest in computational economics as a promising research path within the applications of membrane computing, the present paper proposes a model based on PDP systems for an enriched version of the previous producer–retailer problem. This model extends the basic model considered in the previous works, including different economic issues not considered before. All these aspects make the model much closer to real situations. More specifically, it is assumed the presence of a capital market (costs associated to production factors, capacity increase decision mechanisms, depreciation of such production capacity and dividends payment). These elements are explained in detail in Sect. 3, right after introducing some prerequirements setting the context of this work in Sect. 2. The formalization of the model is then presented in Sect. 4. Later, in Sect. 5, the simulation results for different scenarios are discussed. Finally, the main conclusions of this work are outlined in Sect. 6.
2 Preliminaries
This section starts introducing the topic of computational modelling, discussing benefits and drawbacks of some widely spread approaches, such as ordinary differential equations. Then PDP systems are described, this being the choice made in the framework of membrane computing to model the economic processes presented in the following sections.
2.1 Modelling approaches
Traditionally, ordinary differential equations have been the main modelling tool for biological systems. Several drawbacks are associated to this family of techniques: (a) the complexity of the resulting models prevents from using exact solutions, requiring numerical approaches; (b) the introduction of new variables or any other modification, improvement or extension of the model usually requires a reconstruction of the model from scratch and (c) difficulties arise if the processes modeled present a strong discrete nature or the number of copies of the objects involved is small.
On the contrary, membrane computing [12, 16] presents a more recent approach, showing several advantages for modelling biological and nonbiological systems. From the interaction of the elements of this modelling framework (objects, membranes, multisets and evolution rewriting rules) it arises a modelling tool with a high degree of generality applicable to many different situations. This framework captures the idea of modularity or extension, allowing the gradual addition or modification of evolution rules or even changes in the membrane structure in a relatively easily way, without changing the type of P system. Besides, parallelism is introduced in a natural way in the model, allowing the operation of multiple elements simultaneously, and there are no limits to the number of variables interacting simultaneously.
The following section will present PDP systems, used to model the phenomena studied along this work.
2.2 Population dynamics P systems

\(G=(V,S)\) is a directed graph with \(m \ge 1\) nodes, \(V=\{e_1,\ldots ,e_m\}\), \( S \subseteq V \times V\).

\(\varGamma \) and \(\varSigma \) are alphabets such that \(\varSigma \subsetneq \varGamma \).

\(T \ge 1\) is a natural number.
 \(\forall k, 1 \le k \le m, \varPi _k =(\varGamma ,\mu ,M_1,\ldots ,M_q,{\mathcal {R}},i_{\text {in}})\), where:

\(\mu \) is a rooted tree with \(q \ge 1\) nodes labelled with elements of \(\{1,\ldots ,q\} \times \{0,+,\}\).

\(\forall i, 1 \le i \le q, M_i \in M_f(\varGamma )\) (i.e., they are finite multisets over \(\varGamma \)).
 \({\mathcal {R}}\) is a finite set of (skeleton) rules, of the type:where \(u,v,u',v' \in M_f (\varGamma ), 1 \le i \le q, \alpha ,\alpha ' \in \{0,+,\}\), and p is a probability function with domain \(\{0,\ldots ,T\}\). The sum of probabilities of rules whose left hand side (LHS) is \(u[v]_i^\alpha \) is 1 at each instant \(t \, (0 \le t \le T)\).$$\begin{aligned} u[v]_i^\alpha {\mathop {\longrightarrow }\limits ^{\scriptstyle {p}}} u'[v']_i^{\alpha '} \end{aligned}$$

\(i_{\text {in}}\) is a node of \(\mu \).


\(\forall j, 1 \le j \le m, E_j \in M_f (\varSigma )\).
 \(R_\text {E}\) is a finite set of environment rules of the type:where \(x,y_1,\ldots ,y_h \in \varSigma , \{(e_{j},e_{j_i})\in S, 1 \le j \le m, 1 \le i \le h\}\) (object x in environment j is consumed, and sent to \(h \le q\) environments), and \(p_1\) is a probability function with domain \(\{0,\ldots ,T\}\). Also, at each instant \(t \, (0 \le t \le T)\), the sum of all probability function values associated to rules whose LHS is \((x)_{e_j}\) must be 1.$$\begin{aligned} (x)_{e_j} {\mathop {\longrightarrow }\limits ^{\scriptstyle {p_1}}} (y_1)_{e_{j_1}},\ldots ,(y_h)_{e_{j_h}} \end{aligned}$$

There are no skeleton rules conflicting with rules of the environment; i.e., there is no simultaneous occurrence of PDP system skeleton rules \(u[v]_i^\alpha {\mathop {\longrightarrow }\limits ^{\scriptstyle {p}}} u'[v']_i^{\alpha '}\) in the skin membrane of a system \(\varPi _k\) and environment rules of the type \((x)_{e_j} {\mathop {\longrightarrow }\limits ^{\scriptstyle {p_1}}} (y_1)_{e_{j_1}},\ldots ,(y_h)_{e_{j_h}}\), such that \(x \in u\).

Each environment \(e_j\) contains exactly one system \(\varPi _k\).
2.3 Economic processes modelling
Differential equation systems have been traditionally used for modelling economic processes. Although these techniques are predominant, many efforts have been made to investigate other modelling techniques such as multiagent systems [7]. Due to the good performance of membrane computing for modelling the dynamics of biological systems, several attempts have been made to follow this approach in modelling economic processes [1, 8, 10, 11, 13, 14, 15].
Comparing economic and biological processes, a parallelism can be identified between them (see [13]). An economic interpretation can be assigned to different elements of the membrane computing models. The objects in a multiset can represent elements of different nature: monetary units, goods, authorization for transactions, depreciation representation or even production capacities. Membranes can be entities such as producers, consumers, markets or generic places for transactions. Multiset rewriting rules can model a huge variety of processes such as purchase transactions, production of goods or depreciation phenomena involving objects coming from a different or the same membrane.
3 Enhanced producer–retailer model
This section will explore in certain detail the interactions taking place in the reference problem known as the “producer–retailer” problem.
3.1 General description
Informally, the producer–retailer problem can be described as a one good market with several players interacting with each other. In this scenario, a set of producers \(P_i\) transform raw material a produced by a generic source S into units of good d, and a set of retailers \(R_j\) receive orders \({\bar{d}}\) from a generic consumer C. The producers and the retailers try to match units of d with units of \({\bar{d}}\) by means of transactions.
Each player is characterized by a parameter: \(P_i\) has a production capacity, \(R_j\) has a storage capacity, S has a raw material production rate and C has also a demand \({\bar{d}}\) generation rate. Each of these transactions implies the exchange of monetary units associated to the existence of prices. Thus, \(u_S\) represents a monetary unit owned by S. This money has been obtained from \(P_i\), who paid a price for each unit of a. Similarly, \(u_i\) stands for a monetary unit owned by \(P_i\), obtained from \(R_j\), who paid a price for each unit of d. Likewise, certain units of \(v_j\) are owned by \(R_j\), who paid a price for each unit of \({\bar{d}}\). Finally, units represented by object \(u_C\) are owned by C.
Simultaneously, there exist budget restrictions associated to the total number of monetary units owned by each player, introducing the possibility of lack of money, hence making impossible to apply certain rules. Real economy dynamically adjusts its parameters internally to maintain its activity cycle after cycle. Therefore, to get our model closer to real situations, more aspects must be modelled. Firstly, variations of \(P_i\)’s and \(R_j\)’s capacities will be allowed (associated to capital stock depreciation and investment decisions). Secondly, a cyclic monetary flow in the systems will be considered and sources of randomness will be added.
3.2 Production side
In economic theory, the number \(Y_i\) of goods produced by \(P_i\) is a function \(f_i\) (production function) of the socalled factors of production that specifies how factors are transformed into goods. These factors are the physical inputs used in the production process. Typically, they are classified into three main categories: raw material, labour of workers and capital stock.
For simplicity, we make some assumptions. All \(P_i\) have access to the same technology, thus they share the same production function: \(\forall i (f_i=f)\). Again for the sake of simplicity, labour is not considered as a factor in our model, obtaining the following production function: \(Y_i=f({\text {raw}}\_{\text {material}},{\text {capital}}_i)=f(a,b_i)\), where a represents the total amount of raw material available for production and \(b_i\) represents the total production capacity of \(P_i\). Additionally, we also consider the simplest form for f, where only one unit of a and \(b_i\) are consumed to produce one unit of d. This exchange rate can be easily changed to consider more complex situations.
3.3 Demand side
In the context of the socalled al behaviour model, each individual consumer tries to maximize a utility function \(U_i\). This \(U_i\) quantifies in monetary units the happiness of individuals, associated to their preferences about the simultaneous consumption of multiple disposable goods. Utility is a function of the units of good d obtained (consumption) and the cost of opportunity of time not dedicated to work (leisure): \(U_i=U_i (\text {consumed}\_\text {inputs}) = U_i (\text {consumption,leisure})\).
For simplicity, we make some assumptions. Labour (as complementary to leisure) is not considered in our model. Besides, all consumers have the same utility function (same preferences or standardized behaviour) \(\forall i (U_i=U)\). This gives rise to the concept of representative consumer, that we can model as a generic consumer C. We can, therefore, calculate the sum of the utility functions of all the consumers, obtaining a socalled aggregate demand of d (each unit is denoted by \({\bar{d}}\)). Thus, it is obtained a simplified utility function: \(U_i=U=g({\bar{d}})\).
3.4 Ownership of production factors and stakeholders
In macroeconomic models, factors are property of the aggregate consumer C. Thus, \(P_i\) (and \(R_j\) as intermediate producers) must hire these factors out from its owners paying an amount of money for them (producer’s costs). This will be modeled as costs associated to production capacities.
Typically, C is a stakeholder for \(P_i\) and \(R_j\). Therefore, C expects to receive dividends depending on the benefits obtained by \(P_i\) and \(R_j\). Benefits not distributed remain in the company allowing to pay costs of factors. The initial number of monetary units owned by \(P_i\) and \(R_j\) can be interpreted as the initial investment of C. Finally, to make our system closed, C must also be stakeholder of S. For simplicity, S generates units of a without any production capacity (no production costs).
With the elements mentioned above, multiple monetary flows get enabled: from \(u_i\) and \(v_j\), then to \(u_C\), and from \(u_S\) to \(u_C\). This implies additional flows with respect to those, previously considered, associated to the transactions between \(P_i\), \(R_j\), S and C.
3.5 Investment decisioncapacity increase
Each \(P_i\) has to decide what to do with the surplus obtained after purchasing transactions. This is known as the investment—saving decision. There are two possible choices: (1) to dedicate part of it to accumulate more production capacity (i.e., making the choice of capital stock increase); or (2) to keep capacity unchanged, leaving earnings accumulated as savings. This decision should be based on specific facts, and the model to design should detect situations as the lack of production capacity (in these case, capacity should be increased) or the excess of goods (so that capacity should remain unchanged).
3.6 Capital stock depreciation
In macroeconomic theory, there exists a phenomenon suffered by capital stock called depreciation. Typically, it is modeled as: \(K_t=K_{t1}D_{t1}+I_t\), where: \(K_t\) is the capital stock value (production capacity) at time t; \(K_{t1}\) is the capacity at time \(t1\); \(D_{t1}\) is the depreciation of \(K_{t1}\) and \(I_t\) is the investment at time t. For simplicity, we assume a constant depreciation rate \(\delta \), such that \(D_{t1}= \delta K_{t1}\). Thus, the previous equation can be written as: \(K_t= (1\delta ) K_{t1}+I_t\). In our model, depreciation is considered as a fixed reduction of the multiplicity of \(b_i\). Therefore, unless we have a mechanism for increasing capacity, after a finite period of time, production capacity will be exhausted.
4 Model formalization
In this section the formalization of the model is described, along with the interpretation of the elements included, parameters involved and an analysis of the different modules of rules taking part in the evolution of the system.

\(G=(V,E)\), with \(V=\{e_1\}\) and \(E=\{(e_1,e_1)\}\).

\(\varGamma =\{b_i,d_i,u_i,c_j,{\bar{d}}_j,v_j,{\bar{e}}_j,f_{j,i},g_i,y_i,z_i,m_i,h_i: 1 \le i \le k_1, 1 \le j \le k_2 \} \cup \{R_1\} \cup \{C,S,{\bar{d}},a,u_C,p,q\}\).

\(\varSigma =\emptyset \).
 \(\varPi _1=(\varGamma ,\mu ,M_1,M_2,{\mathcal {R}}_{\varPi _1},i_{\text {in}})\), where

\(\mu = [ [ \, ]_2 ]_1\).

\(M_1=\{C,S,R_1\}\cup \{g_i,u_i^{7 k_{i,1}k_{10}}:1\le i\le k_1\}\cup \{v_j^{7 k_{j,3}k_{10}}:1\le j\le k_2\}\).

\(M_2=\{c_j^{k_{j,3}}: 1\le j\le k_2 \}\cup \{b_i^{k_{i,1}}: 1\le i\le k_1\}\).

\(i_{\text {in}}\).

\({\mathcal {R}}_{\varPi _1}\) is described in Sect. 4.2, Modules of rules.


\(R_\text {E}=\emptyset \).

C: aggregate generic consumer.

S: raw material supplier.

a: unit of supplied raw material provided by S.

p: randomness generator for a provision by S.

\({\bar{d}}\): unit of aggregate demand from C.

q: randomness generator for \({\bar{d}}\) generation by C.

\(u_C\): monetary unit owned by C.

\(b_i\): unit of production capacity of \(P_i\), \(1 \le i \le k_1\).

\(h_i\): unit of production capacity of \(P_i\) before depreciation, \(1 \le i \le k_1\).

\(d_i\): unit of good supplied by \(P_i\), \(1 \le i \le k_1\).

\(u_i\): monetary unit owned by \(P_i\), \(1 \le i \le k_1\).

\(c_j\): unit of capacity of \(R_j\), \(1 \le j \le k_2\).

\({\bar{d}}_j\): unit of good demanded by \(R_j\), \(1 \le j \le k_2\).

\(v_j\): monetary unit owned by \(R_j\), \(1 \le j \le k_2\).

\({\bar{e}}_j\): units demanded by \(R_j\) and authorized for transaction, \(1 \le j \le k_2\).

\(f_{i,j}\): authorization for \({\bar{d}}_j\) to be exchanged with \(d_i\), \(1 \le i \le k_1\), \(1 \le j \le k_2\).

\(y_i\): unit (in idle state) of aborted purchase transactions considered for capacity increase, \(1 \le i \le k_1\).

\(m_i\): randomness generator for \(y_i\), \(1 \le i \le k_1\).

\(z_i\): activated unit of aborted purchase transactions considered for capacity increase, \(1 \le i \le k_1\).

\(R_1\): for technical reasons.

\(g_i\): for technical reasons, \(1 \le i \le k_1\).
4.1 Model parameters

\(k_1\): total number of producers.

\(k_2\): total number of retailers.

\(k_3\): raw material inserted into the system by S—min value of range.

\(k_4\): raw material inserted into the system by S—max value of range.

\(k_5\): aggregate demand inserted into the system by C—min value of range.

\(k_6\): aggregate demand inserted into the system by C—max value of range.

\(k_7\): price fixed by S for each unit of a.

\(k_8\): failed purchases considered for increasing capacity—min value.

\(k_9\): failed purchases considered for increasing capacity—max value.

\(k_{10}\): cost of capital stock per cycle.

\(k_{11}\): depreciation rate of capital stock.

\(k_{12}\): step of capacity increase.

\(k_{13}\): dividend percentage.

\(k_{i,1}\): initial production capacity of \(P_i\), \(1 \le i \le k_1\).

\(k_{i,2}\): price fixed by \(P_i\) for each unit of \(d_i\), \(1 \le i \le k_1\).

\(k_{j,3}\): initial capacity of \(R_j\), \(1 \le j \le k_2\).

\(k_{j,6}\): price fixed by \(R_j\)\(\underline{\hbox {for each order of good } j}\), \(1 \le j \le k_2\).
4.2 Modules of rules
Along this section, the main modules of rules taking part in the PDP system designed are explained. However, before entering into details about each particular module, a brief explanation about the way certain randomness is included in some sets of rules is provided.
4.2.1 Introducing randomness in the model
An elegant “PDPway” of generating randomness in rewriting rules is proposed. Consider the following generic set of rules:
\([ s ] \rightarrow [ s a^{NL} w^{2L} ]\) \([ w ] \xrightarrow {0.5} [ \# ]\) \([ w ] \xrightarrow {0.5} [ a ]\)
Its aim is to generate approximately N units of object a in the range \([a^{NL},a^{N+L}]\). First, we generate \(a^{NL} w^{2L}\), thus guaranteeing that at least a number of objects a equals to the lower limit of the range is generated. Secondly, two possible rules are applied to this new symbol w, each one with probability 0.5, transforming w into one unit of a or clearing it. This introduces the random effect introducing variation among different executions of the system, what may lead to any number of objects a within the interval above. In our model, this strategy will be used at the beginning of each cycle to produce an amount of a generated by S, and similarly to generate a number of objects \({\bar{d}}\) produced by C; the same technique will also be applied in the investment decision mechanism. Further developments of the model could consider more sources of variability, possibly involving prices, probabilities of performing transactions between \(P_i\) and \(R_j\), etc.
4.2.2 Module 1: Production factor and demand generation
4.2.3 Module 2: Producer’s costs
4.2.4 Module 3: Producer’s and retailer’s operations
4.2.5 Module 4: Purchase transactions
4.2.6 Module 5: Dividends distribution
4.2.7 Module 6: Capacity depreciation
4.2.8 Module 7: Capacity increase decision
4.2.9 Technical and cleaning rules
5 Simulation results
The model presented in the previous section was translated into PLingua [9] language, and a custom application based on MeCoSim [17] has been prepared to perform virtual experiments simulating different instances of the system under study, thus generating the corresponding PDP systems.
Parameters user for simulation
Param  Values  Description 

\(k_1\)  2  Total number of producers 
\(k_2\)  3  Total number of retailers 
\(k_3\)  59  Units of a inserted into the system by S—min value of range 
\(k_4\)  62  Units of a inserted into the system by S—max value of range 
\(k_5\)  59  Units of \({\bar{d}}\) inserted into the system by C—min value of range 
\(k_6\)  62  Units of \({\bar{d}}\) inserted into the system by C—max value of range 
\(k_7\)  11  Price fixed by S for each unit of a 
\(k_8\)  3  # failed purchases considered for increasing capital—min value 
\(k_9\)  5  # failed purchases considered for increasing capital—max value 
\(k_{10}\)  2  Cost of capital stock per cycle 
\(k_{11}\)  0.1  Depreciation rate of capital stock 
\(k_{12}\)  1  Step of capacity increase 
\(k_{13}\)  0.01  Dividend percentage 
\(k_{i,1}\)  (65, 35)  Initial production capacity of \(P_i\), \(1 \le i \le k_1\) 
\(k_{i,2}\)  (13, 13)  Price fixed by \(P_i\) for each unit of \(d_i\) 
\(k_{j,3}\)  (50, 30, 20)  Initial capacity of \(R_j\), \(1 \le j \le k_2\) 
\(k_{j,6}\)  (15, 15, 15)  Price fixed by \(R_j\) for each order of good j, \(1 \le j \le k_2\) 
6 Conclusions
In our work we have proposed an enhanced model for the classical producer–retailer problem. Not only basic interactions between producers and retailers are considered (production of goods from raw material, reception of orders from consumers and purchase transaction to match this goods and orders). In addition to those essential processes, a number of phenomena has been considered to get the model closer to the complexities of real world. We have taken advantage of modularity, one of the main benefits of membrane computing for modelling complex systems. Hence, the successive addition of complexity does not imply the necessity to build the model from scratch every time. Many real economic world interactions have been included in the model as new layers to the basic ones: capital stock depreciation, capacity increase decision mechanism, costs of capital (rents for its owners), dividend payments, and a general idea of making monetary units flow across the system. Additionally, randomness has been introduced, in several steps of the model, by means of a smart mechanism used in PDP world.
As described in previous sections, these ideas have been materialized through a model based on a specific P system framework (more specifically, PDP systems). This model has been then translated into PLingua and simulated using MeCoSim, where the system evolution has been analysed in depth. Some remarkable facts are that the final system can evolve autonomously without any exogenous influence, as it could be the provision of a certain number of elements at the beginning of each cycle. Although initial values of variables are settled, they change their values reaching an equilibrium point. Once this stability point has been reached, the system varies slightly around it. From the previous results, we can derive that multiple economic issues can be modeled using membrane computing. Therefore, more efforts must be done in this direction.
Notes
Acknowledgements
This work was partially supported by Grant numbers 61472328 and 61320106005 of the National Natural Science Foundation of China, and by the project TIN201789842P of the Spanish Ministry of Economy, Industry and Competitiveness.
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