ning; ant colony optimization; lot-sizing. 1. Chair of ... Collaborative operations planning [12] addresses a core probl
A collaborative ant colony metaheuristic for distributed multi-level uncapacitated lot-sizing Tobias Buer1, ∗
Jörg Homberger2
Hermann Gehring3
A b s t r a c t. The paper presents an ant colony optimization metaheuristic for collaborative planning. Collaborative planning is used to coordinate individual plans of self-interested decision makers with private information in order to increase the overall benefit of the coalition. The method consists of a new search graph based on encoded solutions. Distributed and private information is integrated via voting mechanisms and via a simple but effective collaborative local search procedure. The approach is applied to a distributed variant of the multi-level lot-sizing problem and evaluated by means of 352 benchmark instances from the literature. The proposed approach clearly outperforms existing approaches on the sets of medium and large sized instances. While the best method in the literature so far achieves an average deviation from the best known non-distributed solutions of 75 percent for the set of the largest instances, for example, the presented approach reduces the average deviation to 7 percent. Keywords: Group decisions and negotiations; metaheuristics; collaborative planning; ant colony optimization; lot-sizing
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Chair of Logistics, University of Bremen, 28334 Bremen, Germany corresponding author Faculty of Geomatics, Computer Science and Mathematics, Stuttgart University of Applied Sciences, 70174 Stuttgart, Germany Information Systems Research Group, University of Hagen, 58084 Hagen, Germany
This paper is a postprint. The final publication is available at Taylor & Francis Online via http://dx.doi.org/10.1080/00207543.2013.802822. Please cite as: Buer, T., Homberger, J., Gehring, H. (2013): A collaborative ant colony metaheuristic for distributed multi-level uncapacitated lot-sizing. In: International Journal of Production Research 51.17, S. 5253–5270. DOI: 10.1080/00207543.2013.802822
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Introduction and literature review
A coalition of decision makers with private information that requires collaborative production planning is considered. Each of the decision makers, hereafter referred to as agents, is selfish and seeks to implement his or her local optimal production plan to minimize his or her local costs. However, the agents could improve their situation by coordinating their individual production plans. The cost savings due to a better global production plan could be allocated to the agents in order to overcompensate them for deviating from their local optimal production plans. From a more general point of view, the problem of finding an appropriate allocation of cooperative gains is dealt with by cooperative game theory. The special situation of joint planning of selfish agents in case of private information considered here is the subject of collaborative operations planning. Collaborative operations planning [12] addresses a core problem of supply chain management namely the coordination of planning activities across company borders – that is, between agents. Collaborative operations planning seeks to find a compromise between hierarchical and central planning methods. During hierarchical planning, agents create their plans in a given sequence. The resulting (local optimal) plan of one agent defines the planning input of the subsequent agent. Such a sequence of local optimal plans usually leads to an inefficient global plan. Central planning, on the other hand, tries to reach a global efficient plan by using a central planning authority which solves an integrated problem and knows all relevant information. That is, the central planning authority depends on sensitive data (e.g. unit costs, capacity used, warehouse stock) which has to be revealed by the agents although it is usually classified as confidential. To overcome these deficits, collaborative planning aims to achieve a superior plan compared to hierarchical planning whereas less exchange of private information is required compared to central planning. Collaborative operations planning relies on a mediator that pursues no own goals with respect to the planning outcome and whose actions are transparent [15]. Depending on the practical context, a mediator might be a joint enterprise of the planning agents, a professional society or, as [15] proposes, a piece of software. If the source code of this software as well as the initialization of the components like e.g. a pseudorandom number generator is known to all agents, then the computational results are deterministic and reproducible by each agent. Although this paper focuses on the algorithmic planning decisions and not on the enabling organisational infrastructure, the requirements regarding the existence of a mediator for collaborative planning seem not out of reach. The state of the art in collaborative planning is discussed by [17] and by [33]. Overviews of automated negotiation approaches relevant to collaborative planning are given by [28] and [35]. Applications of collaborative planning to supply chains are discussed by [13], and [14, 15], for example. The proposed collaborative planning approach is applied to a distributed variant of the well-known multi-level uncapacitated lot-sizing problem (MLULSP) introduced by [46]. The distributed MLULSP (DMLULSP) was presented by [20] and assumes several local and selfish agents with private information instead of a single agent with full information. The DMLULSP covers some important features of real world problems. There are several final products, a multilevel production structure, and a trade-off between inventory and setup costs. Coordination is difficult because there are agents with private information and conflicting objectives. Finally, the problem is computationally challenging, as it is NP-hard for general product structures [2]. As to computational experiments, a variety of benchmark instances is available in the literature which allow for comparative tests including other approaches.
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Lot-sizing problems are of high relevance in modern supply chains [38]. Surveys of the rich literature on lot-sizing problems are given for example by [10], [24], or [39]. Integration of lotsizing with cooperative game theory is discussed by [8, 9]. Metaheuristic solution approaches for the non-distributed MLULSP are amongst others presented by [6, 7], [32], [29], [1], [18], [41], [40], [44], and [45]. The literature discusses at least two distributed lot-sizing variants. The one studied by [26, 27] is based on auctions and has been tested on small size instances. For the other, the distributed MLULSP, several metaheuristic approaches that integrate concepts from group decision making are known in the literature: a simulated annealing approach (SA) [20], an approach based on an evolutionary strategy (ES) [22] and one based on ant colony optimization (ACO) [21, 23]. These approaches differ by the used metaheuristic principle, the applied negotiation or voting mechanisms, and the relevant objective functions (minimizing total global cost or maximizing fairness). ES is especially useful for maximizing fairness. SA and ACO strive to minimize total global cost. In terms of solution quality, ACO outperforms SA on medium sized instances (40 and 50 items with 12 and 24 periods) but not on small or large instances (500 items, and 36 or 52 periods). Furthermore, ACO is significantly slower than SA on small, medium, and large instances. The aim of this paper is to present a new collaborative ant colony metaheuristic (CACM) for solving the DMLULSP under global total cost minimization. Compared to [23] and the other mentioned approaches, the proposed CACM incorporates the following new conceptual and methodical ideas: 1. Encoded solutions are represented by a new and simplified search graph. Let m denote the number of items to produce and let n denote the number of periods. The search graph proposed by [19] and used in [23] requires m(n + 1) nodes and 12 mn(n + 1) directed edges while the new search graph uses 2mn + 1 nodes but only 4mn − 2 edges. The new search graph facilitates learning of pheromone values and therefore makes it easier to discover promising paths which lead to high quality solutions. 2. Individual preferences of the agents – which are private – are aggregated by a voting mechanism and by a new collaborative local search procedure. Usually, to be able to reach a (near) optimal solution fast, an ant colony metaheuristic requires heuristic information on the problem, for example, the costs of the agents. However, this operational information is sensitive and therefore private. The new approach exploits this information indirectly, which enables to guide the search via heuristic information and, nevertheless, keeps operational information private. 3. The new approach does without rank based voting mechanisms. This reduces computational effort and increases robustness, that is, there are less opportunities for undesired strategic behavior of the agents. As it is shown below, the incorporation of these concepts has a high positive impact on the solution quality of the resulting approach. The rest of the paper is organized as follows. In Section 2 the considered distributed multi-level lot-sizing problem (DMLULSP) is characterized. Section 3 describes the developed collaborative ant colony metaheuristic for solving the DMLULSP. Subject of Section 4 is the evaluation of the new approach by means of a computational study with 352 benchmark instances. Finally, concluding remarks are given in Section 5.
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Table 1: Notation for the DMLULSP. Problem parameters m n I T si hi ti rij Γ+ (i) ⊂ I Γ− (i) ⊂ I dit Mit
number of items number of production periods set of items, I = {1, . . . , m} set of possible production periods, T = {1, . . . , n} setup costs per period for item i ∈ I inventory holding costs per period and per unit of item i ∈ I lead time required to assemble, manufacture, or purchase item i ∈ I number of items i required to produce one unit of item j with i, j ∈ I, i 6= j set of all direct successors of item i ∈ I set of all direct predecessors of item i ∈ I exogenous demand (unit of quantity) for item i ∈ {j ∈ I|Γ+ (j) = ∅} in period t ∈ T P a sufficiently large number, i ∈ I, t ∈ T , e.g. Mit = t0 >t dit0
Decision variables dit lit xit yit
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2.1
endogenous demand (unit of quantity) of item i ∈ {j ∈ I|Γ+ (j) 6= ∅} in period t ∈ T inventory (unit of quantity) of item i ∈ I at the end of period t ∈ T lot-size (unit of quantity) of item i ∈ I in period t ∈ T binary setup decision, yit = 1 if item i ∈ I is produced in period t ∈ T and yit = 0 otherwise
A distributed multi-level uncapacitated lot-sizing problem
Classical centralized problem formulation
The distributed lot-sizing model studied in this paper extends the well-known multi-level uncapacitated lot-sizing problem (MLULSP, cf. [46], [34], [6]). Therefore, the MLULSP is presented first using the notation given in Table 1. The MLULSP assumes a single decision maker who is aware of all information required for planning, especially, the setup costs and inventory holding costs per item and period. A formal description of the MLULSP is given by the formulas (1) to (8).
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min f nd (y)
=
5
XX
(si · yit + hi · lit )
(1)
i∈I t∈T
s. t.
lit = li,t−1 + xit − dit , li,0 = 0 , lit ≥ 0 , dit =
X
rij · xj,t+ti ,
∀i ∈ I, ∀t ∈ T, ∀i ∈ I, ∀i ∈ I, ∀t ∈ T \ {0},
(2) (3) (4)
∀i ∈ {j ∈ I | Γ+ (j) 6= ∅}, t ∈ T,
(5)
∀i ∈ I, ∀t ∈ T, ∀i ∈ I, ∀t ∈ T, ∀i ∈ I, ∀t ∈ T.
(6) (7) (8)
j∈Γ+ (i)
xit − Mit · yit ≤ 0 , xit ≥ 0 , yit ∈ {0, 1} ,
The goal of the MLULSP is to minimize the total costs f nd of a single, central decision maker in the non-distributed scenario. These are expressed by the objective function (1) which sums up the setup costs and the stockholding costs for all items i ∈ I over all periods t ∈ T . The inventory balance is guaranteed by (2). For all items, the inventory of the first period t = 0 is zero (3) and for remaining periods non-negative (4). For each period, the demands dit for the level zero items i ∈ I with Γ+ (i) = ∅ are given. The demands for the remaining items are determined by (5). These constraints ensure that the production of item j in period t + ti triggers a corresponding demand dit for all i ∈ Γ− (j), that is, a demand for each item i preceding item j in the bill of materials. Without loss of generality, rij = 1 is assumed in (5). The lot-size xit is non-negative (7). If xit > 0, that is, item i is produced in period t, then yit = 1, otherwise yit = 0. This is enforced by the constraints (6) and (8). [43] and [42], for example, discuss the implications of alternative formulations of capacitated and centralized lot sizing models, in particular with respect to tighter lower bounds based on linear programming relaxations. It is an interesting but nontrivial question wether these tighter model formulations can also be applied to the distributed lot-sizing problem with asymmetric information discussed in the subsequent section. Because the heuristic solution approaches presented in this paper solve such a distributed model and for that purpose do not calculate linear programming lower bounds, the question is not studied further and the more descriptive formulation of (1) to (8) is preferred. [2] have shown that the MLULSP is NP-hard for general multi-level product structures, i. e., for product structures where each item can have multiple successors and predecessors like in Figure 1 [31]. Figure 1 shows an example for a graphical representation of a product structure with m = 40 which follows the literature [3]. The production dependencies between items are indicated by arrows. Furthermore, items are classified by decreasing production level from top to bottom. Each end product i ∈ I with Γ+ (i) = ∅ is located at the highest level which is level 0. Each common part is located at the lowest level at which it is used anywhere in the product structure [6]. The allocation of items to a group of agents also shown in Figure 1 is described next.
2.2
Extension to a distributed group decision problem formulation
Following [20], the MLULSP is now extended to a distributed group decision problem which is called distributed multi-level uncapacitated lot-sizing problem (DMLULSP). Instead of a single decision maker or agent who is responsible to produce all items I, the responsibility is jointly assigned to a group of agents A. Each agent a ∈ A is responsible to produce the set of items
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1
level 0
agent a1
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level 1
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level 2
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(produ ing items 1-20)
level 3
level 4
agent a2
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(produ ing items 21-40)
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level 5
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Figure 1: Example of a product structure and a partition of the items to two agents [20]. Ia with a∈A Ia = I and a∈A Ia = ∅. For instance, an agent might represent a profit center or an independent company and the agents might interact in a supply chain. All agents are autonomous and self-interested. Therefore, the individual objective function fa of agent a ∈ A is to minimize his or her total local costs for producing the items Ia , that is, S
T
min fa (y) =
XX
(si · yit + hi · lit ).
(9)
i∈Ia t∈T
Furthermore, the DMLULSP assumes asymmetric information regarding the cost parameters si and hi . That is, for all items i ∈ Ia produced by an agent a (a ∈ A), only agent a knows the values of si and hi ; all other agents a0 (a0 ∈ A, a0 6= a) are not aware of the values of si and hi (i ∈ Ia ). Agent a does not want to reveal these to other agents during (collaborative) planning, therefore these cost parameters are denoted as private information of agent a. If this information would be common knowledge, price negotiations between agents might be negatively affected. On the other hand, symmetric information or public information, i.e., information available to all agents, is assumed regarding the bill of materials. This assumption can be justified by some kind of common industry knowledge or joint development of products (e.g. collaborative engineering). The DMLULSP consists of the constraints (2) to (8) and the objective function (10) which minimizes the total global costs:
min f (y) =
X
fa (y).
(10)
a∈A
In comparable scenarios, the total global costs are also considered of interest by [14], [47], [25], and [12, 11]. In the following, the function (10) is simply referred to as total cost function or central cost function, the function (9) is also denoted as individual or local cost function of agent a ∈ A. Usually, the minimization of the individual cost functions conflicts with the objective of minimizing the total costs. To resolve these conflicts and support the agents in agreeing on
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a joint production plan in order to produce all items I, the solution approach proposed in the following section uses collaborative local search and different voting mechanisms.
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A collaborative planning metaheuristic based on ant colony optimization
3.1
Overview
A metaheuristic solution approach based on the ant colony optimization metaheuristic is presented that especially takes into account the distributed nature of the DMLULSP. The distributed solution approach is denoted as collaborative ant colony metaheuristic (CACM); an overview is given by Algorithm 1. The metaheuristic CACM works on an encoded representation of a DMLULSP solution which is described in Section 3.2. Components of an encoded solution are represented by a search graph G which is traversed by artificial ants in order to construct encoded solutions. Section 3.3 introduces the search graph G and Section 3.4 presents the stochastic ant construction procedure. Each constructed solution is improved by a collaborative local search heuristic (cf. Section 3.5). Both, the used graph structure and the local search heuristic incorporate new concepts compared to the ant metaheuristic of [23]. After a set B of encoded solutions has been constructed, the agents negotiate about the acceptance of a solution in B by casting votes according to one of several voting rules described in Section 3.6. Finally, the pheromone information related to the arcs of the search graph is updated according to the current accepted solution e∗ (Section 3.7). That is, in the next iteration of CACM, ants use a different probability distribution in order to construct a new encoded solution. Algorithm 1: Collaborative ant colony metaheuristic (CACM) Input: no. of items m, no. of periods n, set A of agents, no. of generated solutions s, ballot size b, min and max pheromone values τ min and τ max , evaporation rate ρ, intensification rate σ Output: jointly accepted encoded solution e∗ initialize search graph G and pheromone values τ // Section 3.3 e∗it ← 0, i = 1, . . . , m; t = 1, . . . , n for s ← 1 to s do B ← {} for b ← 1 to b do e ← constructEncodedAntSolution(G, τ ) // Section 3.4 e ← collaborativeLocalSearch(e, A) // distributed, Section 3.5 B ← B ∪ {e} s ← s + 1 + m(n − 1) end e∗ ← voting(B, A) // distributed, Section 3.6 ∗ min max τ ← updatePheromoneValues(e , τ, τ , τ , ρ, σ) // Section 3.7 end return e∗ CACM requires several input parameters. The problem data include the number of items m ∈ N, the number of periods n ∈ N, and the set A of agents. The termination of CACM
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is controlled by the number of generated solutions s ∈ N. The ballot size b ∈ N specifies the number of solutions which can be negotiated by the agents A in a single election. The remaining parameters influence the update of the pheromone information on the arcs of the search graph G. The minimum and maximum pheromone values are defined by τ min , τ max ∈ N. The pheromone evaporation rate and the pheromone intensification are given by ρ and σ with 0 ≤ ρ, σ ≤ 1. The solution approach CACM can deal with asymmetric information distributed among the agents. CACM is executed by a neutral mediator. In order to construct encoded solutions, the mediator has to be aware of the items I to produce and of the possible production periods T . The agents do not have to reveal private information like costs or free production capacities, neither to the mediator nor to other agents. To control the search process in order to find a jointly accepted solution e∗ , the mediator has to interact with the agents in both steps marked as distributed in Algorithm 1. CACM outputs a jointly accepted solution e∗ .
3.2 3.2.1
Representation of a solution Solution encoding
For the MLULSP, the decision variables xit represent the lot-size of item i produced in period t. The binary decision variables yit represent the setup decisions which indicate whether item i is produced in period t at all. Because of the structure of the MLULSP, it is possible to determine the optimal lot-sizes xit if the optimal setup decisions yit are known [6]. This is transferable to the DMLULSP. Therefore, the proposed CACM focuses its search effort to approximate the set of optimal setup decisions. However, CACM does not operate on the yit directly, but indirectly on an encoded representation of the setup decisions. For each binary setup decision variable yit , there is an encoded binary setup decision variable eit following the suggestion of [18]. Both variables are related as follows:
(eit = 0) ⇔ (yit = 0), (eit = 1) ⇒ (yit = 0 ∨ yit = 1).
(11) (12)
Accordingly, the variables eit = 1 indicate a possible, but not mandatory, production of item i ∈ I in period t ∈ T . Another ant metaheuristic approach that also uses encoded decision variables is suggested by [16], for example. Whether a possible production takes place if eit = 1, can be deduced by the following decoding rule. 3.2.2
Solution decoding
An encoded solution e ∈ {0, 1}m×n is transformed to a DMLULSP solution in a two steps procedure. The product level of an item i determines the order in which demand dit , lot-size xit , and inventory lit are calculated. The procedure starts with the lowest level items (final products) and advances with items of increasing product level. That is, the demands for the final products (level 0) are determined first, the demands for level 1 items are calculated next, and so on [6, 18]. Hence, the items are considered in the sequence of their non-decreasing product level. In case item i is a final product (level 0), then dit is given for all 1 ≤ t ≤ T . For each non-final product i the demand can be calculated according to equation (5) using the demands of all
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1 v1,1
1 v1,2
1 v1,3
1 v2,1
1 v2,2
b
b
b
b
b
0 v1,1
0 v1,2
0 v1,3
0 v2,1
0 v2,2
1 1 v2,3 = vm,n
b
v
0 0 v2,3 = vm,n
Figure 2: Example of a search graph for two items and three periods. successors of i indicated by Γ(i). All dkt , k ∈ Γ(i) have already been calculated, because of the used calculation sequence. By means of the second decoding step, the setup decisions, lot-sizes, and stored quantities are calculated. Note, an encoded solution that allows for each item a production in the first period, that is, ei1 = 1 for all i = 1, . . . , m, represents a feasible decoded solution in any case. As there are no capacities the total required demand for future periods can be produced in the first period, which results in a feasible solution.
3.3 3.3.1
Definition of a search graph Search graph
The ant colony optimization principle is guided by the image of an ant that constructs a feasible solution by traversing a graph which consists of solution components specific to the problem. A solution is represented by the path chosen by the ant. In order to enable such an artificial ant to construct an encoded solution for the DMLULSP, a directed graph G = (V, E) with a set V of nodes and a set E of directed edges is used (see Figure 2). The node set V consists of three disjunct subsets, i. e., V = {v} ∪ V 1 ∪ V 0 . The starting point of an ant is the initial node v. For each of the two possible values of an encoded decision variable eit there exists a node. The encoded decision to produce item i in period t (eit = 1) is represented by node vit1 ∈ V 1 . The encoded decision eit = 0 is represented by node vit0 ∈ V 0 . Therefore, V 1 ∩ V 0 = ∅ and |V 1 | = |V 0 |. The nodes in V 1 are called black nodes, the nodes in V 0 are called white nodes. The set of directed edges E is defined as follows. The initial node v is associated with 1 0 1 0 ) and (v, v1,1 ). Furthermore, each node v ∈ V \ {vm,n , vm,n } acts as two directed edges (v, v1,1 the origin of exactly two directed edges. From a black node two edges originate, one points to another black node and one points to a white node; analogously, from a white node one edge points to a black node and one edge points to another white node (see Figure 2). More a precisely, let a, b ∈ {0, 1}, a 6= b and v a ∈ V a , v b ∈ V b . The pair of edges originating from vi,t is a a a b defined as (vi,t , vi,t+1 ) and (vi,t , vi,t+1 ), provided that t < n. To ease presentation, two subsets of directed edges E 1 and E 0 are introduced. E 1 contains those edges that point to a black node and E 0 contains those edges that point to a white node.
E = E 1 ∪ E 0 with E 1 ∩ E 0 = ∅, E 1 := {(u, v) ∈ E|u ∈ V ∧ v ∈ V 1 }, E 0 := {(u, w) ∈ E|u ∈ V ∧ w ∈ V 0 }.
(13) (14) (15)
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An encoded solution of the DMLULSP is represented by all nodes v ∈ V 1 ∪ V 0 on any path 1 0 or at node vm,n . P through G that starts at node v and ends either at node vm,n 3.3.2
Initialization of pheromone trail
The sequence of directed edges traversed by an ant depends on the pheromone value τ(u,v) placed on each edge (u, v) ∈ E. Higher pheromone values τ(u,v) on an edge (u, v) increase the probability that an ant traverses edge (u, v). [19] proposed an ant metaheuristic approach for a (non-distributed) MLULSP on a different search graph and successfully pursued the idea to initially construct each possible solution with equal probability (’balanced path’). In contrast to that idea, the goal of this approach is to initially construct unbalanced paths that predominantly consist of black nodes. Following the max-min ant system proposed by [36], a minimum and a maximum pheromone value for each edge (u, v) ∈ E are defined as τ min and τ max . The initial pheromone value τ(u,w) for each edge (u, w) ∈ E 0 is set to 1, and the initial pheromone value for each (v, w) ∈ E 1 is set to τ max − 1.
3.4 3.4.1
Construction of a solution Component selection probability
In order to construct an encoded solution e, the graph G is traversed by an ant along the directed edges. The traversal of G starts at the initial node v (see Figure 2). Until the ant 1 0 1 0 arrives at node vm,n or vm,n , the ant decides randomly at each node u ∈ V \ {vm,n , vm,n } which 1 0 of the two possible edges (u, v) ∈ E and (u, w) ∈ E to choose. The random decision depends on the pheromone intensities τ(u,v) and τ(u,w) of the edges (u, v) and (u, w). The probability to choose edge (u, v) ∈ E is
p(u,v) =
τ(u,v) with τ(u,v) + τ(u,w) = τ max and (u, v) ∈ E 1 ∧ (u, w) ∈ E 0 . max τ
(16)
Due to the pheromone initialization described in Section 3.3.2, the initial path chosen by the first ant is highly likely to consist of black nodes only when τ max is sufficiently large. Given the solution decoding rule, this means that for each item a production is possible (but not mandatory) in the first period and, therefore, a feasible solution is generated immediately in all probability. 3.4.2
Heuristic information
Typically, ant colony approaches from the literature additionally bias the selection probability of a component (here, an edge (v, w) ∈ E) via heuristic information. In the present case, however, the DMLULSP is a distributed problem with asymmetric information which is solved by a mediator. Because the mediator is neutral, he or she cannot take into account heuristic information that would depend on an individual agent and therefore privilege (or penalize) this agent. For this reason, equation (16) does without local heuristic information.
3.5
Collaborative local search heuristic
An encoded solution e constructed by an ant is improved by a collaborative local search heuristic (CLS, cf. Algorithm 2). In this heuristic, a move is only executed, if each agent a ∈ A agrees. A move is defined as a bit shift, that is, for all i ∈ I and all t ∈ T, t ≥ 2, the value of eit is
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shifted from zero to one or from one to zero, respectively. Each shift leads to a new solution e0 . The solution e0 is submitted to all agents a ∈ A. If and only if e0 has lower local costs fa (e0 ) compared to fa (e) for each agent a ∈ A, the next move is executed on e0 and so on. Algorithm 2: Collaborative local search (CLS) Input: encoded solution e, set of agents A Output: possibly modified encoded solution e foreach t ← 2, . . . , n do foreach i ← 1, . . . , m do e0 ← e e0it ← ¬eit s←s+1 veto ← f alse foreach a ∈ A do if fa (e0 ) > fa (e) then veto ← true
// bit shift
// complete approval voting // each agent a can veto e
end if veto = false then e ← e0 end end return e
In CACM (cf. Algorithm 1), the counter of generated solutions s is increased by 1+m(n−1), as there are m(n − 1) possible moves and each move applied to a generated ant solution leads to a new solution. Furthermore, the final solution e generated by CLS is in CACM again submitted to all agents. This time, the agents compare e to the jointly accepted solution e∗ and decide via voting whether e should become the new jointly accepted solution e∗ .
3.6
Distributed decision making by means of voting
Voting in order to update a jointly accepted solution e∗ can be realized by one of the following voting rules.
3.6.1
Complete approval voting
In approval voting each agent can vote for as many candidate solutions in B as he or she wants and the solution with the most approvals wins [4]. For the problem at hand, a modified and restricted version of approval voting is used which is denoted by the term complete approval voting. An encoded candidate solution e is only accepted as winner, if all agents a ∈ A approve e. Therefore, by accepting e, no agent a increases his or her current local costs fa (e∗ ), i. e., fa (e) ≤ fa (e∗ ), ∀a ∈ A. For complete approval voting, the ballot size is restricted to b = 1. Although this voting rule is quite simple, it has not been used in the previous approaches of [20, 22] and [23].
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Adapted Borda maximin voting rule
Each agent a ∈ A ranks the candidate solutions e ∈ B according to his or her preferences. A ranking of e ∈ B by agent a is given by eap eap−1 . . . ea1 where candidate solution eap is preferred most by agent a and receives p points. Solution eap−1 is ranked second and receives p − 1 points, candidate solution ea1 is preferred least and receives 1 point. Given all rankings by all agents a ∈ A, the Borda maximin rule selects that candidate solution e0 which maximizes the lowest number of points assigned by an agent to a candidate solution. Finally, the solution e0 identified by Borda voting is compared to the jointly accepted solution ∗ e by means of complete approval voting. The winning solution becomes the new jointly accepted solution e∗ . 3.6.3
Rawls or minimax voting rule
The minimax voting rule, introduced by [30], minimizes the maximum local cost of an agent. In contrast to the voting rules Borda maximin and complete approval, the Rawls voting rule requires the agents to reveal their local cost to the mediator. For each encoded candidate solution e ∈ B, each agent a ∈ A submits his or her local costs fa (e) to the mediator: e0 = min max{fa (e)}. e∈B a∈A
(17)
Finally, the solution e0 identified by Rawls voting is compared to the jointly accepted solution e∗ by means of complete approval voting. The winning solution becomes the new jointly accepted solution e∗ .
3.7
Pheromone update
As the search advances, the pheromone information on each arc is continuously updated. First, the pheromone τ(u,v) of an edge (u, v) ∈ E evaporates by the evaporation rate ρ. Second, τ(u,v) is increased again by στ max , if and only if edge (u, v) is part of the path P chosen by an ant (that implies u ∈ P ∧ v ∈ P ). Furthermore, equation (18) takes into account the smallest and largest possible pheromone values τ min and τ max , respectively:
τ(u,v) ←
min{(1 − ρ) · τ
+ σ · τ max ; τ max } if u ∈ P ∧ v ∈ P , min max{(1 − ρ) · τ } otherwise. (u,v) ; τ (u,v)
(18)
In contrast to common approaches in the literature, the pheromone update rule (18) does not consider the value of the objective function of a solution. In the distributed planning problem at hand, the agents do not reveal their individual costs to the mediator (cf. Section 2) who controls the search process and updates the pheromone values. Therefore, the pheromone update depends largely on the encoded solution jointly accepted by the agents.
3.8
Variants of the metaheuristic for evaluation purposes
Two variants of the CACM approach are now introduced which are required mainly for the evaluation of CACM in Section 4.
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The first variant is denoted as CACMex , i.e., CACM excluding collaborative local search (CLS). The approach CACMex is identical to CACM but simply skips the call of the function collaborativeLocalSearch (cf. Algorithm 1). The second variant is denoted as hierarchical ant colony metaheuristic (HACM). HACM acts on the additional assumption that the agents determine their individual production plans sequentially and not simultaneously like in CACM. The planning sequence of the agents is determined by the level of the items to produce according to the bill-of-materials. Agents that are responsible to produce lower level items plan before agents with higher level items. That is, the agents responsible to produce the end products (items on level 0) establish their production plan first. Afterwards, the agents responsible to produce items on the next higher level consider demand for lower-level items as given as well as non-negotiable and establish their production plans. With this procedure, HACM solves the DMLULSP in a hierarchical manner by solving a sequence of non-distributed subproblems. A subproblem is the problem of a single agent a to find the minimum cost setup decisions for the set Ia of items he or she is responsible to produce. Thereby, agent a considers previous decisions by other agents as fixed. The subproblem is solved by a reduced variant of CACM (cf. Algorithm 1). Instead of a set of agents only agent a is considered, i.e. A := {a}. Consequently, a voting mechanism is no longer required and agent a accepts a new solution e0 if it reduces the local costs of the current solution e, i.e., if fa (e0 ) < fa (e). Furthermore, HACM uses the same kind of search graph as CACM. However, the graph contains only those nodes that represent setup decisions with respect to the items in Ia . After heuristically solving a subproblem, the solution is transferred to the next agent a0 . a0 considers the decisions of a and all previous agents as fixed and for his part now has to solve his subproblem and so on until all subproblems are solved. The HACM approach, however, is only reasonable if the allocation of items to agents is given in such a way that the local costs of an agent a1 may not be changed by decisions made by agent a2 , if a2 plans after a1 . In Figure 1 for example, the allocation of items to agents a1 and a2 satisfies this property. However, if agent a1 instead of a2 had to produce item 40, the property would not be satisfied. All of the test instances used for the computational tests (cf. Section 4) allow for the usage of HACM. Of course, despite these technical limitations of the applicability, a hierarchical planning approach like HACM is probably better suited for planning scenarios with highly unequal distributions of bargaining power between the agents than for a scenario with agents at eye level and approximately equal bargaining power.
4 4.1 4.1.1
Evaluation Set-up of computational study Goals
The metaheuristic CACM is evalauted by means of a computational benchmark study. Three goals are pursued. First, the impact of the three voting rules from Section 3.6 is studied. Second, the effect of the new collaborative local search heuristic CLS from Section 3.5 is tested. Third, the performance of CACM is compared to results from the literature, that is, to the best known solutions for the non-distributed MLULSP as well as to other heuristics for the distributed MLULSP.
14
T. Buer, J. Homberger, H. Gehring
Table 2: Characteristics of used benchmark instances
4.1.2
class
group
|A|
m
n
no. of instances
s
s1 s2 s5
1 2 5
5 5 5
12 12 12
96 96 96
m
m1 m2 m5
1 2 5
l
l1 l2 l5
1 2 5
40, 50 12, 24 40, 50 12, 24 40, 50 12, 24 500 500 500
36, 52 36, 52 36, 52
40 40 40 40 40 40
Instances
All in all, 528 benchmark instances are used in this study4 . [37], [5], and [6] introduced 176 instances for the non-distributed MLULSP. In these instances, a single agent (|A| = 1) is responsible to produce all items and the instances are divided into three groups of small, medium, and large instances denoted as s1, m1, and l1 (cf. Table 2). Based on these MLULSP instances, [20] introduced 176 instances where the items are jointly produced by two agents (|A| = 2) and 176 instances were the items are jointly produced by five agents (|A| = 5). Therefore, the set of items of the MLULSP instances was partitioned into two and five subsets, respectively, and each subset was assigned to an agent. Demand and cost data of the original MLULSP instances were adopted and not modified. The 352 DMLULSP instances are divided in six groups denoted as s2, s5, m2, m5, l2, and l5 (cf. Table 2). In instance group s5, each agent produces the same number of items, while in the remaining groups (s2, m2, m5, l2, and l5) this is not the case. 4.1.3
Measure of solution quality
In the following, the objective function value f (y) of the DMLULSP is considered in relation to the objective function f nd (y) of the non-distributed MLULSP which has been suggested by [12, 11]. For a given instance, the computed solution y for the distributed MLULSP is compared to the best-known solution y bk for the non-distributed MLULSP and the percentage gap G(y) is calculated as:
G(y) =
f (y) − f nd (y bk ) 100. f nd (y bk )
(19)
The best-known values f nd (y bk ) have been gathered from [6], [29], [18, 20], and [44]. These values are optimal for the instances of group s1. The total cost of an optimal MLULSP solution is a lower bound for the total cost of an optimal solution of the DMLULSP. For test instances without proven optimal solutions, the total cost of the best-known solution of the MLULSP 4
Instances are available at http://www.dmlulsp.com.
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15
Table 3: Values of parameters during the evaluation of metaheuristics CACM, CACMes , and HACM τ min τ max ρ σ s
class s
class m
class l
1 100 5% 5% 50,000
1 1,000 5% 5% 200,000
1 1,500 5% 5% 400,000
is not a guaranteed lower bound of an optimal DMLUSLP solution. Nevertheless, one usually expects that the best-known MLULSP solution has lower total costs than the best-known DMLULSP solution. This is true for the used instances and solution approaches at hand. 4.1.4
Implementation and hardware
CACM was implemented in JAVA (JDK 1.6). The computational experiments were executed on a Linux personal computer with Intel Core 2 Duo processor 1.83 GHz and 2 GB of main memory. To compare the heuristics in Section 4.5, all experiments were executed on this computer. 4.1.5
Parameters
By means of a preliminary study, the parameters of CACM, CACMex , and HACM shown in Table 3 have been determined. For all instances, the pheromone evaporation rate ρ, the pheromone intensification factor σ, and the minimum pheromone value τ min per edge are constant. Then again, the number of generated solutions s and the maximum pheromone value τ max per edge grow with increasing instance size. The ballot size b is evaluated together with the voting rules in the next section.
4.2
Effect of voting rules and ballot size
The effects of the Borda and Rawls voting rules on the performance of CACM are studied by means of the benchmark instances from groups s2 and s5. Both voting rules are applied using ballot sizes b of 1, 10, 20, and 100. The results are given in Table 4. The configuration of Borda voting and Rawls voting with a ballot size of b = 1 results in equivalent voting procedures and, therefore, leads to identical results (see Table 4). Especially in case of scenarios with five agents, the results for the Borda voting rule are slightly better than the results for the Rawls voting rule. However, in the five agents scenario, both voting rules lead to the best results for ballot sizes of only one. For the two agent scenario, ballot sizes of one lead to the second best results. All in all, both voting rules seem to result in higher total costs compared to the degenerated case (b = 1). Therefore, the parameter ballot size b is set to one, that is, the function voting(B, A) in Algorithm 1 is in fact deactivated. The only voting mechanism applied is complete approval voting which is used during collaborative local search. A positive side effect of doing without Rawls and Borda voting is that no local
16
T. Buer, J. Homberger, H. Gehring
Table 4: Effect of Borda and Rawls voting Agents
G
Ballot size b Borda
Rawls
|A| = 2
1 10 20 100
2.21 2.14 2.40 2.46
2.21 2.13 2.31 2.55
|A| = 5
1 10 20 100
8.31 8.78 8.62 9.22
8.31 8.98 9.21 9.41
Table 5: Effect of collaborative local search G
Heuristic s1 CLS CACMex CACM
s2
s5
3.93 5.29 9.53 1.65 4.96 9.15 0.07 2.21 8.31
cost information has to be revealed (Rawls) and the opportunity of strategic bidding, which is intrinsic in ranked based voting, is reduced (Borda).
4.3
Effect of collaborative local search
In this test, the effect of collaborative local search (CLS, cf. Section 3.5) is studied. We show that CLS is able to contribute to increase the solution quality of CACM and simultaneously to reduce the runtime of CACM significantly. The CACM approach including CLS is compared with CACM excluding CLS. The latter is denoted as CACMex . As described previously, only complete approval voting is used. Furthermore, CLS is used as stand-alone heuristic which follows Algorithm 1 without the constructEncodedSolution procedure. For CLS, the initial encoded solution e is set to the unit matrix. The three variants are compared by means of the instances from groups s1, s2, and s5. The instances of s1 correspond to the non-distributed MLULSP for which optimal solutions are known. Consequently, Table 5 depicts the average gap G with respect to the optimal MLULSP solution. CACM solves 80 of 96 instances to optimality. For the remaining 16 instances, the average optimality gap is only 0.07 percent although CACM does not use heuristic information specific to the MLULSP to control the search process. Furthermore, CACM outperforms the standalone CLS and CACMex both in the non-distributed (s1) and distributed (s2, s5) scenario.
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Figure 3: Convergency of CACM and CACMex measured on instance m40 with two agents. This result is also confirmed by additional tests performed in Section 4.5. Thereby, CACM outperforms CACMex for all instance groups. For the largest instances l2 and l5 the advantage of CACM over CACMex is outstanding (cf. Tables 7 and 8). Finally, Figure 3 shows exemplarily for a medium sized instance with two agents, that the heuristic CACM converges faster than the heuristic CACMex . The heuristic CACM converges faster than CACMex , because it is able to learn the relevant pheromone values faster. Pheromone information is adapted based on a jointly agreed candidate solution e∗ . During the first iterations of CACM these candidate solutions are in general much better than the candidate solutions used in CACMex , because CACM uses collaborative local search to improve a solution generated by an ant. Consequently, the relevant pheromone values are adopted faster with respect to the real preferences of the agents. Hence, the next ant which constructs a solution can exploit the structure of the pheromone values and increase the chance to find a superior solution.
4.4
Effect of cost distribution among agents
We now look at the costs each agent bears in the distributed scenario and the effects of this cost distribution on the the performance of the approaches CACM and HACM. For each DMLULSP test instance the distribution of costs to agents is calculated in the following manner. The instance is solved heuristically assuming all items were produced by a single agent, i.e., a non-distributed MLULSP is solved. This single-agent solution y gives us the setup decisions for each item and each period. Now, we presume an identical solution would have been computed for the corresponding distributed scenario with two and five agents, respectively, and compute the cost share fa (y)/f (y) · 100 of each agent a. These two (five) values for a two-agent (five-agent) test instance are denoted as cost distribution of the respective instance. Table 6 shows the resulting cost shares in columns two to six averaged over all instances of an instance group. Consider instance group l5 for example. If the solution for the singleagent scenario computed by a central decision maker would have been realized in the five-agent
18
T. Buer, J. Homberger, H. Gehring
Table 6: Distribution of the total costs among the agents and performance of heuristics. G Instance group
a1
s2 s5
a2
a5
CACM
HACM
44 56 — 27 17 18
— — 18 20
2.11 9.17
5.06 11.09
m2 m5
66 34 — 41 17 15
— — 14 13
1.98 8.06
3.15 8.57
l2 l5
81 67
— 8
5.99 6.62
3.61 6.92
19 10
a3
— 8
a4
— 7
scenario as well, then the agents a1 , a2 , a3 , a4 , and a5 would have to bear costs of 67, 10, 8, 8, and 7 percent of the total costs, respectively (on average over all instances of group l5). Additionally, Table 6 states average gaps G per instance group computed by CACM and HACM. For HACM, the hierarchical planning sequence is the same as the numbering of the agents, i.e. a1 plans before a2 and a2 before a3 etc. The results shown in Table 6 indicate that the performance of CACM and HACM depend on the underlying cost distribution. The instance groups s2, s5, and admittedly more roughly speaking, m2, m5, and l5 feature a uniform cost share (at least, with respect to agents a2 to a5 in m5 and l5). On these instance groups with an almost even cost distribution CACM outperforms HACM. On the other hand, on instance group l2 with a strong uneven cost distribution, the hierarchical planning approach HACM clearly outperforms CACM. HACM not only benefits from the strongly uneven cost distribution but also from the fact that agent a1 bears the lion’s share of the total costs and is the first agent in the planning sequence. Compared to group l2, the cost distribution of group l5 is less uneven, but agent a1 still bears a significant share of the total costs. CACM is able to outperform HACM on the l5 group, however, the advantage is less compared to instances with a more uniform cost distribution like s2, s5, m2. Recall that CACM is based on complete approval voting. In this voting system each vote has equal weight and each agent has equal voting power. Consequently, CACM (with complete approval voting) seems best suited for instances were all agents are affected by approximately equal costs. A strong disproportionate cost share among the agents reduces the performance of CACM and benefits HACM which does not use negotiation at all. Although, the application of collaborative planning does not require that all agents have equal bargaining power, measured for example by an equal cost share, a strong imbalance of the bargaining power suggests to prefer stronger hierarchical planning approaches. Table 6 supports this predominantly accepted fact and furthermore quantifies acceptable imbalances of bargaining power (measured as cost share) to the extent of the solution approaches and test instances at hand.
4.5
Comparison with heuristics from the literature
CACM is compared to three approaches from the literature designed for the DMLULSP denoted here as SA, ES, and ACO which are based on the metaheuristics simulated annealing,
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19
Table 7: Comparison of different heuristics by means of average percentage gap G from bestknown non-distributed solutions. G Instance group s2 s5 m2 m5 l2 l5 mean G
|A|
ES
SA
ACO
CACM
CACMex
HACM
2 5 2 5 2 5
5.11 19.54 47.23 96.15 84.39 162.13
1.42 2.32 7.11 12.13 38.11 74.68
— — 5.73 8.15 — —
2.11 9.17 1.98 8.06 5.99 6.62
4.35 9.21 3.29 8.76 16.33 22.18
5.06 11.09 3.15 8.57 3.61 6.92
51.03
16.02
—
5.65
9.44
6.93
evolutionary strategy, and ant colony optimization, respectively. SA was introduced by [20], ES is presented in [22], and ACO is discussed in [21, 23]. Some of these previous computational experiments were repeated or broadened to take into account additional instances (see Section 4.1.4). In addition, the approaches CACMex (cf. Section 4.3) and HACM (cf. Section 4.4) are use for comparison, too. Table 7 shows the average gap G over 176 two-agents instances (groups s2, m2, l2) as well as over 176 five-agent instances (groups s5, m5, l5). Due to long runtimes, [21, 23] compared their approach only by means of the medium sized instances of groups m2 and m5. The detailed results of CACM, CACMex , and HACM for medium and large instances are given in the appendix in Table 9 and Table 10. With respect to the average gap over all instances, the procedure CACM is the best approach. In detail, it outperforms all other approaches on the medium (m2, m5) and the fiveagent large instances (l5). In particular, CACM reduces the average gap G of the five-agent large instances (group l5) from 75 percent to 7 percent. However, on the two-agent large instances (l2) CACM is inferior to HACM because of the extreme unequal cost distribution as suggested in Section 4.4. The best results for the small instances (s2, s5) are computed by SA. With respect to the required runtime, CACM and HACM are the fastest approaches. Table 8 shows the average runtime per instance for each instance class. Although CACMex and CACM both terminate after generating an identical number of solutions, CACM is faster and finds superior solutions. CACMex requires approximately twice as much time, because all solutions are computed by means of an ant that traverses through the search graph. This is computationally more costly than the combined ant and collaborative local search (CLS) approach used by CACM. For each solution generated by an ant in CACM, the collaborative local search phase (CLS) computes m(n − 1) solutions. Therefore CLS consumes most of the total runtime required by CACM – this corresponding time required by CLS-phase is reported in brackets in the fifth column of Table 8. Pheromone adaption is based on a solution improved by CLS and therefore the learning of the pheromone values is faster (cf. Section 4.3) which leads to better solutions faster. A similar effect is true for a comparison of CACM and ACO. On the one hand, ACO does not apply something similar to CLS. On the other hand, the used search graph of ACO involves significantly more nodes and directed edges to represent a potential solution of a problem instance. Therefore, the learning of appropriate pheromone values to guide the search process is probably more difficult and also more time consuming. Drawing conclusions by a comparison with ES and SA is difficult, because different metaheuristic solution concepts
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T. Buer, J. Homberger, H. Gehring
Table 8: Computation times in seconds for different DMLULSP heuristics. Instance class s m l 1 CPU 2 CPU
ES1 34 122 4100
SA2
ACO2
6 42 2610
— 85 —
CACM2 0.4 (0.4) 18 (17) 1095 (1085)
CACMex,2
HACM2
1 42 2287
0.4 17 1074
2.4 GHz, 2 GB RAM 1.8 GHz, 2 GB RAM
are applied. Nevertheless, it is striking that CACM uses in the CLS-phase a simpler voting mechanism than ES and SA but in exchange voting is applied much more frequently. That is, the preferences of the agents as well as the gathered information about the feasible solution space are integrated into the search process significantly earlier which potentially improves the search. On the other hand, the agents have to exchange information more frequently by using CACM which might become a performance issue for a real world distributed system.
5
Conclusions
A collaborative solution approach based on the metaheuristic ant colony optimization was presented. It is used to coordinate joint production planning of a coalition of self-interested decision makers with asymmetric information. The search process is executed by a mediator who is not aware of the agents local costs. Furthermore, the agents do not have to reveal private cost information during search. Nevertheless, to control the search process, the mediator tries to elicit preferences by means of voting and a new collaborative local search procedure. The results are used to update the pheromone information which guide the solution construction process. Solutions are constructed by means of a new graph structure based on encoded solutions which is less complicated and therefore easier to search. The new approach CACM is evaluated by means of 356 benchmark instances from the literature and outperforms all existing distributed planning approaches on the sets of medium and large instances. For future research, it appears interesting to develop metaheuristics for collaborative lotsizing problems that explicitly take into account differences in the bargaining power of the agents. By this, potential planning conflicts between the agents could be resolved in order to improve the quality of a joint solution.
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Detailed Computational Results Table 9: Results per instance of groups m2 and m5. instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 median mean stand. dev. median G mean G stand. dev. G no. of best
m1, |A| = 1
m2, distributed to |A| = 2 agents
best-known
CACM
CACMex
m5, distributed to |A| = 5 agents
HACM
CACM
CACMex
HACM
194571 165110 201226 187790 161304 342916 292908 354919 325212 385939 179762 155938 183219 136462 186597 340686 378845 346313 411997 390194 148004 197695 160693 184358 161457 344970 352634 356323 411338 401732 185161 185542 192157 136757 166041 289846 337913 319905 366848 305011
198654.45 167159.30 201436.75 189432.95 168115.90 347533.80 296308.90 363276.35 333256.15 397906.50 180144.25 156130.95 183316.80 140267.00 188307.30 350363.75 383953.00 355522.90 421956.45 398145.05 148697.75 201034.60 160924.90 186187.90 161967.00 353513.20 369629.50 363559.75 433067.10 416743.90 192901.15 188634.85 197132.00 138717.25 169387.00 294543.90 344460.85 327621.70 377363.70 306785.75
198882.25 171049.85 206163.35 194624.15 169448.75 361556.75 306968.15 365991.35 338205.10 399733.05 186273.30 158806.55 184841.90 143509.90 193936.05 355287.90 384291.00 359519.30 422378.55 413240.20 155820.90 202997.00 161849.00 185426.70 161967.00 355496.55 366283.90 366807.75 429202.50 416224.40 190285.65 191133.70 196301.35 138538.55 173854.65 297076.30 348805.45 327910.30 379477.30 311204.05
201111.70 171486.90 205828.80 192141.90 171232.00 357774.10 297279.35 365881.15 340158.25 403863.50 187453.30 157343.20 183316.80 143273.10 188832.50 356713.70 393565.40 362010.00 422673.80 406909.50 153633.55 202926.20 161290.90 187520.35 161967.00 357377.20 369877.00 364774.25 431887.60 418036.95 195513.15 190227.60 197589.55 139620.70 172662.30 295794.25 345452.15 331926.50 379710.05 308950.90
207655.95 174637.30 215672.95 204519.15 168236.50 371034.90 312728.90 374513.25 360186.00 427922.75 198732.35 168125.95 203307.80 154127.50 202756.50 374680.00 418806.15 391312.00 467870.25 447712.55 153985.00 204575.00 170716.95 190854.45 169077.40 373222.00 384679.85 385051.35 459952.45 430166.95 198846.05 197505.30 203624.90 142564.05 181454.00 309364.45 362379.50 342949.55 396312.90 318313.65
208241.75 175216.15 214475.45 204219.60 173070.50 386138.70 319108.55 386293.40 371456.10 467683.70 194685.00 175130.15 206459.35 159405.10 202062.65 386120.80 432113.90 396115.20 480909.65 456849.10 156042.30 204828.00 172666.05 189842.45 169109.70 381963.50 394554.15 398472.60 474879.75 436106.80 199090.65 196739.70 205218.55 140784.90 182648.50 314808.00 366854.50 345055.65 404507.30 334211.65
208247.35 174803.05 217148.80 206384.90 171153.90 371768.95 312516.70 374964.25 362002.10 429155.40 201047.75 169405.35 203307.80 155005.80 204165.30 377264.80 421202.30 392701.40 470057.20 450545.85 154732.70 204627.00 172166.70 191636.40 169112.70 376412.35 388249.85 385597.30 460670.15 430335.75 200145.65 198291.65 205281.35 143993.00 183848.25 309271.30 363750.50 345287.60 397393.40 319545.80
247990.33 268851.56 97813.78
247990.33 268851.56 97813.78
251619.83 271784.26 98077.92
250811.53 271889.68 98504.08
262518.70 285503.36 105915.14
264641.73 291603.49 111348.89
263210.05 286829.96 106116.96
1.98 1.94 1.98
3.17 3.19 3.19
3.43 3.33 3.33
7.60 7.92 8.03
8.79 8.89 9.26
8.15 8.27 8.37
33
8
2
31
7
3
DOI: 10.1080/00207543.2013.802822
25
Table 10: Results per instance of groups l2 and l5. instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 median mean stand. dev. median G mean G stand. dev. G no. of best
l1, |A| = 1
l2, distributed to |A| = 2 agents
best-known
CACM
CACMex
l5, distributed to |A| = 5 agents
HACM
CACM
CACMex
HACM
591585 816043 908616 929929 1145749 7639920 3921407 2694924 1880021 1502194 59121845 13422827 4718146 2908634 1737525 468463630 18677678 7308193 3519932 2278214 1187090 1341584 1400480 1382150 1657248 12671808 7159416 4148783 2889151 2183815 101497679 18028225 6780986 4055536 2559885 755506278 33309777 10464662 5116338 3391440
598331.10 983688.85 1032319.15 1012680.90 1268583.15 8155591.50 4079292.90 2852269.45 1920319.95 1603257.30 59771274.85 13547690.90 4968953.90 3045043.25 1899527.05 474478284.15 18926673.55 7486783.70 3677452.80 2484885.50 1246882.50 1537395.30 1503905.30 1501658.40 1816316.70 13098643.70 7475896.35 4422111.25 3060915.15 2346916.65 101738189.20 18369193.00 7263804.65 4338786.85 2859674.40 765975781.25 33938382.50 10887201.85 5468818.55 3527875.50
649815.30 882560.10 1002479.25 1014434.25 1275614.45 8141180.55 4417355.45 3169689.05 2247017.65 1844421.15 60261769.50 14137662.60 5359517.85 3512087.30 2120197.60 485854801.40 19634749.10 7870083.65 4190663.00 2863554.50 1307211.40 1555485.20 1516499.35 1585219.65 1958928.95 14520512.30 8506418.65 5397638.50 3646522.50 2844732.65 115401660.20 20507693.40 8235842.20 5280449.40 3393103.75 856832025.90 37139993.55 11911232.70 6605788.30 4539781.55
632167.10 851634.20 955825.00 1020921.95 1190696.45 8078568.80 4038692.80 2805242.00 1880795.40 1545564.95 59346473.65 13544671.45 4982895.30 3018060.00 1814838.20 473084843.70 18798681.60 7451655.45 3709874.15 2318807.75 1238992.40 1386940.40 1478737.50 1497207.45 1727320.20 12986878.40 7489992.25 4341520.95 2949831.45 2279640.80 101621006.75 18266307.30 7306427.60 4216924.75 2663137.50 768119762.15 33583756.05 10690165.45 5414239.30 3423618.05
640030.95 822896.10 950025.65 974096.60 1216926.25 8177967.70 4229914.15 2980847.25 2062996.65 1607679.70 61490718.35 13781645.70 5142033.00 3238236.40 1882382.40 477152556.90 19051497.65 7686675.85 3875200.35 2540246.90 1243071.00 1405066.95 1421940.10 1421346.45 1764525.40 13487365.65 7929655.90 4729313.00 3105981.85 2330602.00 103712670.75 19008087.90 7486497.70 4453129.20 2766811.95 776564429.50 34262496.20 11022996.20 5699592.35 3663302.05
672543.00 877847.60 1032810.40 1112956.85 1326778.20 8353241.90 4551409.00 3392878.65 2334214.50 1874798.50 71766810.30 15327938.40 5368936.65 3557514.15 2198144.35 525026690.90 20652282.55 8051786.35 4320221.30 3036054.30 1413818.55 1623301.30 1729325.15 1827069.00 2264489.15 14496647.95 8533151.35 5574582.10 3889864.50 2977978.50 127131963.35 23438263.45 8335693.75 5332454.55 3549355.25 863134674.00 37526573.85 12037228.05 6662996.45 4661680.65
682026.20 878116.00 1042237.20 1005603.75 1253179.60 8352017.45 4242115.30 2954997.50 2037780.40 1582942.50 59487021.05 13599258.80 5095968.45 3172715.95 1836227.65 473499841.40 18801404.35 7623376.05 3827338.20 2458638.45 1315914.35 1516627.70 1544275.85 1473113.60 1815437.40 13286789.85 7777793.30 4648287.65 3034770.70 2301789.15 101817608.05 18545396.70 7396442.30 4362216.35 2707332.60 768119905.70 33587297.30 10797449.40 5654083.85 3574782.40
3602664.15 40154281.32 138206216.15
3602664.15 40154281.32 138206216.15
4304009.22 43578409.84 151202639.22
3566746.10 40093832.91 138390529.51
3769251.20 40674586.42 139816427.65
4435815.15 45524424.22 155130476.31
3701060.30 40217753.01 138399898.46
5.58 5.89 5.89
14.21 14.64 15.32
3.95 3.84 3.84
6.60 6.55 6.67
21.85 22.24 22.52
7.76 7.58 7.74
7
0
33
12
0
28
