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Logist. Res. (2011) 3:159–175 DOI 10.1007/s12159-011-0054-9 OR IGINAL PAPER Modeling production networks with discrete processes by means of communities of autonomous units • • Hans-Jo ¨ rg Kreowski Sabine Kuske Caroline von Totth Received: 10 September 2010 / Accepted: 18 March 2011 / Published online: 19 April 2011 Springer-Verlag 2011 Abstract Communities of autonomous units are devices factors. Logistic processes with central control are not for the visual modeling of interactive logistic processes. realistic and flexible enough to deal with this situation—in The framework is founded on rule-based graph transfor- particular, if parties with different interests are involved. mation and allows specifying autonomous units in such a The use of interactive processes with decentralized and way that they can run in parallel and make their decision autonomous control is more adequate and may solve the about future actions independently of each other. The problem, but can also cause new difficulties. How can usefulness of the framework is demonstrated in this paper one guarantee that the autonomous processes cooperate by modeling a new variant of production networks with properly? How can interactive processes be described in a discrete production processes. One of the main results suitable way? How can one analyze their behavior and shows that the visual model is correct with respect to a prove desired properties like conflict freeness, termination more traditional quantitative mathematical model. in time, and reachability of goals? To answer these ques- tions, formal modeling methods for cooperating autono- Keywords Production networks Autonomous units mous processes are indispensable. In logistics, there are Discrete production processes Stability Graph various modeling methods like business process models, transformation UML, Petri nets, multi-agent systems, particle swarms and systems of equations and inequalities. However, most of them do not provide both: a formal framework to prove 1 Introduction properties on one hand and features to express decentral- ization, interaction, cooperation, and autonomy on the Logistic systems and networks become more and more other hand. Logistic modeling of today often relies on dynamic and structurally complex due to the fast changes testing and simulation which allow to track down unwanted behavior and to guarantee that a finite collection of in customers’ demands, the wide spectrum of employed technologies, the growing globalization, and other such inputs works properly, but cannot make sure that desired properties hold for the whole system and all its runs (cf., e.g., [12]). As an alternative approach, we offer communities of autonomous units as devices for formal and visual modeling of interacting logistic processes (cf. [10, 11, 13]). H.-J. Kreowski (&) S. Kuske C. von Totth In this paper, we demonstrate their usefulness by intro- Department of Mathematics and Computer Science, University ducing and studying a new variant of production networks of Bremen, P.O. Box 330 440, 28334 Bremen, Germany e-mail: kreo@informatik.uni-bremen.de where the usual mathematical models are accompanied by specifications as communities of autonomous units. S. Kuske Production networks are investigated in many variants e-mail: kuske@informatik.uni-bremen.de in logistics and control theory mainly with continuous C. von Totth production processes (cf., e.g., [1–3, 6, 14–16]). A major e-mail: caro@informatik.uni-bremen.de 123 160 Logist. Res. (2011) 3:159–175 topic is the stability meaning that the quantities in a pro- 5. From the point of view of logistics, our framework duction network do not grow beyond any bound. While provides the basic elements of a language to model there are many applications for which the assumption of interactive logistic processes with autonomous control continuous inflow, outflow, production, and distribution is in a visual way. This allows to enhance the usual simulation of processes by a visual simulation on adequate, there are others for which a stepwise processing is more realistic (e.g. if production and distribution are graph transformation tools. Furthermore, as the pro- cess semantics is formally defined, one can prove done from time to time only). In this paper, we model production networks with dis- properties of the running processes like termination and correctness. There is also the future perspective of crete production processes in two ways. We start with a quantitative mathematical model as it is quite common in tool support because several research activities on logistics. Based on the mathematical model, two results can automatic verification of graph transformation are be proven that stress the basic properties of this kind of going on. production networks. The first result states that production The paper is organized in the following way. In the processes are free of loss because all quantities at all pro- following section, the basic notions of production networks duction sites that are delivered or put in during the process and their discrete processes are introduced by means of are stored or distributed or put out eventually. The second quantitative modeling. In Sect. 3, a sufficient condition for result concerns deterministic production networks for the stability of deterministic production networks is given. which we can provide a stability criterion. But this suffi- Section 4 treats production networks as communities of cient condition applies only to particular production net- autonomous units where this rule-based and visual works. Therefore, one may wonder how the other networks modeling framework is introduced, too. In Sect. 5,itis run and work. For this purpose, we propose a second model discussed shortly how the visual simulation of production of production networks by means of communities of networks is implemented on the graph transformation autonomous units. engine GrGen.NET (cf. [8]). Section 6 deals with a The modeling framework of communities of autono- generalization of production networks by rules that change mous units offers the following features of interest: distribution rates, which are constant in the basic model. 1. It is rule-based so that the operational semantics Section 7 concludes the paper by pointing out some topics provides the running processes. In other words, of future research. production processes do not have to be defined on a metalevel, but are given by the framework. Moreover, the operational semantics is independent of any 2 Production networks and discrete production processes particular implementation so that it is easier to understand than programing code. In this section, our most elementary notions of production 2. As it is based on graphs, it is visual so that one can look at the running processes and see what happens— networks and their processes are introduced. It follows the at least in principle. In practice, one needs visual concept of supply and production networks as studied, for simulation tools which are available for the framework example, in [5, 9], but we replace continuous production in prototypical form (see Sect. 5 for more details). processes by stepwise processing. 3. It supports autonomy in form of control conditions. A production network consists of a set of production Each autonomous unit has a control condition to sites, which are numbered from 1 to n (without loss of decide which rule is applied next out of all possible generality). Some of them are also input sites and some rule applications. But one can also employ control output sites. Each site has got a maximum production rate. Each input site has a maximum input rate in addition, and conditions on the community level to coordinate the interacting processes of the units and to avoid chaotic each output site a maximum output rate accordingly. All of behavior in this way. them serve as upper bounds. But for greater flexibility, 4. It provides parallelism so that autonomous units can each of them may be infinite so that no restriction is run simultaneously. Parallel processing may be possi- imposed in this case. The dynamics of a production net- ble or mandatory. In the latter case, synchronization is work is given by production processes, which consist of organized by a special type of control condition. stepwise inputs, changes of production states, and outputs. Moreover, the framework of graph transformation A production state provides a quantity per site. To change a provides means to find out which activities can be current state, an input rate, a production rate, and an output performed in parallel without conflicts and whether rate are chosen for each site. The input rate is added to the required parallelism is possible at all. current quantity, the output rate is part of the production 123 Logist. Res. (2011) 3:159–175 161 rate which is subtracted from the quantity, and the differ- Remarks ence of the production rate and the output rate is distributed. 1. A production site j with maxin [ 0 is called an input The fraction each site gets is established by the distribution site and, accordingly, an output site if maxout [ 0. matrix of the network. This yields a follow-up state so that 2. The graph underlying PN is given by G(PN)) = such steps can be iterated. To start a process, an initial state ([n], E) with E ¼fði; jÞ2½n j d [ 0g meaning that ij is chosen. Moreover, some conditions are required. The the sites are the nodes and there is an edge from site input rates, the production rates, and the output rates should node i to site node j whenever the distribution rate d is ij never exceed their maxima. The production rates chosen for greater than 0. some next step should not exceed the current quantities. And 3. The introduced notion is oriented on the concepts of the distribution rates should make sure that the amount of production networks in [5, 9]. Many variations and site quantites that is distributed equals the sum of fractions extensions are possible, like lower bounds for input, that arive at the sites. If one wants to emphasize the network production and output in addition to the upper bounds, aspect, then one may consider the graph underlying a pro- or variable distribution rates rather than invariant ones. duction network with the sites as nodes and an edge from This is further discussed in Sects. 6 and 7. site i to site j whenever the (i, j)-entry of the distribution matrix is greater than 0. Example 1 A sample production network SAMPLE is given In a more concrete setting, one may think of production by the components of Fig. 1. It has seven production sites: sites as various plants that are connected by roads or tracks 2 and 5 are the input sites with 100 and 200, respectively, so that raw material is delivered, processed material is as maximum input rates and 1, 3, and 6 are the output sites transported from plants to other plants according to some with unbounded output. The distribution matrix and the distribution plan, and finished material is taken away by vector of maximum production rates complete the quanti- trucks or trains within a certain period of time. This tative model. The underlying graph is depicted in Fig. 2.It establishes a production cycle, and the iteration of such is extended in Sect. 4 where a visual representation of steps defines a production process. entire production networks is presented. Before production networks and their processes are defined formally, some notational conventions are needed. Definition 2 (Production process) Let PN be a produc- The set of natural numbers is denoted by N, Nnf0g is tion network. denoted by N and [k] denotes the subset f1; ...; kg of N. [ 0 1. A production state is a vector of site quantities q 2 R . The set of real numbers is denoted by R; we use R to 2. Let q be a production state, in; p; out 2 R vectors of describe the set of non-negative real numbers with 0. input rates, production rates, and output rates, respec- Given a set X and n 2 N, the set of all n-vectors x ¼ tively, with in B maxin, p B max, out B maxout and ðx ; ...; x Þ with x 2 X for i 2½n is denoted by X .If X is 1 n i out B p B q. Then, the follow-up state q is defined by ordered by B, then the order is extended to X where x B y for x; y 2 X means x B y for i 2½n. Accordingly, the set X i i q ¼ q þ in p þ d ðp out Þ for j 2½n: j j j ij i i of all infinite sequences y ¼ðy Þ with y 2 X for i 2 N is j i i i2N i¼1 denoted by X . The set of all n,n-matrices a ¼ða Þ ij i;j2½n n9n with a 2 X for i; j 2½n is denoted by X . The extra ij symbol 1 denotes infinity and is greater than every real number, i.e. r\1 for r 2 R. Definition 1 (Production network)A production network PN consists of • a set [n]of production sites for some n 2 N, • vectors maxin; max; maxout 2ðR [ f1gÞ the entries of which are called maximum input rates, maximum production rates, and maximum output rates, respec- tively, and nn • a distribution matrix d 2 R with d ¼ 1 ij j¼1 for i 2½n. Fig. 1 An example of a production network 123 162 Logist. Res. (2011) 3:159–175 ð0; 0; 0; 0; 0; 0; 0Þ!ð0; 100; 0; 0; 200; 0; 0Þ !ð50; 100; 0; 100; 200; 0; 150Þ !ð50; 125; 62:5; 100; 225; 137:5; 150Þ As in each step of a production process each quantity is partly kept, partly put out, and partly distributed, one may expect that the overall quantity in a production network is fully established by the initial quantities, all the inputs, and all the outputs and that there is no loss. To state this precisely, some notations are needed concerning the summation of inputs, outputs, and site quantities. n N Let q 2ðR Þ be a production process with q ! q for j 2 N . Then, Q ¼ q denotes j1 j [ 0 k kj j¼1 in ;p ;out Fig. 2 The underlying graph of the production network SAMPLE j j j the overall quantity of state q for k 2 N. And, for P P P P k n k n k 2 N , In ¼ in and Out ¼ [ 0 k ji k j¼1 i¼1 j¼1 i¼1 3. This construction is called a production step from q to out denote the cumulated inputs and the cumulated out- ji q and is denoted by puts, respectively, up to step k. Moreover, In = Out = 0. 0 0 q ! q : n N in;p;out Theorem 1 Let q 2ðR Þ be a production process with q ! q for j 2 N . Then, the following holds: j1 j [ 0 4. A production process is an infinite sequence of in ;p ;out j j j production states q 2ðR Þ such that, for every Q ¼ Q þ In Out for all k 2 N: k 0 k k k 2 N there are vectors in ; p ; out 2 R with [ 0 k k k The proof is omitted. The theorem can be shown by q ! q . k1 k in ;p ;out k k k induction on the number of steps of a production process. Our notion of production networks covers various special Remarks cases one encounters in the literature like networks with a 1. It should be noted that, given a production state, a single input site or a single output site (or both) and like acyclic production step is always defined for any choice of networks where distributed quantities never come back to the input rates, production rates, and output rates. There- distributing site. One may also get rid of the bounds and of the fore, it causes no problems to assume that production restrictions they impose on the free choice of input rates, processes run forever. production rates, and output rates by setting them to 1.Orone 2. But if one is interested in finite processes, then one can may require additional properties of production processes like consider just the prefixes q ; ...; q of a production 0 k constant input rates or output rates proportional to the pro- process q 2ðR Þ for some k 2 N: duction rates, or exhaustive production rates that use up the site 3. In particular, one may consider production processes quantities up to the maximum production rates. These three properties together define a kind of deterministic production q 2ðR Þ as finite if there is an activity bound k 2 N network that is further considered in the next section. such that all input, production, and output rates for Instead of restrictions, one may also relax the notion of pro- l [ k are 0 and the quantity vectors become invariant, duction networks. For example, the condition that the distribu- i.e. q = q . Then, only the sequence up to k matters at l k tion rates of one site to all sites sum up to 1 may be replaced by all and q can be considered as final state. 4. In a production step, the difference of the production d 1: ij rate and the output rate is distributed to the neighbor j¼1 sites. Hence, it may be called distribution quantity. This would mean that a certain part of the distribution quantity gets lost in each step. Or the condition may be Example 2 Given the production network SAMPLE of Example 1, it may start with the initial state dropped completely allowing an increase in quantities (0, 0, 0, 0, 0, 0, 0), put in the maximum input in each step, while distributed. There may be even cases where negative choose always each current quantity as production rate, put quantities are meaningful. this out at site 1 and 6 and half of it at site 3. Then, one gets There are at least two further aspects that could be subject to generalization. Instead of having only one quantity per site a production process with the following first three steps: 123 Logist. Res. (2011) 3:159–175 163 and step meaning that only one kind of material or goods is fixed if the output factors are fixed such that there is measured, one may consider a vector of quantities reflecting only a single production process for each initial state. a variety of products. Moreover, it may be meaningful to 2. As the production rates use up the site quantities up to replace the static distribution matrix by a dynamic one. The the maxima, they are called exhaustive. latter case is further discussed in Sect. 6. Theorem 2 Let PN be a deterministic production net- work with the constant input vector in = maxin, the dis- 3 Deterministic production networks and stability tribution matrix d and the vector of output factors a. Let m 2 R be a solution of the system of linear equations In most applications of production networks, a site has a ðE ðdðaÞÞ Þ x ¼ in bounded storage capacity so that the question of stability becomes important. A production network is stable if the where E is the unit matrix, d(a) is given by d(a) = d ij ij site quantities of each production process do not exceed a (1 - a ) for i; j 2½n and (d(a)) is the transposed matrix. fixed bound. It will be shown in this section that deter- Let q 2ðR Þ be a production process of PN with ministic production processes are stable if a certain system q B m B max. Then PN is stable. of linear equations has non-negative solutions. Again, the proof is omitted. It can be carried out by In this context, a production network is called deter- induction on the number of production steps. ministic if its inputs are constant, its production rates are chosen exhaustively meaning that the site quantities are Remarks used up to the limit of the maximum production rates, and 1. It is worth noting that the unique production process of if its outputs are certain fractions of the production rates. the deterministic production network becomes con- Definition 3 Let PN be a production network. stant if the initial state is chosen as the solution of the N system of linear equations. In other words, one can 1. A production process q 2ðR Þ in PN is stable if show q = m for all k 2 N. n k there is an upper bound vector m 2 R such that 2. Moreover, Theorem 2 still holds if one relaxes the q B m for each k 2 N. assumption of constant inputs. Let PN be an arbitrary 2. PN is stable with respect to an upper bound vector production network and q 2ðR Þ be a production n n þ m 2 R if each production process q 2ðR Þ in PN þ þ process with the input vectors in for k 2 N , k [ 0 with q B m is stable. exhaustive production rates and output rates that are 3. PN is deterministic with respect to some vector of determined by a vector a of output factors. Further- output factors a 2 R with a B 1 for j 2½n if every more, let m be chosen as in the theorem. In other production process q 2ðR Þ with q ! q k kþ1 þ words, the assumptions of the theorem are fulfilled up in ;p ;out kþ1 kþ1 kþ1 to the constant input condition. Production networks for k 2 N is subject to the following further conditions: with variable input turn out to be stable if all other • in = maxin, k?1 assumptions are fulfilled. • p = min (q , max ) for j 2½n, and (k?1)j kj j • out ¼ a p for j 2½n. ðkþ1Þj j ðkþ1Þj Example 3 Consider the production network SAMPLE of Example 1 as a deterministic one with the constant input Remarks vector maxin and the vector a = (1, 0, 0.5, 0, 0, 1, 0) of output factors. Then, Theorem 2 applies to SAMPLE. Its 1. This means that the input is constant, the production matrix E - (d(a)) and constant input vector in are rates are uniquely determined, and the output rates are 0 1 0 1 1 0:50 0 0 0 0 0 B C B C 01 0 0:25 0 0 0 100 B C B C B C B C 00 0:625 0:25 0 0 0:25 0 B C B C B C B C E ðdðaÞÞ ¼ 0 0:50 1 0:25 0 0 in ¼ 0 B C B C B C B C 00 0 0:25 1 0 0 200 B C B C @ A @ A 00 0:125 0:25 0 1 0:75 0 00 0 0 0:75 0 1 0 123 164 Logist. Res. (2011) 3:159–175 and the solution of the corresponding system of linear 1. Choose Choose an input, an output, and a production equations is given by: rate subject to the conditions of Definition 2. 2. Output Subtract the output rate from the production 850 1700 m ¼ 65:38; m ¼ 130:77; 1 2 rate. 13 13 3. Distribute Distribute the remaining distribution quan- 1540 1600 m ¼ 118:46; m ¼ 123:08; 3 4 tity to the neighbor sites according to the distribution 13 13 matrix. 3000 2280 m ¼ 230:77; m ¼ 175:38; 5 6 4. Calculate the new quantity as follows: 13 13 (a) Subtract Subtract the production rate from the m ¼ 173:08: actual quantity q. (b) Add Add to the obtained quantity the input rate as Stability is a very important property of a production well as all amounts obtained from the neighbors network because it makes sure that there will never be a in their distribution steps. shortage of storage capacity provided that the capacity is chosen according to the stability bound. If a network is Production networks together with their actual states can unstable, it means that the input quantities are not distributed be modeled as graphs in a natural way where the production in such a way that all inputs are put out eventually, so that it sites are represented as nodes labeled with their actual piles up at some of the sites. To avoid this effect, the quantities and the non-zero values d of the distribution ij distribution rates should be adaptable to the waiting quantities matrix d are represented as edges from i to j labeled with d . ij at the receiving site following the principle that a site should Consequently, the steps of production processes can be get less input whenever its current quantity is high. The idea modeled as graph transformations. Since the common to readjust distribution rates and to get a more balanced environments of communities are graphs and the actions of distribution of quantities in this way is further considered in units are graph transformation rules, communities of Sect. 6 where the production sites can decide about the autonomous units are well suited to specify production net- quantities they deliver to neighbor sites in dependence of the works so that the described behavior of the sites can be quantities that are present there. For this purpose, we remodel directly modeled by the autonomous units of the community. production networks as communities of autonomous units in More precisely, the ingredients of autonomous units are Sect. 4 , which allow to dynamize the distribution rates of taken from an underlying graph transformation approach each site by adding new rules to the site unit. providing a class G of graphs, a class R of graph transfor- mation rules together with an operator ¼) that specifies how 4 Production networks as communities of autonomous to apply the rules to graphs, a class C of control conditions, units and a class X of graph class expressions for specifying goals or environment properties, i.e., every expression x of X In this section, we show how production networks can be specifies a set SEM(x) of graphs in G. The environments that modeled as communities of autonomous units introduced in are transformed by communities belong to G; the actions [10]. Communities of autonomous units are rule-based and performed by the units correspond to applications of rules in graph-transformational devices to model interactive pro- R; the decisions of the units are made according to control cesses that run independently of each other in a common conditions in C, and the goals are specified with an expres- environment. An autonomous unit has a goal that it tries to sion from X. This leads to the following definition. reach, a set of rules the applications of which provide its Definition 4 (Autonomous units) An autonomous unit is a actions, and a control condition which regulates the choice of system aut = (g, R, c) where g 2 X is the goal, R R is a actions to be performed. Each autonomous unit decides set of graph transformation rules, and c 2 C is a control about its activities on its own depending on the state of the condition. environment and the possibility of rule applications, but without direct influence of other ongoing processes. Autonomous units are meant to work within a commu- The autonomous units of a community can act sequen- nity of autonomous units that modify the common envi- tially, in parallel, or concurrently (cf. [11, 13]). For modeling ronment together. Every community is composed of an production networks by communities of autonomous units, a overall goal that should be achieved, an environment parallel semantics is suitable because each production site of specification that specifies the set of initial environments a network can be naturally modeled by an autonomous unit the community may start working with, a set of autono- that acts in parallel with all other units. More precisely, in mous units, and a global control condition to restrict the every production step of a production process, each unit possibilities of interaction among the units. The overall performs the following actions: goal may be closely related to the goals of the autonomous 123 Logist. Res. (2011) 3:159–175 165 units in the community. Typical examples are the goals q are chosen as 12, 36, 15.5, and 5, respectively. The graph admitting only successful semantic sequences w.r.t. one or consists of a node equipped with a j-labeled loop, and for all autonomous units in the community. every x 2fmaxin; max; maxout; qg there is an x-edge pointing to a node equipped with a loop labeled with a real Definition 5 (Community) A community is a system number (or with 1) representing the quantity of x. Since the Com ¼ðGoal; Init; Aut; CondÞ, where Goal 2 X is a graph drawing of large production networks would lead to rather class expression called the overall goal, Init 2 X is a graph complex graphs we choose the more compact graphical class expression called the initial environment specifica- representation of production sites on the right of Fig. 3. tion, Aut is a set of autonomous units, and Cond 2 C is a There, the site attributes are listed in the site node itself. control condition called the global control condition. Accordingly, the representation of a production network Communities for production networks consist of one PN with respect to a production state q 2 R is the edge- unit per production site. The initial environment specifi- labeled graph env(PN)(q) that is constructed in the fol- cation specifies production networks whose number of sites lowing way. corresponds to the number of units in the community. The 1. Take the underlying graph G(PN) defined in the control condition requires to run all units infinitely long in remarks after Definition 1 where an edge (i, j) has i as parallel. In this paper, the goal specifies stability. source, j as target and d as label. ij In the following, we present a concrete graph transfor- 2. Extend each site node as described above. mation approach that is suitable for modeling production networks. Using the compacted representation, the example pro- duction network SAMPLE of Sect. 2 is represented by the 4.1 A class of graphs for production networks graph in Fig. 4. Production networks can be suitably represented as edge- 4.2 A rule class for modeling the actions of production labeled directed graphs consisting of nodes connected via sites directed labeled edges. More precisely, let R be a set of labels. An edge-labeled directed graph over R is a system The class R of graph transformation rules chosen in this G = (V, E, s, t, l), where V is a set of nodes, E is a set of paper is based on the double pushout approach, which is edges, s, t: E ! V are the source and target mappings which assign to each edge its source and target node, respec- tively, and l : E ! R is a mapping assigning a label to each edge in E. For representing production networks, we must require that R contains the elements of R as well as the symbols 1, maxin, max, maxout, and q. A production site j together with its maximum input rate maxin , its maximum produc- tion rate max , its maximum output rate maxout , and its j j current quantity q is represented by the edge-labeled graph shown on the left of Fig. 3 where maxin , max , maxout , and j j j (a) (b) Fig. 3 A production site as a directed edge-labeled graph (left) and Fig. 4 The compacted graph for the example production network of its compacted depiction (right) Sect. 2 123 166 Logist. Res. (2011) 3:159–175 well studied in the literature (cf., e.g., [4, 7]). Every graph because it is the difference of the production rate and the transformation rule of this class consists of three edge- output rate. labeled directed graphs L, K, and R such that K is a sub- A third example of a graph transformation rule is the rule graph of L and R. Formally, a graph G = (V, E, s, t, l)isa distribute(j, i) given in Fig. 7. The left-hand side and the 0 0 0 0 0 0 subgraph of a graph G = (V , E , s , t , l ), denoted by gluing graph contain the two production sites j and i that are 0 0 0 G G ,if V is a subset of V , E is a subset of E , connected with a d -edge where d is the corresponding ji ji 0 0 0 s(e) = s (e), t(e) = t (e), and l(e) = l (e) for all edges e in entry in the distribution matrix. The distribution quantity dq E. The graphs L, K and R are called left-hand side, gluing of site j is given in the left-hand side, the gluing graph, and graph and right-hand side, respectively. Rules are depicted the right-hand side. The right-hand side consists of the same in the form L ? R where the nodes and edges of the gluing production sites, but site i is additionally equipped with a graph K are indicated by identical positions and node value g which corresponds to the fraction d dq. ji colors. Graphs are transformed via applications of graph An example of a graph transformation rule is the rule transformation rules. Roughly spoken, a rule r ¼ðL choose(j) in Fig. 5. Its left-hand side and its gluing graph K RÞ is applied to a graph G by replacing an image of consist of the production site j. The right-hand side consists the left-hand side L with the right-hand side R such that the of the same site plus additional values for an input rate in, image of the common part K is not changed. Formally, the an output rate out, and a production rate prod. On the right image of a graph L in G is the image of a graph morphism g of the rule, the constraints that must be satisfied by the from L to G. More precisely, for two graphs values of in, out, and prod are listed. H = (V , E , s , t , l ) and G = (V , E , s , t , l ), a H H H H H G G G G G A second example of a rule is outputðjÞ given in Fig. 6. graph morphism g: H ? G is a pair of structure-preserving All three graphs of the rule consist of the production site j mappings g : V ? V and g : E ? E , i.e., V G H E G H plus a chosen production rate p and a chosen output rate o. g (s (e)) = s (g (e)), g (t (e)) = t (g (e)), and V G H E V G H E Additionally, the right-hand side contains a node, the value l (g (e)) = l (e) for all e 2 E . The image gðGÞ H is H E G G of which corresponds to the distribution quantity of site j also called a match of G in H. Fig. 5 Rule choose(j) Fig. 6 Rule output(j) 123 Logist. Res. (2011) 3:159–175 167 Fig. 7 Rule distribute(ij) In more detail, an application of a rule r ¼ðL K RÞ þ R Þ where ? denotes the disjoint union of graphs and to a graph G consists of the following steps: the inclusions are the natural extensions of the inclusions in the rules r ; ...; r . 1 n 1. A graph morphism g: L ? G is selected subject to the For example, the parallel rule r þ þ r where 1 7 following two application conditions: r = choose(j) for j ¼ 1; ...; 7 can be applied to the pro- (a) the dangling condition: the removal of g(L) - duction network in Fig. 4. This application models the g(K) from G yields no dangling edges, and parallel choice of the rates for each production site. (b) the identification condition: if two nodes or two The presented rules model the above-explained steps edges of L are identified (i.e., mapped to the same choose, output, and distribute of a production step. The graph element) in the match of L, they must be step subtract that subtracts the production rate from the in K. current quantity is modeled with the rule subtract(j) given in Fig. 9. The step add which adds the quantities from the 2. g(L) - g(K) is removed from G, yielding the graph Z. neighbors and the input rate to the current quantity is 3. R is added to Z yielding H by merging K with g(K). modeled by the rule add(j) given in Fig. 10. This means that every item of R that is also in the It is worth noting that since the left-hand sides of gluing graph K is merged with its image in Z and the choose(j), outputðjÞ, and distribute(j, i) are equal to the rest of R is added disjointly so that sources, targets, and corresponding gluing graphs, nothing is deleted in their labels are kept. applications. Rules with deletion are subtract(j) and For example, for j = 2, the rule choose(j) in Fig. 5 can add(j). be applied to site 2 of the production network in Fig. 4 by selecting a value p for the production rate, a value o for the 4.3 Graph class expressions for initial production output rate, and a value g for the input rate such that the networks and goals conditions of the rule are satisfied. If one chooses p = 10.7, o = 0, and g = 100, the resulting graph is Graph class expressions serve to specify the set of initial depicted in Fig. 8. This rule application models the step environments of a community and the goals. choose mentioned before, i.e., it models the choice of an Typical examples of graph class expressions are con- input, an output, and a production rate by site 2. crete single graphs, sets of graphs, or sets of labels. Every To the graph in Fig. 8, rule outputð2Þ can be applied and graph as well as every set of graphs specifies itself, and afterward rules distribute(2,1) and distribute(2,4) (because every set D of labels specifies all graphs that are only only sites 1 and 4 are neighbors of site 2). An application of labeled with symbols in D. rule distribute(j, i) models the distribution of the amount The initial environment of a community modeling the d dq from site j to site i where dq is the distribution processes of a production network PN consists of the edge- ji quantity calculated in the application of the rule outputðjÞ. labeled graph representing PN where each site has an arbitrary initial quantity. Concretely, for a production For modeling production processes adequately, the rule choose should be applied in each production step in parallel network PN with the components [n], maxin, max, maxout, and d, we use the graph class expression envðPNÞ¼ with each site. This is possible by combining rules to parallel rules by building the disjoint unions of their fenvðPNÞðqÞj q 2 R g where env(PN)(q) are the edge- respective components. Formally, for rules r ; ...; r with labeled graphs introduced in subsection 4.1. For each node 1 n r ¼ðL K R Þ, their parallel composition r þ þ i 2½n, its set of neighbors is defined by NðiÞ¼ i i i i 1 r yields the rule ðL þ þ L K þ þ K R þ fj 2½nj d [ 0g. n 1 n 1 n 1 ij 123 168 Logist. Res. (2011) 3:159–175 Fig. 8 A graph resulting from an application of the rule choose(j) Fig. 9 Rule subtract(j) The goal of each autonomous unit modeling a produc- Fig. 10 Rule add(j) tion site j can be specified with the graph class expression unit, i.e., after every step of a production process. This is bound where bound 2 R is some fixed vector. It specifies expressed by the term stable with SEMðstableÞ¼ all graphs env(PN)(q) where the quantity of site j does not SEMðbound Þ where n is the number of units in the exceed bound , i.e., SEMðbound Þ¼fenvðPNÞðqÞ j j j2½n j q bound g. A transformation of the unit is said to be community. It is worth noting that one can guarantee suc- j j successful if the resulting graph meets the goal. cessful production processes if one models deterministic net- In the presented modeling of production networks, the works, takes a non-negative solution of the equation system in global goal of the community expresses stability. It requires Theorem 2 as bound, and chooses the initial state q and the that the goal of every unit will be fulfilled after every run of the maximum production rate max such that q B bound B max. 123 Logist. Res. (2011) 3:159–175 169 4.4 Control conditions for a correct behavior Apart from the autonomous units of a community, the of production networks community itself may be provided with a global control condition. As global control conditions we use the parallel In many cases, rule application is highly non-determin- operator || and infinite sequential composition. More con- istic—a property that is generally not desirable. On one cretely, the global control condition aut jjjjaut 1 k hand, there can be several rules that are applicable to the requires that the autonomous units aut ; ...; aut run in 1 k current graph. On the other hand, there may be several parallel each one exactly once. Moreover, the control matches for one and the same rule. Hence, a graph condition c prescribes to apply the control condition transformation approach provides a class of control con- infinitely often. The combination of both control conditions ditions so that the degree of non-determinism of rule is used in the community presented in the following application can be reduced. Typical examples of control subsection. conditions are regular expressions over rules. It is well known that each regular expression over rules specifies a 4.5 The community for production networks possibly infinite set of rule sequences. Hence, every reg- ular expression reg over rules can be used as a control Based on the ingredients presented in the previous subsec- condition of an autonomous unit that allows all transfor- tion, we can now define the community C(PN) for a pro- mation processes in which the rules are applied in the duction network PN in a straightforward way as in Fig. 11, same order as they occur in at least one of the rule i.e., CðPNÞ¼ ðstable; envðPNÞ; fsiteðjÞj j 2½ngÞ where sequences specified by reg. for j 2½n the autonomous unit siteðjÞ¼ ðbound ; For modeling the behavior of production sites, regular fchooseðjÞ; outputðjÞ; subtractðjÞ; addðjÞg [ fdistributeðj; iÞj expressions are augmented by the condition i 2 NðjÞg; c Þ with c ¼ chooseðjÞ; outputðjÞ; R j j i2NðjÞ as_long_as_possible and the parallel composition operator distributeðj; iÞ; subtractðjÞ; addðjÞ! is given in Fig. 12. ?. More concretely, the regular expression r (where r is Summarizing, the following observation relates a run- some rule of an autonomous unit) means to apply r exactly ning step in the community with a process step in the once. The operator as_long_as_possible denoted by an mathematical model. exclamation mark can be applied to single rules with the Observation A running step of the community C(PN) effect that the rule must by applied as long as possible. The has the form parallel composition operator ? is applied to a set R of rules and requires to apply all rules in this set in parallel, envðPNÞðqÞ¼) envðPNÞðq Þ which corresponds to the application of the parallel rule 0 0 where q is obtained from q by q ¼ q þ R r. Sequential composition of control conditions is r2R n in p þ d ðp out Þ for j 2½n: j j ij i i i¼1 denoted by a semicolon, i.e., the expression c ; ...; c 1 k prescribes to execute the control conditions c ; ...; c As a consequence of this observation, we get the fol- 1 k exactly in this order. Finally, non-deterministic choice is lowing result. expressed by the symbol |, i.e., the expression c jjc 1 k Theorem 3 Each production process q in PN corre- means to apply one of the conditions c ; ...; c . 1 k sponds to an infinite run of the community C(PN), i.e., for Concretely, the behavior of production site j can be every k 2 N there are vectors in ; p ; out 2 R with [ 0 k k k modeled by applying the rules of the previous subsection q ! q if and only if envðPNÞðq Þ¼) k1 k k1 according to the control condition in ;p ;out k k k envðPNÞðq Þ. chooseðjÞ; outputðjÞ; distributeðj; iÞ; subtractðjÞ; addðjÞ! i2NðjÞ where NðjÞ½n is the set of neighbors of site j as defined in the previous subsection. In words, this control condition prescribes to apply at first the rule choose(j) and then the rule outputðjÞ. Afterward the rule distribute(j, i) is applied in parallel with all neighbors i of site j. In the next step, subtract(j) is applied and then add(j) as long as possible. It can be shown that the transformation processes that obey this control condition model the above-described steps choose, output, distribute, subtract, and add in the required order. Fig. 11 Community C(PN) 123 170 Logist. Res. (2011) 3:159–175 Fig. 12 The autonomous unit site(j) This shows that C(PN) models PN correctly. RAM, having found 1920 matches (many of these being parallel matches across the network) and performed 1920 Remark The production process q is stable if for every graph rule applications in that time (see Fig. 13). k 2 N the graph env(PN)(q )isin SEM(stable). [ 0 k In order to simulate production runs on larger networks, we have written an additional graph grammar, which cre- 4.6 Modeling deterministic production networks ates random production networks for test purposes (cf. Fig. 14). A graph with 402 nodes is generated in 655 ms; Deterministic production networks can be modeled by 3000 production steps are completed after another replacing the constraints on the right side of rule choose in 21840 ms (i.e., some 21 seconds), with over 4 million Fig. 5 with the properties of deterministic networks pre- matches found and rewrite steps executed in that time. sented in Definition 3. Obviously, in this case, all modeled This type of simulation is valuable as a visual way to processes are stable if a solution of the equation system in model and debug production networks or detect flaws in Theorem 2 is chosen for bound and if this solution is existing ones, altering them until they become stable. In between the state q of the initial environment env(PN)(q ) 0 0 particular, the simulation includes a visual debugger which and the maximum production rate max of PN, i.e., allows to view in detail every step performed by the sys- q B bound B max. tem, from the matching of rule patterns to nodes to the assignment of new values to variables. Additionally, the declarative nature of graph transformation rules makes 5 Visual simulation the modeling less error prone, and the production process model easily scalable, e.g., by introducing different mate- In order to simulate deterministic process runs on the rial types. Other possible extensions to the current model sample production network in Fig. 1, we have imple- are sketched in the conclusion. mented the general production community C(PN) (cf. Fig. 11) using the graph transformation engine GrGen.NET 6 Production networks with variable distribution rates (cf. [8]). The GrGen.NET graph model is based on typed, In this section, the notion of production networks is attributed, directed multigraphs with inheritance. The base extended by allowing to change the distribution rates such types at the core of this model are Node and Edge, and the that the distribution matrix becomes variable. This serves primitive attribute data types int, float, double, string, two purposes. On one hand, the modeling of production boolean, and object, the latter denoting a .NET object. networks becomes more flexible and more realistic. On the We made use of the subpattern matching capability of other hand, it is demonstrated how the modeling frame- GrGen.NET, using the iterated subpattern in order to work of communities of autonomous units can be used to simulate parallel rule application. GrGen.NET also does not specify aspects of autonomy. provide autonomous units; however, it allows to structure An autonomous unit decides about its next action by rule application by embedding imperative calls to other rules choosing a rule application out of all possible ones. The into the declarative right-hand side of a rule. Furthermore, such calls may be controlled using, for example, regular decision depends on the control condition which may be the conjunction of several conditions of different kinds. A expressions. We made use of this feature to emulate auton- typical case is a condition that establishes some order or omous units very closely to our original specification. priority among the rules. For example, the control condition The simulation runs very fast, with our example network of the unit site(j) requires that first choose(j) must be applied SAMPLE completing 41 steps and reaching the maximal then output(j) followed by distribute(j,i) and finally, production rate at all seven sites in 156 milliseconds on an subtract(j) followed by some add(j)’s. Consequently, the Intel Core i5 M520 CPU with 2.40 GHz and 6 GB of 123 Logist. Res. (2011) 3:159–175 171 Fig. 13 Community C(SAMPLE) in GrGen.NET: all sites have reached their saturation point after 41 steps order of rule applications is fixed by this condition. But the waiting time is reduced, meaning that the further pro- rules are generic since they contain variables that must be cessing is not delayed for too long and that the chance for instantiated before the actual application can take place. stability is improved. While the actual values of the left-hand side variables are uniquely determined by the match of the rule in the envi- 6.1 The rule to change the distribution rates ronment, the unit can choose and decide about the values of the right-hand side variables. A second kind of control Let j 2½n be some production site and i ; ...; i for some 1 k condition is given by constraints for these values. So far, we k 2 N be its neighbors that can be reached from j by a have used only two extreme cases: Either the choice is transport edge each. Then, the rule adjust(j) has the form totally free within certain limits like the choice of given in Fig. 15. The left-hand side contains the sites p, q, g, and o in the rule choose(j), or it is computed j; i ; ...; i and the connecting edges. Moreover, for each 1 k uniquely like p - o in output(j) and q - p in subtract(j). neighbor i , there is a set of variables Var in the left- l l How further decisions in between the two extreme cases can hand side so that the actual values are available whenever be designed and used is demonstrated in the following. the rule is applied. Consequently, there is a unique To make the distribution matrix variable, we enrich matching for each environment graph. But to apply the each production site by a new rule, the application of b rule, the new distribution rates d for l 2½k must be ji which changes the current distribution rates of the edges chosen or computed. Some possibilities are discussed in outgoing of the considered site. The new distribution rates the next subsection where different sets of variables are are chosen due to proper constraints. To allow a variety used. of possibilities, the new rule is designed in a generic way. And some examples of constraints are provided. The first 6.2 Constraints for changing the distribution rates one is free choice. The second one reflects the quantities that wait for processing at the neighbor sites. The third The simplest possibility is to allow a free choice. Then, the one takes the maximum production rates into account only constraint to be considered is that distribution rates for additionally. In the latter two cases, the intention is to the site j must sum up to 1. The sets of variables may be deliver the distribution quantities in such a way that the empty in this case. 123 172 Logist. Res. (2011) 3:159–175 Fig. 14 A network with 400 nodes in GrGen.NET after 3000 production steps contain the quantities accordingly. This idea is reflected in ðconstraint 1Þ d ¼ 1 ji the following constraint: l¼1 b q ðconstraint 2Þ d ¼ ji But this is not really a good idea because free choice ðb q Þ l¼1 l may lead to quite chaotic processes. for m 2½k and some b 2 R with q \b þ i As indicated at the end of Sect. 3 and the intro- The differences b q are in converse order to the duction of this section, a much better idea is to choose order of the quantities so that the larger the quantity is, the the new distribution rates in such a way that the smaller the difference grows. The division by the sum chances for stability grow. A production process is makes sure that the new distribution rates sum up to 1: unstable if there is at least one production site at which k k k X X the quantities grow beyond any bound. The cause of b q ðb q Þ i i m m¼1 m d ¼ ¼ ¼ 1 P P ji this effect is that the site gets more delivered than it k k ðb q Þ ðb q Þ i i m¼1 m¼1 l¼1 l l¼1 l redistributes over time. To avoid the unbounded growth, one may shorten the delivered quantities by making the Note that the new distribution rates reflect the distribution rates smaller in inverse proportion to the differences between the current site quantities in lessened amount of material piled up at the sites. The three form if the upper bound b is chosen larger. We require that following constraints are examples how a production b is larger than maxfq j l 2½kg to avoid that any site unit can autonomously control its distribution fol- distribution rate becomes 0. lowing this general principle. In the exceptional case that all quantities at neighbor sites are equal, the bound b must be greater—at least a The site j may consider the current quantities at the neighbor sites and assume that the smaller the quantity is, bit—because otherwise the sum of all differences would be the faster the processing runs. The sets of variables must 0 and the quotient would not be defined. 123 Logist. Res. (2011) 3:159–175 173 Fig. 15 Rule adjust(j) The reflection of the waiting time becomes more sophisticated if one replaces the quantities in constraint 2 by the quotients of quantities and maximum production rates. b w ðconstraint 3Þ d ¼ with ji ðÞ b w l¼1 w ¼ for m 2½k and some b 2 R i þ max with w \b The latter quotient may be called waiting number because the smallest integer greater or equal is the Fig. 16 Application of rule adjust(4) with constraint 2, with minimum number of steps to process the current b = 600 quantity. Clearly the sets of variables must contain the quantities and the maximum production rates. 6.3 Example The last explicit example takes into account that it may not always be reasonable to forget the old distribution The production site 4 of SAMPLE has the neighbors 2, 3, 6, totally so that one may like to mix the old rates with new and 5 and distributes a quarter of the production rate to ones. A weighted average will do this job: each of them due to the distribution matrix (cf. Figs. 1 and 4). Let us assume in addition the following current quan- rd þ sd ji m ji tities: q = 450, q = 300, q = 250, and q = 500. Then, 2 3 5 6 ðconstraint 4Þ d ¼ for m 2½k ji r þ s one can apply the rule adjust(4) using constraint 2 with and r; s 2 R with r þ s [ 0 and þ b = 600. Figure 16 shows the rule application restricted to 0 0 some distribution rates d .. .d the significant part of the network. In Fig. 17, the same is ji ji 1 k depicted for constraint 3 with b ¼ . While the new dis- which may be chosen as one of tribution rates in the first case are smaller the larger the the three cases above quantities are, the second case reflects the waiting numbers Each of the four constraints (and other similar ones) w = 3, w = 2, w = 1, and w ¼ . While, for example, 2 3 5 can be used as control condition in the autonomous site 2 gets a larger fraction from site 4 than site 6 in the first unit site(j) after it is enriched by the rule adjust(j). As case, it is the other way round in the second case. the control condition concerns only this rule, it may be The considerations in this section exemplify how placed beside the right-hand side of the rule (cf. communities of autonomous units may be modified and Fig. 5). extended to cover new aspects and features. Concerning the 123 174 Logist. Res. (2011) 3:159–175 nice to know whether this works and for which production network and which variability. 3. To make the model more flexible, one may enhance the notion of production networks by relaxing, modifying, or specializing various assumptions like the following: • There may be lower bounds of input, production, and output rates in addition to upper bounds. • There may be different kinds of materials and information flows through the network rather than a single homogeneous kind of quantities. • There may be particular time conditions for production and transportation at each site rather than the common-step assumption. Fig. 17 Application of rule adjust(4) with constraint 3, with b ¼ • There may be more information about the produc- tion like costs, prices, etc. to refine the basis for the autonomous decision making and planning at the variable distribution rates, we have taken into account production sites, or one may also involve produc- some measures that reflect something like the waiting time tion goals into the consideration. with a look-ahead of 1 (meaning that we access only the information provided by direct neighbors). In a similar 4. Another possible modification would be to assume that way, one could involve larger look-aheads or criteria other the produced and distributed material consists of a than waiting time like pheromone traces (cf. [2]). We are number of atomic items such that only integer division also convinced that further principles of planning in pro- is possible. In this case, the graph-transformational duction networks (cf. [1]) can be realized in this way. model may be particularly suitable as the atomic items could be represented by atomic graph components explicitly. 7 Conclusion We think that communities of autonomous units provide a suitable framework to model production networks with In this paper, we have modeled and investigated a variant respect to the points 3 and 4 at least. As the framework is of production networks with discrete production processes. equipped with a well-defined syntax and semantics, it The usual quantitative modeling based on matrices and offers the perspective of further tool support beyond the vectors has been supplemented by a visual modeling visual simulation. For example, it should be possible to employing the rule-based and graph-transformational employ a verifier like a model checker, SAT solver, or framework of communities of autonomous units. It has theorem prover eventually to prove properties of produc- turned out that the community version models production tion processes like stability automatically. networks correctly with respect to their mathematical description so that all results for one of the models apply to Acknowledgments We are grateful to the anonymous reviewers for the other and conversely. Therefore, one gets both: On one their valuable comments. The authors would like to acknowledge that their research is partially supported by the Collaborative Research hand, one can prove results like the stability of determin- Centre 637 (Autonomous Cooperating Logistic Processes: A Para- istic production networks provided that certain systems of digm Shift and Its Limitations) funded by the German Research linear equations are solvable; on the other hand, the visual Foundation (DFG). simulation is supported. The attempt to bring two styles of modeling together may therefore be considered as prom- ising. 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Logistics Research – Springer Journals
Published: Apr 19, 2011
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