Membrane Computing Applications in Computational Economics Eduardo Sánchez Karhunen

Transcription

1 Membrane Computing Applications in Computational Economics Eduardo Sánchez Karhunen BWMC 2017 Sevilla, February 3, 2017

2 Contents 1. Preliminaries 2. Producer Retailer problem: Initial Model Description. Formalization. Implementation in P Lingua & MeCoSim. Simulation & Results discussion. 3. Producer Retailer problem: Enhanced Model Description. Formalization. Implementation in P Lingua & MeCoSim. Simulation & Results discussion. 4. Further developments

3 Motivation Success of MC modeling biological systems Translation to unexplored field: Economic Modeling Replication of Păun s Producer - Retailer Problem results: Selection of the proper type of P System Economic processes modeling Implementation in P-Lingua & MeCoSim Simulate & discuss results Extension of the original model with new economic processes: Identification and modeling of processes Implementation & simulation Further developments

4 Why not extend to other fields? Computational economics: Computational modeling of economic systems (ODEs, ABM, ) Up-to-date efforts: Polish authors: Korczynski (2005) Păun s efforts: Membrane computing as a framework for modeling economic processes. In Proc. SYNASC 05, Timisoara, Romania, IEEE Press, 2005, Păun Gh. and Păun R. (2005) Păun Gh. and Păun R. Membrane Computing and Economics. In Păun Gh., Rozenberg G., Salomaa, eds. (2010) A. Handbook of Membrane Computing. Oxford University Press, 2010,

5 Păun s proposals Encourage researchers of other areas to use P Systems. Suggests modeling of some processes: Production of goods Order of goods Purchase transactions: Preferences between pairs (producer, retailer) Geographical barriers No distinction between counterparts Monetary unit exchange Capacity increase

6 Producer Retailer Problem

7 Model Entities Actors: Producers: (b i, u i ) -> (capacity, money) Retailers (c j, v j ) -> (capacity, money) Generic sources: Of raw material (u S, generation rate) Of demand or Generic consumer (u C, demand rate)

8 Model interactions Good s and order s flows: ҧ ҧ Producers generate good d from raw material. Retailers receive order d from generic consumer. d and d are matched Purchase P i,j = P(producer i, retailer j) CYCLIC PROCESS Monetary flows: Monetary unit exchange (u S, u C, u i, u j ) A set of prices. External monetary injection: Key role for system evolving.

9 Summarized actors & interactions

10 Păun s proposed system dynamic Presents a system behavior simulation:

11 Proposal - drawbacks Păun sketches the model: No indications about: Type of P System to be used. The sequence of steps of the cyclic behavior. The competing set of rules to be used. Probabilities associated to rules in a strange way. Non-as-usual Randomness introduced in a naive way.

12 Reproducing Păun s system evolution Define a so-called: Initial Model Steps: Select a type of P System -> PDP System. Probabilities associated to rules. Success in ecosystem modeling. Define the steps of the cycle: Associated to the transactions. Formalize the model. 2 1 skin Stablish the set of rules: Following Păun s guidelines. Avoid problems associated to strange probabilities.

13 Defining steps of cycle INITIALIZATION Aggregate demand creation Raw material disposability CLEANING 1 PRODUCTION Cleaning and technical rules 5 2 Orders generation Production of goods TRANSACTION AUTHORIZATION 4 3 Purchase transactions Purchase authorizations generation

14 ҧ Model Formalization (I) Π = (G, Γ, Σ, T, R E, μ, R Π, f r R Π, M 1, M 2 ) Where: G = (V, E) with V = e 1 and E = (e 1, e 1 ). PDP System of degree (2,1) Working alphabet: Γ = b i, d i, u i, c j, d j ҧ, v j, e j ҧ, f i,j 1 i k 1, 1 j k 2 {R 1, R 2 } C, S, d, ҧ a, u C, u S Where: C: aggregate generic consumer. S: raw material supplier. ҧ d: unit of aggregate demand from C. a: unit of supplied raw material provided by S. u C : monetary unit owned by C. u S : monetary unit owned by S. b i : unit of production capacity of producer i.1 i k 1. ҧ d j : unit of good demanded by retailer j. 1 j k 2. v j : monetary unit owned by retailer j.1 j k 2. e j : unit of good demanded by retailer and authorized for transaction unit of d j ҧ. 1 j k 2. f i,j : authorization for k 1, 1 j k 2. R 1, R 2 : for technical reasons. ҧ d j to be exchange with d i. 1 i d i : unit of good supplied by producer i. 1 i k 1. u i : monetary unit owned by producer i.1 i k 1. c j : unit of capacity of retailer j.1 j k 2.

15 Model Formalization (II) Σ =. R E =. Π = Γ, μ, M 1, M 2, R Π where: o Membrane structure: μ = [ [ ] 2 ] 1. o M 1 = C, S, R 1, R 2 k {b i,1 k i, u i,2 k j,3 i : 1 i k 1 } {c j : 1 j k 2 } Initial multisets contain basically: b i k i,1, u i k i,2 : producers initial parameters. c j k j,3 : retailers initial capacities. Where: k i,1 : initial production capacity of producer i. 1 i k 1. k i,2 : initial monetary units of producer i. 1 i k 1. k j,3 : initial capacity of retailer j. 1 j k 2.

16 Model Parameters Goal: maximize model parametrization k 1 : total number of producers. k 2 : total number of retailers. k 3 : units of raw material inserted into the system by S. k 4 : allowed deviation from k 3. k 5 : units of aggregate demand inserted into the system by C. k 6 : allowed deviation from k 5. k 7 : price fixed by S for each unit of a. k 8 : price fixed by C as an estimation of each order of good. k i,1 : initial production capacity of producer i. 1 i k 1. k i,2 : initial monetary units of producer i. 1 i k 1. k j,3 : initial capacity of retailer j. 1 j k 2. k m,4 : discrete prob. distribution of units of raw material inserted into the system by S. 1 m 3. k m,5 : discrete prob. distribution of units of aggregate demand inserted into the system by C. 1 m 3. k i,6 : price fixed by producer i for each unit of d i. 1 i k 1. k j,7 : price fixed by retailers j for each order of good. 1 j k 2.

17 Aggregate demand creation Raw material disposability Set of rules Initialization Cleaning and technical rules Orders generation Production of goods Step 1.a: raw material disposability Purchase transactions 4 3 Purchase authorizations generation r 1 R 1 s k 1,4 2 a k 3+k 4 + s R 1 2 r 2 R 1 s k 2,4 2 a k 3 + s R 1 2 r 3 R 1 s k 3,4 2 a k 3 k 4 + s R 1 2 k 3 : units of raw material inserted into the system by S. k 4 : allowed deviation from k 3. k m,4 : discrete prob. distr. of units of raw material inserted. k 5 : units of aggregate demand inserted by C. r 4 R 1 s 2 1 k 1,4 k 2,4 k 3,4 a k 3 2 k 4 s R k 6 : allowed deviation from k 5. k m,5 : discrete prob. distr. of units of aggr. demand inserted. Step 1.b: generic demand creation r 5 R 2 c 2 r 6 R 2 c 2 r 7 R 2 c 2 r 8 R 2 c 2 k 1,5 k 2,5 k 3,5 ҧ d k 5+k 6 u C (k 5 +k 6 ) k 8 c R ҧ d k 5 u C k 5 k 8 c R ҧ d k 5 k 6 u C (k 5 k 6 ) k 8 c R k 1,5 k 2,5 k 3,5 ҧ d k 5 2 k 6 u C (k 5 2 k 6 ) k 8 c R 2 2 +

18 Aggregate demand creation Raw material disposability Set of rules Production Cleaning and technical rules Orders generation Production of goods Purchase transactions 4 3 Purchase authorizations generation Step 2.a: producer operation r 9 a b i u i k u S k 7 d i i k 1 Step 2.b: retailer operation r 10 dҧ k c j u j,7 C 2 + d j ҧ k 0 v j,7 j 2 1 j k 2 k 1 : total number of producers. k 2 : total number of retailers. k 7 : price fixed by S for each unit of a. k j,7 : price fixed by retailers j for each order of good.

19 Aggregate demand creation Raw material disposability Set of rules Auth. & Trans. Cleaning and technical rules Orders generation Production of goods Step 3: Purchase auth. generation Purchase transactions 4 3 Purchase authorizations generation r 14 r 15 r 16 r 17 r 18 r 19 d 1 ҧ 2 d 1 ҧ dҧ dҧ dҧ dҧ 3 2 e 1 ҧ f 1,1 2 e 1 ҧ f 1,2 2 e 2 ҧ f 2,1 2 e 2 ҧ f 2,2 2 e 3 ҧ f 3,1 2 e 3 ҧ f 3,2 2 Geo-barriers Non-preferences Preferences k 1 : total number of producers. k 2 : total number of retailers. k i,6 : price fixed by producer i for each unit of d i. Solution: f i,j follows the probability distribution of the desired transactions probabilities. Step 4: Purchase transactions k 0 r 20 d i e j ҧ f j,i v i,6 j k bi c 2 j u i,6 i 2 1 i k 1, 1 j k 2

20 Aggregate demand creation Raw material disposability Set of rules Cleaning Cleaning and technical rules Orders generation Production of goods Purchase transactions 4 3 Purchase authorizations generation Step 5: cleaning rules Eliminate non-exhausted authorizations: r 26 f i,j i k 1, 1 j k 2 k 1 : total number of producers. k 2 : total number of retailers. Unauthorize non-exhausted e j ҧ : r 27 e j ҧ ҧ 0 d 2 j 2 1 j k 2 Signaling a new cycle: r 30 r 1, r 2 2 r 1, r 2 2 0

21 P - Lingua Set of rules has been implemented in P Lingua. An example for each set of rules: Initialization: / r2 / s, R 1 2 s, a k 3 + R 1 2 : k 2,4 ; Production: / r9 / b i, a, u i k us k 7 d i 2 : 1 1 i k{1} Authorization: / r18 / dn 3 2 en 3, f{3,1} 2 : 0.15 Transaction: / r20 / d i, en j, f j, i, v j k{i, 6} 2 b i, c j, u i k{i, 6} 2 : 1 1 i k 1, 1 j k{2}

22 Simplified trace

23 Simulation parameters Simulation tool: MeCoSim Parameters: same as Păun s paper Parameter Value/s Description k 1 2 Total number of producers k 2 3 Total number of retailers k 3 60 Units of raw material inserted into the system by S k 4 1 Deviation from k 3 k 5 60 Units of aggregate demand inserted into the system by C k 6 1 Deviation from k 5 k 7 11 Price fixed by S for each unit of a k 8 14 Price fixed by C as an estimation of each order of good k i,1 (65,35) Initial production capacity of producer i. 1 i k 1 k i,2 {750,400) Initial monetary units of producer i. 1 i k 1 k j,3 (50,30,20) Initial capacity of retailer j. 1 j k 2 k m,4 (0.01,0.95,0.03) Values of discrete probability distribution of units of raw material inserted into the system by S k m,5 (0.03,0.90,0.04) Values of discrete probability distribution of units of aggregate demand inserted into the system by C k i,6 (12,13) Price fixed by producer i for each unit of d i k j,7 (13,14,15) Price fixed by retailer j for each order of good j. 1 j k 2

24 MeCoSim definition Parameter Value Description k 1 k 2 k 3 <@r,1> Index 1 = 1 <@r,8> Index 2 = 2 Captures number of producers based on the number of rows in table Producer_input Captures number of retailers based on the number of rows in table Retailer_input Units of raw material inserted into the system by S k 4 Deviation from k 3 k 5 <9,$1$-2,2> Units of aggregate demand inserted into the system by C k Index 1 = 6 Deviation from k 5 [3..<@r,9>+2] k 7 Price fixed by S for each unit of a k 8 k i,1 <1,$1$,$2$+3> Price fixed by C as an estimation of each order of good Initial production capacity of producer i. 1 i k 1 k i,2 Index 1 = [1..k{1}] Index 2 = [1..2] Initial monetary units of producer i. 1 i k 1 k j,3 Index 1 = [1..k{2}] Initial capacity of retailer j. 1 j k 2 <8,$1$,4> Index 2 = 3 k <10,$1$,$2$-3> m,4 k m,5 Index 1 = [1..<@r,10>] Index 2 = [4..5] k i,6 <1,$1$,6> Index 1 = [1..k{1}] Index 2 = 6 k j,7 <8,$1$,5> Index 1 = [1..k{2}] Index 2 = 7 Values of discrete probability distribution of units of raw material inserted into the system by S Values of discrete probability distribution of units of aggregate demand inserted into the system by C Price fixed by producer i for each unit of d i Price fixed by retailer j for each order of good j. 1 j k 2

25 Simulation results monetary units Producers monetary units Retailers monetary units

26 Simulation results - capacities Producers capacities Retailers capacities

27 Simulation results - comparison Păun's evolution Initial model evolution

28 Enhanced Model Summarized behavior of Initial Model: A steady increase of monetary units owned by producers, retailers and generic consumer. Nearly stable producer s and retailer s capacities. Monetary units obtained by raw source of material get out of circulation. Why? Producers & retailers capacities are fixed and no changes are allowed. Raw material and aggregate demand are initially settled and remain unchanged during the system evolution. Artificial exogenous injection of monetary units into consumer C at the beginning of each cycle. This flow is necessary to maintain system evolving.

29 Enhanced Model Getting closer to real situations: Allowing variations of producers and retailers capacities: Capital stock depreciation. Investment or capital increase decision. Remove external injection of monetary units: Payment of rents to the owners of the production factors. Raw material source is owned by the aggregate consumer. Aggregate consumer is stakeholder of producers and retailers, thus implying dividends payments. Inclusion of randomness in a PDP-way: Raw material generation. Aggregate demand generation. Mechanism of capacity increase decision.

30 Producer Retailer Enhanced Model

31 Model Entities Actors: Producers: (b i, u i ) -> (capacity, money) Retailers (c j, v j ) -> (capacity, money) Generic sources: Of raw material (u S, generation rate) Of demand or Generic consumer (u C, demand rate)

32 Model interactions Good s and order s flows: Producers generate good d from raw material. ҧ Retailers receive order d from generic consumer. d and ҧ d are matched Purchase P i,j = P(producer i, retailer j) CYCLIC PROCESS External monetary injection: Removed.

33 Additional interactions Monetary flows: Initial Model monetary exchange due to prices. Rents payments to owners: Generic Consumer. Dividends payments to stakeholders: Generic Consumer. Raw material source owners: Generic Consumer. Capacity variations: Producers capacity depreciation. Producers capacity increase decision: nonsatisfied demand from retailers. CYCLIC PROCESS

34 Defining steps of cycle INITIALIZATION Capacity cost payments Aggregate demand creation Raw material disposability EVOLUTION 1 PRODUCTION Dividend payments Capacity increase decision Capacity depreciation 5 2 Orders generation Production of goods TRANSACTION AUTHORIZATION 4 3 Purchase transactions Purchase authorizations generation BASIC MODEL

35 Model Formalization (I) Π = (G, Γ, Σ, T, R E, μ, R Π, f r R Π, M 1, M 2 ) Where: G = (V, E) with V = e 1 and E = (e 1, e 1 ). Working alphabet: Γ enhanced = Γ initial / u S, R 2 g i, y i,, m i, z i, h i : 1 i k 1 } { p, q Where: C: aggregate generic consumer. S: raw material supplier. ҧ d: unit of aggregate demand from C. a: unit of supplied raw material provided by S. u C : monetary unit owned by C. b i : unit of production capacity of producer i.1 i k 1. d i : unit of good supplied by producer i. 1 i k 1. u i : monetary unit owned by producer i.1 i k 1. c j : unit of capacity of retailer j.1 j k 2. ҧ d j : unit of good demanded by retailer j. 1 j k 2. v j : monetary unit owned by retailer j.1 j k 2. e j ҧ : unit of good demanded by retailer and authorized for transaction unit of ҧ d j. 1 j k 2. PDP System of degree (2,1) f i,j : authorization for k 1, 1 j k 2. R 1 : for technical reasons. ҧ d j to be exchange with d i. 1 i p: randomness generator for a provision by S. q: randomness generator for ҧ d generation by C. h i : unit of production capacity of producer i before depreciation.1 i k 1. y i : unit (in idle state) of aborted purchase transactions considered for capacity increase. 1 i k 1. m i : randomness generator for y i. 1 i k 1. z i : activated unit of aborted purchase transactions considered for capacity increase. 1 i k i k 1. g i : for technical reasons. 1 i k 1

36 Model Formalization (II) Σ =. R E =. Π = Γ, μ, M 1, M 2, R Π where: o Membrane structure: μ = [ [ ] 2 ] 1. o M 1 = C, S, R 1 k g i, u i,1 k 10 7 k j,3 k 10 7 i : 1 i k1 }, {v j : 1 j k2 o M 2 = k j,3 c j : 1 j k 2 k {b i,1 i : 1 i k 1 } Initial multisets contain basically: b i k i,1, u i k i,1 k 10 7 : producers initial parameters. c j k j,3, v j k j,3 k 10 7 : retailers initial parameters. They need same initial amount of monetary units to pay initial capacity costs. Where: k i,1 : initial production capacity of producer i. 1 i k 1. k j,3 : initial capacity of retailer j. 1 j k 2.

37 Model Parameters Goal: maximize model parametrization k 1 : total number of producers. k 2 : total number of retailers. k 3 : raw material inserted into the system by S minimum value of range k 4 : raw material inserted into the system by S maximum value of range. k 5 : aggregate demand inserted into the system by C minimum value of range. k 6 : aggregate demand inserted into the system by C maximum value of range. k 7 : price fixed by S for each unit of a. k 8 : number of failed purchases considered for the analysis of increasing capital stock minimum value. k 9 : number of failed purchases considered for the analysis of increasing capital stock maximum 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 producer i. 1 i k 1. k i,2 : price fixed by producer i for each unit of d i. 1 i k 1. k j,3 : initial capacity of retailer j. 1 j k 2. k i,6 : price fixed by retailers j for each order of good. 1 j k 2.

38 Set of rules Initialization From Naïve randomness: r 5 R 2 c 2 r 6 R 2 c 2 r 7 R 2 c 2 r 8 R 2 c 2 k 1,5 k 2,5 k 3,5 ҧ d k 5+k 6 c R ҧ d k 5 c R ҧ d k 5 k 6 c R k 1,5 k 2,5 k 3,5 ҧ d k 5 2 k 6 c R Generates ҧ d around k 5 To a PDP-way: raw material disposability & generic demand creation: r 1 R 1 s c 2 a k 3 p k 4 k 3 ҧ d k 5 q k 6 k 5 s c R r 2 p r 3 p a 2 + r 4 q Generates [ dҧ k 5, dҧ k 6] Generates [a k 3, a k 4] r 5 q ҧ d 2 +

39 Set of rules Capacity costs Rents for capacity: Generic consumer is the owner of production factors. Agents have enough monetary units to pay for capacity: k r 9 u 10 k i b i 2 b i u 10 C i k 1 k r 10 v 10 k j c j c 2 j u 10 C j k 2 Agents are not able to pay for capacity: r 11 b i 2 + u C k i k 1 r 12 c j 2 + uc k j k 2 k 1 : total number of producers. k 2 : total number of retailers. k 10 : cost of capital stock per cycle.

40 Set of rules Operations Main changes: Generic consumer is the owner of raw material source Producer operation: r 14 a b i u i k u C k 7 d i i k 1 k 1 : total number of producers. k 2 : total number of retailers. k 7 : price fixed by S for each unit of a. Retailer operation: r 15 dҧ k c j u j,6 C 2 + d j ҧ k 0 v j,6 j 2 1 j k 2 k i,6 : price fixed by retailers j for each order of good. k j,7 : price fixed by retailers j for each order of good. Unused capacities: r 16 b i 2 b i 2 1 i k 1 r 17 c j 2 c j 2 1 j k 2 Retired from the operational membrane waiting for their depreciation.

41 Set of rules Auth. & Transactions Purchase authorization generation r 18 ҧ d 1 2 e 1 ҧ f 1,1 2 r 19 d 1 ҧ 2 0 e 1 ҧ f 1,2 2 Geo-barriers r 20 r dҧ dҧ 2 2 e 2 ҧ f 2,1 2 e 2 ҧ f 2,2 2 Non-preferences r 22 r dҧ dҧ 3 2 e 3 ҧ f 3,1 2 e 3 ҧ f 3,2 2 Preferences k 1 : total number of producers. k 2 : total number of retailers. Purchase transactions k 0 r 24 d i e j ҧ f j,i v i,2 j k ui i,2 h 2 i c j 2 k i,2 : price fixed by producer i for each unit of d i. 1 i k 1, 1 j k 2 b i are retired as h i from the operational membrane waiting for their depreciation.

42 Set of rules - Evolution Dividend payment: r 25 v j 2 vj j k 2 Both blocks of rules only applied to producers r 26 u i 2 k 13 u C i k 1 r 27 u i 2 1 k 13 u i i k 1 Capacity depreciation: r 31 h i 2 1 k 11 r 32 h i 2 k 11 b i i k i k 1 k 1 : total number of producers. k 2 : total number of retailers. k 11 : depreciation rate of capital stock. k 13 : dividend percentage.

43 Set of rules capacity increase When strictly necessary only Trigger: non-exhausted f j,i Case a: Enough producer capacity: r 28 f j,i d i 0 di i k 1, 1 j k 2 1 k 11 0 r 29 f j,i h i b 2 i 2 1 i k 1, 1 j k 2 r 30 f j,i h i 2 k i k 1, 1 j k 2 Case b: Not enough producer capacity: r 6 g i 2 0 g i y i k 8 m i (k 9 k 8 ) i k 1 r 7 m i i k 1 r 8 m i y i i k 1 k 1 : total number of producers. k 2 : total number of retailers. k 8 : number of failed purchases considered for the analysis of increasing capital stock min value. k 9 : number of failed purchases considered for the analysis of increasing capital stock max value. k 11 : depreciation rate of capital stock.. Generates [y i k 8, y i k 9 ] r 33 y 0 i 2 z i 2 1 i k 1 0 r 34 f j,i z i k bi i k 1, 1 j k 2

44 Set of rules Cleaning Cleaning rules and technical rules Eliminate non-exhausted authorizations: r 35 f j,i i k 1, 1 j k 2 r 36 z i i k 1 k 1 : total number of producers. k 2 : total number of retailers. Unauthorize non-exhausted e j ҧ : r 13 v j 2 + v j 2 0 r e j ҧ dҧ 2 j 1 j k2 2 Signaling a new cycle: r 38 r 1 2 r j k 2 r 39 g i 2 g i j k 2

45 P - Lingua Set of rules has been implemented in P Lingua. An example for each set of rules: Initialization: / r1 / s, c, r1 2 s, c, a k{3}, p (k{4} k{3}), dn k{5}, q (k{6} k{5}) + r1 2: : 1 Production: / r9 / u i k 10 b i 2 b i, uc k : 1 1 <= i <= k{1}; Transaction: / r24 / d i, en j, f j, i, v j k{i, 2} 2 u i k{i, 2} h i, c j, 2 : 1 : 1 i k 1, 1 j k{2} Capacity increase: / r34 / f j, i, z i 2 b i k : 1 1 i k 1, 1 j k{2}

46 Simplified trace c j b i R C S R v j u i g i 1 STEP 1: Generic demand generation Supply creation Capacity costpayment R 1 g i m i C S p q + 2 v j u i c j b i 1 STEP 1: Generic demand generation Supply creation Capacity costpayment STEP 2: Production of goods Order generation 0 v j u i 0 0 u i 0 c j d i z i C S R 1 d ҧ a u2 C c j g i 1 v j R 1 g i y i u2 C c j b i C S d ҧ a p q u C 1 STEP 5: Dividend payment Capacity depreciation Capacity increasedecision STEP 3: Generation of purchase transaction authorizations v j R 1 g i y i -- u2 C c j u i C S dҧ a p q u C 0 STEP 4: Purchase transactions, h i v j d i R 1 g i y i 0 u2 C c j u i C S d ҧ a p q u C 0 1

47 Simulation parameters Parameters: similar to Păun s paper Parameter Value Description k 1 2 Total number of producers k 2 3 Total number of retailers k 3 59 Units of raw material inserted into the system by S minimum value of range k 4 62 Units of raw material inserted into the system by S maximum value of range k 5 59 Units of aggregate demand inserted into the system by C minimum value of range k 6 62 Units of aggregate demand inserted into the system by C maximum value of range k 7 11 Price fixed by S for each unit of a k 8 3 # failed purchases considered for the analysis of increasing capital stock minimum value. k 9 5 # failed purchases considered for the analysis of increasing capital stock maximum value. k 10 2 cost of capital stock per cycle k depreciation rate of capital stock k 12 1 step of capacity increase k Dividend percentage k i,1 (65,35) Initial production capacity of producer i. 1 i k 1 k i,2 {13,13) Price fixed by producer i for each unit of d i. 1 i k 1 k j,3 (50,30,20) Initial capacity of retailer j. 1 j k 2 k i,6 (15,15,15) Price fixed by retailer j for each order of good j. 1 j k 2

48 MeCoSim definition Parameter Value Description k 1 k 2 k 3 <@r,1> Index 1 = 1 <@r,8> Index 2 = 2 Captures number of producers based on the number of rows in table Producer_input Captures number of retailers based on the number of rows in table Retailer_input Units of raw material inserted into the system by S minimum value of range k 4 Units of raw material inserted into the system by S maximum value of range k 5 Units of aggregate demand inserted into the system by C minimum value of range k 6 k 7 k 8 <9,$1$-2,2> Index 1 = [3..<@r,9>+2] Units of aggregate demand inserted into the system by C maximum value of range Price fixed by S for each unit of a # failed purchases considered for the analysis of increasing capital stock minimum value. k 9 # failed purchases considered for the analysis of increasing capital stock maximum value. k 10 k 11 k 12 k 13 k i,1 <1,$1$,$2$+1> Cost of capital stock per cycle Depreciation rate of capital stock Step of capacity increase Dividend percentage Initial production capacity of producer i. 1 i k 1 k i,2 Index 1 = [1..k{1}] Price fixed by producer i for each unit of d i. 1 i k 1 Index 2 = [1..2] k j,3 <8,$1$,2> Index 1 = [1..k{2}] Initial capacity of retailer j. 1 j k 2 Index 2 = 3 k i,6 <8,$1$,3> Index 1 = [1..k{2}] Price fixed by retailer j for each order of good j. 1 j k 2 Index 2 = 6

49 Simulation results capacities Producers capacities with: Depreciation rate = 0.1 Deactivated capacity increase mechanism. Producers capacities with: Depreciation rate = 0.1 Activated capacity increase mechanism.

50 Simulation results dividends Generic consumer monetary units: Deactivated dividend payment. Generic consumer monetary units: Restored dividend payment.

51 Simulation results producer Initial distribution of capacities: Raw material generation rate [59,62] Raw material generation rate [40,43] System tries to reach an equilibrium point in function of parameters of S

52 Simulation results retailer Initial distribution of capacities: Generic demand generation rate [59,62] Generic demand generation rate [40,43] System tries to reach an equilibrium point in function of parameters of C

53 CONCLUSIONS Initial model: We have been able to reproduce Păun s results using: PDP systems. P Lingua & MeCoSim framework. Inference engine DNDP4. Enhanced model: Initial model has been extended including several real world economic processes. Cost of production factors, dividend payment. Capacity depreciation, capacity increase mechanisms. Removing external injection of monetary units. Model evolves autonomously around an equilibrium point different from the initial conditions.

54 FURTHER DEVELOPMENTS Complete the enhanced model: Macroeconomics interest: behavior of system under perturbation around equilibrium. Introduce mechanisms to adjust prices to some stimulus. Investigate if different patterns of randomness could be generated easily. Future Case of Study: SDGE (Stochastic Dynamic General Equilibrium). Previous techniques can be utilized in this problem. Challenge: generate an emergent optimization behavior.

55 References Gh. Păun, R. Păun: Membrane computing as a framework for modeling economic processes. In Proc. SYNASC 05, Timisoara, Romania, IEEE Press, 2005, Păun Gh and Păun R. Membrane Computing and Economics. In Păun Gh, Rozenberg G., Salomaa, eds. (2010) A. Handbook of Membrane Computing. Oxford University Press, 2010, Pérez-Hurtado I, Valencia L, Pérez-Jiménez MJ, Colomer MA, Riscos A (2010) MeCoSim: A general purpose software tool for simulating biological phenomena by means of P Systems. In: K Li, Z Tang, R Li, AK Nagar, R Thamburaj (eds.) Proceedings 2010 IEEE Fifth International Conference on Bioinspired Computing: Theories and Applications: Vol 1. Changsha: IEEE Press Díaz D, Pérez-Hurtado I, Pérez-Jiménez MJ, Riscos A (2009) A P-Lingua programming environment for Membrane Computing. LNCS 5391: