A Self-Adaptive Memeplex Robust Search Scheme ... - Semantic Scholar

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International Journal of Systems Science

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International Journal of Systems Science Vol. 00, No. 00, 00 Month 20xx, 1–27

A Self-Adaptive Memeplex Robust Search Scheme for solving Stochastic Demands Vehicle Routing Problem Xianshun Chen, Liang Feng, Yew Soon Ong (Received 00 Month 20xx; final version received 00 Month 20xx) In this paper, we proposed a Self-Adaptive Memeplex Robust Search (SAM RS) for finding robust and reliable solutions that are less sensitive to stochastic behaviors of customer demands and have low probability of route failures, respectively, in Vehicle Routing Problem with Stochastic Demands (VRPSD). In particular, the contribution of this paper is three-fold. First, the proposed SAM RS employs the Robust Solution Search Scheme (RS 3 ) as an approximation of the computationally intensive Monte Carlo simulation, thus reducing the computation cost of fitness evaluation in VRPSD, while directing the search towards robust and reliable solutions. Furthermore, a self-adaptive individual learning based on the conceptual modeling of memeplex is introduced in the SAM RS. Finally, SAM RS incorporates a gene-meme co-evolution model with genetic and memetic representation to effectively manage the search for solutions in VRPSD. Extensive experimental results are then presented for benchmark problems to demonstrate that the proposed SAM RS serves as an efficable means of generating high-quality robust and reliable solutions in VRPSD.

Keywords: Vehicle Routing Problems with Stochastic Demands, Self-Adaptation, Memeplexes, Robust Solution Search Scheme, Gene-Meme Co-evolution, Self-Adaptive Individual Learning

1.

Introduction

The Vehicle Routing Problem with Stochastic Demands (VRPSD) involves a set of customers with stochastic demands that must be served by a fleet of vehicles with routes starting and ending at a depot. The VRP is considered as one of the most difficult NP-hard problems due to its complex combinatorial nature; it is the fusion of two NP-hard problems, namely the Traveling Salesman Problem (TSP) and the Bin Packing Problem (BPP). In contrast to the classical VRP, VRPSD is deemed as an even more complex problem due to the uncertainty introduced into VRP by the stochastic demands of the customers. VRP represents the cornerstone of optimization for distribution networks. Although classical VRP is typically treated without stochastic elements in the search (Novoa and Storer 2009), the real-world scenarios in practice have to accept that most of the information and data about customer demands cannot be known beforehand, thus the increasing interest in routing models that are dynamic, stochastic, rich of constraints, etc., and the necessity to develop new efficient and robust search approaches for solving VRPSD. To solve VRPSD, (Secomandi and Margot 2008) classified the existing approaches into dynamic or static class. The markov decision process (MDP) approach of (Secomandi and Margot 2008, Conley 2007, Liang and Lam 2010), the neuro-dynamic programming approach of (Secomandi 2000), etc., are examples of dynamic approaches. In these approaches, routing decisions are made X. Chen is with the School of Computer Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 email: [email protected] L. Feng is with the School of Computer Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 email: [email protected] Y. S. Ong is with the School of Computer Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 email: [email protected] ISSN: 0020-7721 print/ISSN 1464-5319 online c 20xx Taylor & Francis ° DOI: 10.1080/00207721.20xx.CATSid http://www.informaworld.com

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in multiple stages and based on demand realizations. Routes are re-optimized at each stage based on the remaining vehicle capacity and the set of unvisited customers. For routing decisions in static approaches such as chance constraint programming (Shen et al. 2009), robust optimization (Sungur et al. 2008), and stochastic programming with recourse (Christiansen and Lysgaard 2007), once the route decisions are made, they remain unchanged irregardless of the demand realizations. Generally, static strategies have been preferred over their dynamic counterparts because static strategies are useful when a stable reliable solution is desired (Bertsimas 1992) or when re-optimization during route execution is impossible, due to the lack of information (Mendoza et al. 2010). Among the static approaches, most VRPSD research assumes an a priori solution approach. In the first stage, complete a priori routes are designed before any actual demands become known. In the second stage, routes are traversed, demands are revealed, and extra trips to the depot for replenishment are performed if a customer’s demand exceeds current vehicle capacity. The routes order are thus left unchanged. The objective is to find a route sequence that minimizes total expected cost from the original distance traveled and from extra trips to and from the depot. Exact approaches include that of (Bertsimas 1992) who proposed to construct an a priori sequence among all customers of minimal expected total length. (Teodorovic and Pavkovic 1992) developed the a priori sequence based on simulated annealing. (Laporte et al. 2002) then proposed a branch-and-bound method based on the integer L-shaped algorithm. Moreover, (Bianchi et al. 2004) used the length of the a priori tour to approximate the objective, and analyzed several well known metaheuristics for VRPSD. It verified metaheuristics as an effective approach to tackle vehicle routing problems with uncertain demands. (Fleury et al. 2004) presented an evolutionary approach for routing problem with stochastic demands, where a memetic algorithm for VRPSD is introduced. (Novoa et al. 2006) extended a recourse strategy which allows vehicles to serve additional customers from failed routes prior to returning to the depot or to serve customers from failed routes using a new route after returning to the depot. In order to minimize costs further, (Yang et al. 2000) designed a priori routes that may prescribe returns to the depot before the vehicle capacity is depleted (i.e., a form of preventive restocking). Due to the stochastic behavior of customer demands, the feasible solution designed in the first stage may turn out to be infeasible if the final demand of any route exceeds the actual vehicle capacity. This situation is referred to as “route failure” (Juan et al. 2010), and when it occurs, some corrective actions must be introduced to obtain a new feasible solution, which would cause increase in routing cost and control in reality. To address this situation, some authors proposed to construct reliable solutions for VRPSD, which aims at reducing the probability of occurrence on such undesirable situation. For instance, (Tan et al. 2007) described a multiobjective evolutionary algorithm to solve the capacity and time constrained VRPSD, in which the cost associated with the route failures has been considered. More recently, (Juan et al. 2010) introduced the idea to keep a certain amount of surplus vehicle capacity (safety stock) while designing the routes so that if the final routes’ demands exceed their expected values up to a certain limit, they can be satisfied without incurring a route failure. In this paper, we consider the VRPSD with reliable solution construction. In contrast to using Monte Carlo simulation with different safety stock level to arrive at reliable solutions for VRPSD (Juan et al. 2010), which is computationally expensive, we proposed here a Self-Adaptive Memeplex Robust Search (SAM RS) for solving VRPSD. Our contribution is multi-folds. Firstly, SAM RS makes use of the Robust Solution Search Scheme (RS 3 ) to simplify and reduce the computational cost of fitness evaluation in VRPSD while directing the search towards robust and reliable solutions. Furthermore, a self-adaptive individual learning based on the conceptual modeling of memeplex is introduced. Finally, SAM RS incorporates a gene-meme co-evolution model with genetic and memetic representations to efficiently manage the search for solutions in VRPSD. The rest of the paper is organized as follows: Section 2 begins with an introduction of the

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VRPSD. Subsequently, Section 3 discusses the Self-Adaptive Memeplex Robust Search (SAM RS) with respect to the genetic and memetic representations, fitness evaluation, self-adaptive individual learning using memeplex, and the gene-meme co-evolution model, for finding robust and reliable solutions. In Section 4, the empirical assessment of SAM RS on extensive numerical experiments is considered. The superior results obtained by SAM RS in the experiments indicated that it serves as an efficable means of generating high-quality robust and reliable solutions in VRPSD. Finally, the last section end with a brief conclusion of the present research.

2.

Vehicle Routing Problem and Uncertainty in Customer Demands

Here we begin with a description of the VRP problem formulation. The routing design of a solution s to a VRP is a set of K routes: routes(s) = {τ1 , τ2 , . . . , τk , . . . , τK }

(1)

where τk denotes the route visited by vehicle vk , and K is the number of vehicles in s. Route τk defines an ordered set of customers starting and ending with the depot symbolized by 0:

τk = {0, τk1 , τk2 , . . . , τk , . . . , τkm , 0}

(2)

where τk of route τk denotes the th customer visited and m is the number of customers visited by vehicle vk . 2.1.

Capacitated Vehicle Routing Problems

Classical VRPs include Capacitated Vehicle Routing Problem (CVRP). The objective of the CVRP is to minimize the overall distance CCV RP (s) traveled by all K vehicles and is defined as:

CCV RP (s) =

K X

L(τk )

(3)

k=1

=

K X m X d(τk , τk(+1) ) + d(τk1 , 0) + d(0, τkm )] [

(4)

k=1 =1

subjected to constraints: Pm d(τk , τk(+1) ) + d(τk1 , 0) + d(0, τkm ) ≤ Lmax ∀τi ∈ routes(s) P=1 m ∀τk ∈ routes(s) =1 demand(τk ) ≤ Cv

(5)

where d(τk , τk(+1) ) denotes the distance between two consecutive customers τk and τk(+1) , demand(i) denotes the demand of customer i. Lmax is the maximum allowable distance traveled by each vehicle, and Cv is the capacity of each vehicle. In CVRP, the customer demand demand(i) is assumed to be fixed during the actual distribution process. However, in many real world supply chain and distribution systems, most of the information and data about customer demands

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cannot be known beforehand. This kind of VRP is known as Vehicle Routing Problem with Stochastic Demands (VRPSD). 2.2.

Vehicle Routing Problem with Stochastic Demands

The Vehicle Routing Problem with Stochastic Demands (VRPSD) is a stochastic VRP, in which each customer i has a stochastic demand(i) that must be serviced by vehicle vk . In VRPSD, customer demand demand(i) is only revealed at each stop of customer i. During the actual distributionP process, the route τk may fail to meet the capacity constraint at its th customer when Cv < m =1 demand(τk ), at which point vk will have to take a recourse action from τk(−1) to the depot and then back to τk . In this context, the objective of the VRPSD is to find robust solution s = {τ1 , τ2 , . . . , τk , . . . , τK }, that satisfies all customer demands and vehicle capacity constraint Cv , and at the same time minimizes the overall expected distance traveled by all vehicles, CV RP SD (s) as given by:

CV RP SD (s) =

K X

LV RP SD (τk )

(6)

k=1

where LV RP SD (τk ) is the expected distance traveled by vehicle vk . The solution s is robust Li et al. (2011) in the sense that its value does not deviate abruptly as a result of the stochastic behaviors of customer demands in VRPSD. If a solution obtained for VRPSD is highly sensitive to small perturbation in customer demands, its performance becomes unreliable for use in VRPSD (Tsutsui and Ghosh 1997). 2.3.

Safety Stock for Managerial Control over Route Failure in VRPSD

The expected cost of routing in a stochastic setting may not be the sole objective in VRPSD. Other goals may include minimizing route failures during the actual distribution process. This is because whenever a route failure occurs, the vehicle with larger-than-expected accumulated demand has to return to the depot for a refill, which tends to increase the managerial cost. The concept of safety stock (Juan et al. 2010) can be applied to minimize the number of route failures during the actual distribution process. The basic principle behind the safety stock level is that only a certain fraction k of the vehicle total capacity Cv will be considered as available during the routing design phase. In other words, during the VRPSD objective evaluation, the vehicle is modeled to have a lower vehicle capacity Cı , as given by: Cı = k · Cv

(7)

where Cv denotes vehicle capacity of the original VRPSD instance. The parameter k ∈ (0, 1) is known as the safety stock level (Juan et al. 2010). In the routing design phase, the capacity constraint on solution s is then defined as

Cı ≥

X

demand(i), ∀τk ∈ routes(s)

(8)

i∈τk

Here, the objective is to construct reliable solutions (i.e., solutions with a low probability of

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suffering route failures) with optimal or pseudo-optimal total expected costs under the random behaviors of customer demands. This is based on the concept of safety stock which has been previously applied for VRPSD to ensure the managerial control over the frequency of route failure and expected cost penalty (Laporte et al. 1989, Juan et al. 2010). Control over route failures provided by the safety stock level k is effective in that routes will remain feasible as long as the extra demand that might be originated in each route during the routing execution time does not exceed the vehicle reserve capacity. This vehicle reserve capacity, or safety stock, is given by: safety stock = (1 − k) · Cv

(9)

where k denotes the safety stock level associated with routing solution s. The solution reliability can be measured by the probability solution routes will not suffer from any failure (i.e., capacity constraint violation) during the actual distribution phase. 3.

Self-Adaptive Memeplex Robust Search Methodology

In this section, we consider to solve the more realistic VRPSD with safety stock, that reduces the probability of route failure during the actual distribution process. In particular a novel memetic algorithm methodology Self-Adaptive Memeplex Robust Search (SAM RS) is proposed. Memetic Algorithm (MA) represents a subfield of Memetic Computing (MC) (Ong et al. 2010) that is widely established as the synergy of population-based approaches with separate lifetime learning (a.k.a., local search or individual learning) process. Recent studies on MAs have shown that they can converge to high quality solutions more efficiently than their conventional counterparts on a wide range of domains covering problems in combinatorial (Tang et al. 2006, Hasan et al. 2009, Lim and Xu 2005, Lim et al. 2008, Meuth et al. 2010), continuous (Aranha and Iba 2009, Chiam et al. 2009, Kramer 2010, Neri and Mininno 2010), dynamic (Caponio et al. 2008, Lim et al. 2005, Goh and Tan 2007) and multi-objective optimizations (Tan et al. 2005, Ishibuchi et al. 2002, Goh et al. 2009), etc. Algorithm 1 Outline of Canonical MA BEGIN Create an initial population of solutions Pop(gen = 0) = {s1 , s2 , . . . , sı , . . . , sM } While(stopping conditions are not satisfied) gen ← gen + 1 Perform evolutionary operators to generate Pop(gen) from Pop(gen − 1) based on f (sı ): Evaluate f (sı ), ∀sı ∈ Pop(gen) Perform Lifetime Learning Update global best solution sg End While END Alg. 1 outlines the schematic workflow of a canonical MA. The algorithm starts with the initialization of a population Pop(gen = 0) of candidate solutions. At each generation, a population of offspring Pop(gen) is generated using evolutionary search through reproduction operators such

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as crossover and mutation. The Pop(gen) subsequently undergoes lifetime learning and fitness evaluation, and update the global best solution. This workflow of memetic search proceeds until some stopping conditions are reached. Many of the current state-of-the-art MA are adaptive MAs that coordinate the memes while the evolutionary search progresses online, such as the meta-Lamarckian learning (Ong and Keane 2004), cost-benefit (Jakob 2010), or fitness diversity schemes (Neri et al. 2007, Caponio et al. 2009, Tirronen et al. 2008, Caponio et al. 2007), and hyperheuristics Segura et al. (2011). Another class of adaptive MAs is the self-generated MAs and co-evolution MAs (Krasnogor 2004, Smith 2002), which rely on memetic evolution. Lamarckian MA, however, is generally sensitive to uncertainties such as the stochastic behaviors of customer demands in VRPSD, and displays signs of over-searching due to the favor for solutions with higher fitness values. This is particularly so due to the hill-climbing nature of the local search in MAs. However, while the desired robust solutions in VRPSD must exhibit a high tolerance or robustness to uncertainties, they may not be the globally optimal solution. To tackle this problem without sacrificing the search capability of MA, we proposed a self-adaptive memeplex robust search scheme (SAM RS), which is a general approach for optimization in uncertain environments. SAM RS is composed of the following components, namely, geneticmemetic representation, RS 3 fitness evaluation, self-adaptive memeplex individual learning, a gene-meme co-evolution model, and a genetic representation repair. The genetic representation encode vehicle routes and the safety stock level into the geneplex (which represents the routing solution for VRPSD), while the memetic representation encodes the local search instructions (memes) or memeplex (which is a set of mutually assisting memes that carry out behaviors in the individual learning). In particular, the local search instructions self-assemble in the form of memeplex that adapts to refine the solution effectively. The robust solution search scheme (RS 3 )provides a fitness prediction model that effectively induces the sprit of Baldwinian learning Ong et al. (2006) within SAM RS. It directs the MA search towards robust solutions by inducing perturbations into the phenotypic features of the solution (e.g. in the case of VRPSD, the stochastic properties of the customer demands) when evaluating the fitness of individuals. The gene-meme co-evolution model co-adapts the genetic evolution of routing solutions along with the memetic evolution of memeplexes. The genetic representation repair procedure (Chen et al. 2011) restores any infeasible routes in the solution before fitness evaluation. In the subsequent sections, the proposed SAM RS with respect to the designs of solution representation, fitness evaluation, self-adaptive individual scheme, gene-meme co-evolution (e.g., genetic/ memetic crossover and mutation), and genetic representation repair as depicted in Fig. 1 are described.

Figure 1. Outlines of SAM RS and its components

3.1.

Genetic and Memetic Representations in SAM RS

Here we described the genetic and memetic representations of SAM RS. The genetic representation denotes the parametric design space of the VRPSD problem, which include routing design

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{τ1 , τ2 , . . . , τk , . . . , τK } and safety stock level kı associated with solution individual sı . On the other hand, the memetic representation denotes the parametric decision space of the search algorithm, which refers to the instructions (memes) that make up the local search (memeplex) Mı . The relationship of genetic solution and its associated memeplex is that of a cooperative co-evolution, in which memeplex attempts to improve the affiliated genetic solution, i.e., the routing solution. This assures the survival and reproduction of both the high quality memetic and genetic materials through selection pressure. In SAM RS, the encapsulation of instructions and parameters from both the design and decision space through genetic and memetic representations allows an individual in the population to behave more or less like an autonomous agent during the evolutionary and lifetime learning phases of the MA, whose behaviors manifest in the following aspects: • sı is capable of evaluating its fitness under the uncertainty of stochastic customer demands in VRPSD, through the safety stock level and the statistical demand distribution function. • sı is capable of interpreting and executing local search instructions (a.k.a., memes) in memeplex Mı to improve the routing solution during individual learning. • sı is capable of learning and improving the programming structure of the associated memeplex Mı through credit assignment learning mechanism during lifetime learning. • sı imitates and exchanges information on its routing solution, associated safety stock level and memeplex, with other individuals in the population Pop(gen) during the evolutionary phase. Fig. 2 depicts the genetic and memetic representation defined in SAM RS, i.e., genotype and phenotype in genetics, while memotype and sociotype in memetics, respectively.

Figure 2. Genetic and Memetic Representations of sı with the decoded routes, safety stock level kı and search scheme Mı

In the evolutionary phase, sı undergoes independent genetic and memetic transmission, variation and selection operations. This gene-meme co-evolutionary process enables the value of safety stock level kı and behavior of the memeplex Mı in self-adapting to refine individual sı effectively.

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3.2.

Fitness Evaluation based on Robust Solution Search Scheme

The Robust Solution Search Scheme (RS 3 ) is designed for convergence to robust solutions with the stochastic demands modelled as perturbations induced into the phenotypic representation of solutions before fitness evaluation (Tsutsui and Ghosh 1997). Supported by both mathematical models and empirical results in (Tsutsui and Ghosh 1997), the RS 3 is used in our quest towards convergence to robust VRPSD solutions. Making little assumption on the customer demand distribution, the scheme effects implicit averaging of solution fitness (Tsutsui and Ghosh 1997), i.e., averaging the fitness evaluations of each solution vector s and its perturbed vectors si across multiple generations i, which equates to a Monte Carlo simulation on vector s over time. In other words, RS 3 in SAM RS serves as an approximation of the Monte Carlo simulation that evolves towards a true Monte Carlo simulation with increasing generations. It thus constitutes as an efficient substitution of the computationally intensive simulation-based fitness evaluation, especially when dealing with large populations of individual solutions. In many real-world scenarios, historically archived data can be used to model the P DF (demand(i), µi , σi2 ) of each stochastic customer demand. According to (Juan et al. 2010), Log-Normal distribution serves as a natural choice for modeling non-negative customer demands. Thus we considered the Log-Normal distribution as the statistical model for customer demands in the present study (although other distribution may also apply), which is defined as:

2

(ln x−µ ) 1 − 2σ2 i i √ e , X = {x|x = demand(i), 1 ≤ i ≤ N } xσi 2π µ ¶ 1 Var[demand(i)] µi = ln(E[demand(i)]) − ln 1 + , 2 E[demand(i)]2 µ ¶ Var[demand(i)] σi2 = ln 1 + . E[demand(i)]2

P DFX (x, µi , σi2 ) =

(10) (11) (12)

where E[demand(i)] denotes the expected demand of the VRPSD instance and V ar[demand(i)] denotes the demand variance, which can be modeled as:

V ar[demand(i)] = α · E[demand(i)]

(13)

with α ∈ (0, 1) denoting the variance level. The fitness of solution sı is evaluated together with its respective perturbed problem instances {Pi }Ii=1 which is given by: PI f (sı ) =

i=1 fi (sı )

PI =

i=1

PI =

i=1

under capacity constraint

(14)

I PK

k=1 Li (τk )

I

PK

Pm−1

k=1 [

=1

(15) di (τk , τk(+1) ) + di (τk1 , 0) + di (0, τkm )] I

(16)

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m X

demandi (τk ) ≤ Cı ≡ kı · Cv , ∀τk ∈ routes(s), i = 1, . . . , I

9

(17)

=1

where fi (sı ) is the fitness of sı as evaluated in perturbed problem instance Pi , and Li (τk ) is the traveled distance of route τk as evaluated in Pi . di (τk , τk(+1) ) is the distance between two consecutive customers τk and τk(+1) in Pi , demandi (τk ) is the demand of customer τk as sampled in Pi , and Cv denotes vehicle capacity of the VRPSD instance. Parameter kı ∈ (0, 1) is the safety stock level associated with solution sı .

3.3.

Self-Adaptive Individual Learning using Memeplex

In this section, we describe the proposed self-adaptive individual learning. In particular, a structural form of memetic existence, i.e., co-adapted meme complexes or memeplexes, is introduced to the lifetime learning phase of SAM RS. A memeplex is essentially composed of mutually assisting memes (Dawkins 1989), that work together to achieve more than what each meme could accomplish alone (Situngkir 2004, Blackmore 1999). During the lifetime learning phase of a solution sı , the memeplex Mı associated with sı manifest as a set of instructions or memes that carry out local refinement on sı . The proposed memeplex local search structure is unique since it is a self-adaptive local searcher that self-adapts to suit the associated solution during lifetime learning. A memeplex Mı associated with solution sı , can be symbolized by its terminal set {mk }L k=1 of L co-adapted memes. Each meme mk in a memeplex serves as a single instruction for local refinement in the lifetime learning process. In the design of suitable representation for the memes, one can start with modeling the components of the local search in VRP. In this study, 6 move operators {T1 , T2 , T3 , T4 , T5 , T6 } are defined for VRP. Move operators T1 , T2 , T3 are clustering procedures to reassign customers between routes with the objective of reducing the overall traveled distances. T4 denotes a move operator that reduces the total number of routes in the current solution through route merging. Move operators T5 and T6 , on the other hand, serve to reduce the route length defined in the current solution, by reassigning the visiting order of customers in the route. In what follows, each of these move operators are briefly described: T1 : customer τk of route τk ∈ s is removed and reinserted into a separate route τk0 ∈ s, if fitness improvement ∆C(s) > 0. T2 : customer τk of route τk ∈ s is swapped with customer τk0 l of route τk0 ∈ s, τk0 6= τk , if fitness improvement ∆C(s) > 0. T3 : a sub-route τˆk = {τk(+1) , τk(+2) , . . . , τk(+µ) } of route τk ∈ s is swapped with sub-route τˆk0 = {τk0 (l+1) , τk0 (l+2) , . . . , τk0 (l+ν) } of route τk0 ∈ s, τˆk 6= τˆk0 , if fitness improvement ∆C(s) > 0. T4 : two shorter routes τk , τk0 in the current solution are merged to reduce the number of routes and thus the overall distance traveled, if fitness improvement ∆C(s) > 0. T5 : a customer τk ∈ τk is reinserted after a separate customer τk0 l ∈ τk0 , if fitness improvement ∆C(s) > 0. T6 : a customer τk ∈ τk is joined with separate customer τkl ∈ τk by reversing the visiting order of customers between τk and τkl , if fitness improvement ∆C(s) > 0. Taking also into considerations the move operator, as well as others including the acceptance strategy (first improvement or best improvement)) and search depth, meme mk in a memeplex can then be expressed as:

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mk = (< Ti >, < Acceptance Strategy >, < Optional Depth >, . . .)

(18)

The memetic synergy between memes in Mı is represented by matrix Wı , which is given by: 

w11 w12  w21 w22  Wı =  .. ..  . . wL1 wL2

 · · · w1L · · · w2L   . . ..  . .  · · · wLL

(19)

where wij refers to the connectivity (or synergy) from meme mj to mi on solution sı . During the lifetime learning process, memeplex Mı adaptively selects and executes the memes from its terminal set {mk }L k=1 , to locally refine solution sı . The workflow of the lifetime learning on sı with memeplex Mı serving as the adaptive local searcher is illustrated in Proc. 1. Procedure 1 Outline of Self-Adaptive Individual Learning:(sı , Mı , Pı ) procedure BEGIN ∆C(Mı ) ← 0 ∆T( Mı ) ← 0 mprev ← ∅ While TRUE Select meme mj ∈ Mı with activation probability Pactivate (mj , mprev ) (∆Cj , ∆Tj ) ← Improve sı by mi in the context of Pı Assign Credit to meme pair (mi , mprev ) based on feedback (∆Cj , ∆Tj ) ∆C(Mı )+ = ∆Cj ∆T (Mı )+ = ∆Tj If ∆Cj == 0 then ∆T (Mı ) If rand(0, 1) > exp(− ∆C(M ) then ı) Break End If End If mprev ← mj End While END The activity of Mı during the lifetime learning process, as illustrated in Proc. 1, is a series of memetic activations which iterate with each activation mobilizing a meme mj ∈ Mı to locally refine solution sı . The memetic activation probability Pactivate (mj , mi ) is given by: ( Pactivate (mj , mi ) =

w PL ij µ=1 wiµ w PL jj µ=1 wµµ

mi 6= ∅ otherwise

(20)

where i denotes the index of the previously activated meme, wij denotes the preference of selecting mj given mi . When activated, mj deploys its associated move operator and the other encoded instructions including the move operation and move acceptance criteria, to refine solution sı . During the lifetime learning process, as illustrated in Proc. 1, the search performances of

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the memeplex, in terms of {∆Cj , ∆Tj } (where ∆Cj and ∆Tj denotes solution fitness improvement and computational cost made by mj ), are recorded and used as feedback for adjusting the individual meme’s fitness and the memetic synergy between memes. This processed feedback is then assimilated in credit assignment to individual meme fitness and memetic synergy via weight adjustment to Wı , thus allowing individual meme fitness and memetic synergy to synchronize with the structure of individual solution. The detailed steps of synchronizing the individual meme fitness and memetic synergy, as well as the memeplex network adaptation, according to structure of the individual solution, are next described in what follows. wij ∈ Wı is defined for any pair of memes mi and mj , represented by wij and wji , respectively, where wij denotes the degree to which mi ’s activation favors mj ’s activation, thus providing an indication on the preferences of one type of interaction over another. Upon activation of a meme mj ∈ Mı , an immediate reward rj is computed. Credit assignment is then performed via:

wij = γwij + rj = γwij +

∆Cj ∆Tj

(21) (22)

where mi denotes the previously executed meme, while ∆Cj , ∆Tj denote the operational gain and computational cost associated with the functional activity performed by mj , and 0 < γ < 1 is the discount factor of wji . In Eqn. 21, the discount factor γ reflects the influence of a meme’s recent performance over the past. Fig. 3 graphically depicts the credit assignment to the matrix Wı as Mı undergoes a series of memetic activation during its task life cycle, where ri denotes the immediate reward to meme mi ∈ M.

Figure 3. The credit assignment for co-adapted memes as M undergoes a series of memetic activation

Illustrated in Fig. 3, the task life cycle of a memeplex candidate M, is a multiplicity of simple interactions between the weight adjustment for credit assignment processes described by Eqn. 21, and the non-linear interaction process of memetic activation described by Eqn. 20. This self-organizing process allows the gradual learning of compatible memes that exhibits memetic

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stability. 3.4.

Gene-Meme Co-evolution in SAM RS

The evolutionary process in SAM RS is a cooperative co-evolutionary process in which both the associated memetic and genetic materials undergoes transmission, variation and selection mechanisms. This gene-meme co-evolutionary mechanism allows the value of safety stock level kı and the behavior of the memeplex Mı to self-adapt to the routing design of sı , which is depicted in Fig. 4.

Figure 4. Gene-Meme Co-evolution Cycle among sı , Mı in Pop(gen))

The relationship of VRP solutions and its associated memeplex can manifest in the form of cooperative co-evolution, in which memeplex attempts to improve its affiliated genetic solution, i.e., the route design in VRP, while at the same time the high quality of route design in genetic solution assures the memeplex survives through selection pressure, thus giving it higher chances to replicate. 3.4.1.

Genetic and Memetic Crossover

The genetic crossover process between solutions sı and s consists of two separate mechanisms, i.e., crossover between routes(sı ) and routes(s ) and crossover between ki and kj . For crossover between routes(sı ) and routes(s ), the routes in child solution sc are generated from partial routes of parent solutions sı , s . In particular, taking the depot node e0 as the axis center, a bisecting axis ψ of random angle θ is generated to slice the VRP graph into two separate subgraphs containing non-empty customers, H+ and H− , and the child solution sc is a fusion of partial routes in sı and s (e.g. In Fig. 5, sc is created from {τ1ı , τ2ı , τ3ı } ∈ sı and {τ1 , τ3 , τ4 } ∈ s ). The angle θ is selected in such a way as to ensure that both subgraphs are non-empty (Chen et al. 2011). For crossover between kı and k , the safety level kc in child solution sc is generated based on 1-dimensional linear interpolation between parent safety stock levels kı and k , and given as: kc = kı + t · (k − kı ), t ∈ (0, 1)

(23)

For the memetic crossover between Mı and M , the memeplex Mc in child solution sc is generated from the parent memeplexes Mı and M by random exchange of corresponding weight

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Figure 5. Genetic Crossover between sı and s (phenotype)

elements between weight matrices Wı and W . Graphically, this can be depicted by Fig. 6. 3.4.2.

Genetic and Memetic Mutation in SAM RS

The genetic mutation process for solution sı composes of 2 separate mechanisms, i.e., mutation of routes(sı ) and mutation of kı . For mutation in routes(sı ), the mutation operator is performed by one of the four mutation heuristics, namely swap, inversion, insertion and displacement, through random selection, which is purely stochastic in nature and each heuristic has equal probability to be chosen during mutation (Chen et al. 2011). For mutation of kı , a small perturbation δ drawing from a Normal distribution is added to kı , which is given by: kı0 = kı + δ

(24)

For the memetic mutation of Mı , a small set of weights {wij } is randomly selected from Wı , and for each selected wij , a small perturbation δij drawing from a Normal distribution is added.

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Figure 6. Memetic Crossover between Mı and M (top: memotype bottom: sociotype)

3.5.

Genetic Representation Repair

As aforementioned, under the stochastic behaviors of the customer demands, information pertaining to the true demand of a customer is only made known at each customer stop, hence it is possible for routes in solution sı to turn infeasible as a result of capacity constraint violations. The repair of genetic solution sı is thus necessary before fitness evaluation to ensure sı as a feasible solution in the context of sampled customer demand distribution. Here, a Split-Repair procedure (Chen et al. 2011) restores any infeasible routes in sı before fitness evaluation. As illustrated in Proc. 2, the procedure first classify all routes in sı into Γ and Γ0 : Γ = {τk |τk is infeasible route}

(25)

Γ0 = {τk |τk is feasible route}

(26)

Subsequently, a split process is carried out when Γ 6= ∅. In particular, an infeasible route τk is removed from Γ and split into two routes τa and τb by removing the edge between consecutive customers τk and τk(+1) , such that any increase in overall distance traveled δ() due to the split, is at minimum.

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Procedure 2 Outline of Split-Repair (sı ) procedure BEGIN Classify routes in sı into Γ, Γ0 While Γ = 6 ∅ Given τk ∈ Γ For  = 1 to m Do δ() = distance increase by splitting τk into τk and τk(+1) End For J ←  with min{δ()} Split τk into τkJ and τk(J+1) to form τa and τb For τ ∗ ∈ [τa , τb ] If τ ∗ is infeasible then Γ ← Γ + τ∗ Else Γ0 ← Γ0 + τ ∗ End If End For Γ ← Γ − τk End While END 3.6.

Workflow of SAM RS

The basic steps of the SAM RS Memetic Framework are outlined in Alg. 2. The algorithm starts with the initialization of a population Pop(gen = 0) of candidate solutions. Each solution sı is made up of both genetic and memetic materials. The genetic material is composed of the route design {τ1 , τ2 , . . . , τk , . . . , τK } and the associated safety stock level kı in the individual solution sı , while the memetic material refers to the individual learning or local search defined in the memeplex Mı , which self-adapts along with the associated individual solution sı . At each generation, a population of offspring Pop(gen) is generated using evolutionary search, where both genetic and memetic transmission, variation, and selection processes take place through reproduction operations involving crossover, and mutation, and subsequently lifetime learning and RS 3 fitness evaluation. The best solution sopt of a population is then identified according to fitness f (sı ). At this decision point, the evaluation of expected cost CV RP SD (sopt ) is performed using Monte Carlo simulation, which is pitted against CV RP SD (sg ) for updating of the global best solution sg . This workflow of memetic search proceeds until the stopping condition is reached (e.g., the maximum number of fitness evaluations budget allocated).

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Algorithm 2 Outline of SAM RS BEGIN Create an initial population of solutions Pop(gen = 0) = {s1 , s2 , . . . , sı , . . . , sM } While(stopping conditions are not satisfied) gen ← gen + 1 Perform evolutionary operators to solutions to generate Pop(gen) from Pop(gen − 1) based on f (sı ): (sı , s ) ← select parents from Pop(gen − 1) based on f (sı ) Genetic Evolution: Genetic Crossover on {routes(sı ), kı }, {routes(s ), k } with probability pc Genetic Mutation on {routes(sı ), kı } with probability pm Memetic Evolution: Memetic Crossover on Mı , M with probability pc Memetic Mutation on Mı with probability pm Evaluate f (sı ), ∀sı ∈ Pop(gen): Sample instance Pı by parameter perturbation on the original VRPSD instance Genetic Repair sı for infeasible routes Fitness Evaluation: f (sı ) in the context of Pı Self-Adaptive Individual Learning: (sı , Mı ) ∀sı ∈ Pop(gen) (Proc. 1) Update global best solution sg : Select sopt = maxf (sı ) {sı |sı ∈ Pop(gen)} Evaluate CV RP SD (sopt ) by performing Monte Carlo simulation on sopt based on P DF (µi , σi2 ) sg ← sopt if CV RP SD (sopt ) < CV RP SD (sg ) End While END

4.

Empirical Study

In this section, an empirical study on the Self-Adaptive Memeplex Robust Search (SAM RS) for solving VRPSD is presented. In order to analyze our proposed SAM RS, we pit it against, two algorithms, namely CV RP and OBS, which denote recently proposed state-of-the-art methodology (Juan et al. 2010) as the baseline algorithms, for the discovery of robust and reliable solutions in safety stock based VRPSD. The VRPSD benchmarks used in this emperical study are the same as that in (Juan et al. 2010), which were generalized from a set of 55 classical CVRP instances (Ralphs 2010). These problem instances include 14 random instances from (Augerat et al. 1995) set A with sizes ranging from 32 to 80, 13 clustered instances from Augerat set B with sizes ranging from 31 to 78, 16 instances from Augerat set P with sizes ranging from 19 to 101, 11 instances from sets E, M and F by (Christofides and Eilon 1969) and (Fisher 1994). They differ in terms of size (i.e., number of customers), customer distribution, symmetric structure, and vehicle capacity. All the node coordinates and vehicle capacities in the CVRP instances are kept consistent to (Juan et al. 2010), with the deterministic customer demands {di } changed to stochastic demands {demand(i)} with expectation E[demand(i)] = di and variance Var[demand(i)] = α · di .

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17

Experimental Setup

All results presented in this section are the summaries obtained from 30 independent runs for each VRPSD instance under a consistent experimental setup. Each run continues until a maximum of 50,000 fitness evaluations is reached. In each run, the search settings of the memetic algorithm used for solving the benchmark problems are defined as follows: population size of 30, crossover probability of pc = 0.5, mutation rate of pm = 0.5, roulette-wheel selection scheme, and an elite size of 2. The settings are chosen to encourage greater exploration and hence diversity in the population during the search. Statistical results with respect to the following metrics are then reported: BKS

-

C0 C1 C2

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C2

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R0 R1

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R2

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R2

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K1 K2 k1 k2 Gap0 Gap1 Gap2 Gap2

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4.2.

denotes the fixed cost of the best known solution CV RP from the corresponding CVRP instance denotes the expected cost of the best known solution sbks on the VRPSD instance denotes the expected cost for the best solution obtained by OBS on the VRPSD instance denotes the expected cost for the best solution obtained by SAM RS on the VRPSD instance denotes the average of the converged P solution’s expected cost on the VRPSD instance over Crun N R = 30 independent runs. C2 = runs NR denotes the route reliability level associated with the solution having expected cost C0 denotes the route reliability level associated with the solution having expected cost C1 in OBS denotes the route reliability level associated with the solution having expected cost C2 in SAM RS denotes the average of the converged solution’s reliability on the VRPSD instance over P Rrun runs N R = 30 independent. R2 = NR denotes number of routes in solution having expected cost C1 in OBS denotes number of routes in solution having expected cost C2 in SAM RS denotes safety stock level in solution having expected cost C1 in OBS denotes safety stock level in solution having expected cost C2 in SAM RS −BKS denotes the average gap between C0 and BKS over 55 instances. Gap0 = C0BKS × 100% C1 −BKS denotes the average gap between C1 and BKS over 55 instances. Gap1 = BKS × 100% −BKS denotes the average gap between C2 and BKS over 55 instances. Gap2 = C2BKS × 100% C2 −BKS denotes the average gap between C2 and BKS over 55 instances. Gap2 = BKS × 100%

Results and Discussion

Tables 1, 2, 3 show the complete results obtained for all 55 VRPSD instances. The results of CV RP and OBS are obtained directly from (Juan et al. 2010). Each of these tables corresponds to one of the three uncertainty scenarios (low-variance, medium-variance and high-variance), which were previously described in Section 4.1. In the tables, the instance name is given by “[SetId]-n[Size]-k[VehicleCount]”, where “[SetId]” denotes the id of the benchmark set (e.g., A, B, E, P, etc.), “[Size]” denotes the problem size (i.e., the total number of customers served), and “[VehicleCount]” denotes the vehicle count specified in the original classical instance. The values, highlighted in bold, in the tables, report the best results arrived on a particular problem instance by all the algorithms under comparison.

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# 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

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Instance A-n32-k5 A-n33-k5 A-n33-k6 A-n37-k5 A-n38-k5 A-n39-k6 A-n45-k6 A-n45-k7 A-n55-k9 A-n60-k9 A-n61-k9 A-n63-k9 A-n65-k9 A-n80-k10 B-n31-k5 B-n35-k5 B-n39-k5 B-n41-k6 B-n45-k5 B-n50-k7 B-n52-k7 B-n56-k7 B-n57-k9 B-n64-k9 B-n67-k10 B-n68-k9 B-n78-k10 E-n22-k4 E-n30-k3 E-n33-k4 E-n51-k5 E-n76-k10 E-n76-k14 E-n76-k7 F-n135-k7 F-n45-k4 F-n72-k4 M-n101-k10 M-n121-k7 P-n101-k4 P-n19-k2 P-n20-k2 P-n22-k2 P-n22-k8 P-n40-k5 P-n50-k10 P-n50-k8 P-n51-k10 P-n55-k15 P-n55-k7 P-n60-k10 P-n65-k10 P-n70-k10 P-n76-k4 P-n76-k5

BKS 787.08 662.11 742.69 672.47 733.95 833.21 944.88 1146.77 1074.46 1355.80 1039.08 1622.15 1181.69 1766.50 676.09 956.29 553.16 834.92 754.22 744.23 749.96 712.92 1602.28 868.31 1039.36 1276.20 1227.90 375.28 505.01 837.67 524.61 835.26 1024.40 687.60 1164.73 723.54 241.97 819.56 1043.88 691.29 212.66 217.42 217.85 588.79 461.73 699.56 632.69 741.50 944.56 570.27 748.07 795.66 829.93 598.19 633.32

CV RP C0 828.24 689.28 770.58 687.68 773.46 940.65 1076.45 1299.53 1205.57 1517.79 1145.39 1884.98 1352.64 2028.28 726.15 1068.48 589.74 912.97 801.66 800.04 826.93 830.88 1922.49 1070.90 1207.88 1525.45 1370.36 375.28 505.01 838.80 539.21 905.54 1121.13 703.46 1263.53 757.83 247.76 933.10 1101.50 693.52 217.48 233.06 226.34 619.74 466.25 753.27 697.56 828.90 1033.66 588.59 790.46 854.65 906.87 611.81 651.51

R0 0.55 0.60 0.57 0.78 0.41 0.16 0.10 0.20 0.04 0.12 0.09 0.08 0.06 0.08 0.50 0.45 0.49 0.23 0.23 0.31 0.33 0.17 0.06 0.02 0.08 0.04 0.11 1.00 1.00 0.99 0.27 0.07 0.04 0.58 0.46 0.37 0.70 0.07 0.08 0.96 0.85 0.65 0.65 0.50 0.77 0.16 0.10 0.05 0.04 0.39 0.20 0.13 0.05 0.49 0.34

C1 797.74 678.14 762.02 685.36 754.04 839.12 975.12 1169.85 1106.31 1383.90 1073.99 1682.37 1241.51 1828.40 682.65 986.58 564.47 858.91 764.27 752.54 762.26 739.23 1626.89 903.31 1096.78 1311.58 1299.00 375.28 505.01 838.80 537.11 861.77 1056.36 695.49 1191.26 729.97 247.29 866.83 1074.85 693.52 216.96 220.89 224.13 589.39 464.51 721.91 646.65 761.62 991.85 578.61 761.49 811.83 853.83 609.31 648.86

OBS k1 0.96 0.97 0.98 0.96 0.95 0.96 0.96 0.95 0.96 0.95 0.97 0.95 0.96 0.96 0.95 0.94 0.96 0.95 0.94 0.95 0.95 0.95 0.95 0.95 0.96 0.94 0.97 1.00 1.00 1.00 0.98 0.98 0.96 0.97 0.99 0.96 0.99 0.95 0.98 0.98 0.98 0.99 0.97 0.97 0.97 0.96 0.97 0.96 0.95 0.95 0.97 0.97 0.96 0.98 0.98

K1 5 5 6 5 6 6 7 7 9 9 10 10 10 10 5 5 5 7 6 7 7 7 9 10 10 9 10 4 4 4 5 11 15 7 7 5 4 10 8 4 2 2 2 9 5 10 9 11 17 7 10 10 11 5 5

R1 1.00 0.90 0.67 0.93 0.99 0.96 0.93 0.95 0.87 0.94 0.75 0.95 0.99 0.86 0.99 1.00 0.96 0.99 1.00 1.00 0.99 0.98 0.98 0.98 0.92 0.99 0.76 1.00 1.00 1.00 0.65 0.70 0.90 1.00 0.99 1.00 1.00 0.99 0.83 0.96 0.85 0.93 0.96 1.00 1.00 0.90 0.91 0.90 0.79 1.00 0.85 0.92 0.94 0.97 0.85

C2 796.16 675.71 756.07 677.99 748.77 835.83 969.01 1165.63 1115.23 1377.54 1068.74 1667.58 1234.73 1864.78 678.28 990.43 561.70 855.64 761.53 748.64 758.22 730.68 1634.39 908.05 1092.30 1313.42 1300.69 375.00 503.00 835.70 533.02 858.64 1068.57 689.01 1189.82 729.97 244.92 825.25 1063.82 681.03 209.95 218.86 221.92 590.00 461.00 725.36 643.95 757.75 990.06 576.00 763.78 814.37 846.13 599.56 643.42

k2 0.82 0.93 0.95 0.84 0.92 0.88 0.99 0.85 0.82 0.94 0.95 0.80 0.83 0.92 0.96 0.94 0.98 0.98 0.93 0.91 0.91 0.97 0.96 0.91 0.93 1.00 0.98 0.99 1.00 0.95 0.90 0.86 0.95 0.97 0.92 0.95 0.86 0.87 0.97 0.94 0.82 0.86 0.96 0.96 0.84 1.00 0.86 0.94 0.99 0.85 0.82 0.85 0.83 0.87 0.86

SAM RS K2 R2 5 1.00 6 1.00 7 0.90 5 1.00 6 1.00 7 1.00 7 0.95 8 1.00 10 1.00 10 0.93 11 1.00 10 1.00 11 1.00 12 1.00 5 1.00 5 1.00 5 1.00 8 1.00 6 1.00 8 1.00 8 1.00 8 1.00 9 1.00 10 0.85 12 1.00 10 1.00 11 1.00 4 1.00 4 1.00 5 1.00 6 1.00 13 0.94 16 1.00 7 1.00 7 1.00 5 1.00 5 1.00 8 1.00 8 1.00 5 1.00 3 1.00 3 1.00 2 1.00 9 1.00 6 1.00 11 0.99 9 1.00 12 1.00 17 0.97 8 1.00 10 0.96 10 0.98 11 0.99 5 0.98 6 1.00

C2 796.20 677.12 756.94 678.67 749.38 841.18 969.42 1185.19 1119.16 1385.09 1070.97 1699.61 1242.54 1893.68 678.49 995.31 561.75 878.99 762.34 749.02 760.11 732.93 1647.15 910.97 1098.46 1329.13 1304.64 380.00 503.80 839.42 537.89 867.81 1078.48 691.05 1203.69 734.59 250.38 833.31 1067.81 688.55 210.07 219.06 222.43 602.30 461.41 727.86 647.11 762.45 992.44 577.75 773.06 823.53 858.26 612.84 648.22

R2 0.94 0.97 0.96 0.98 0.97 0.96 0.97 0.94 0.95 0.96 0.98 0.97 0.91 0.97 1.00 0.99 0.98 0.95 0.99 0.99 0.99 0.99 0.99 0.92 0.98 0.97 0.97 1.00 1.00 1.00 0.96 0.93 0.96 0.99 0.99 1.00 1.00 0.98 0.98 1.00 0.84 0.84 0.95 1.00 1.00 0.97 0.98 0.94 0.97 1.00 0.98 0.98 0.98 0.98 0.99

Table 1. Results of the 55 VRPSD instances in a low-variance scenario α = 0.05 (The best performing algorithm of the respective VRPSD instance is highlighted in Bold)

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# 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55


BKS 787.08 662.11 742.69 672.47 733.95 833.21 944.88 1146.77 1074.46 1355.80 1039.08 1622.15 1181.69 1766.50 676.09 956.29 553.16 834.92 754.22 744.23 749.96 712.92 1602.28 868.31 1039.36 1276.20 1227.90 375.28 505.01 837.67 524.61 835.26 1024.40 687.60 1164.73 723.54 241.97 819.56 1043.88 691.29 212.66 217.42 217.85 588.79 461.73 699.56 632.69 741.50 944.56 570.27 748.07 795.66 829.93 598.19 633.32

CV RP C0 867.93 726.04 792.49 707.31 801.86 963.15 1115.53 1375.16 1214.44 1576.73 1193.77 2002.01 1403.74 2126.74 753.82 1115.03 608.82 969.74 827.48 827.32 848.27 856.15 2003.64 1086.15 1277.39 1634.31 1522.01 375.49 505.01 858.91 551.21 935.03 1169.79 717.42 1271.72 769.79 249.92 939.56 1178.76 698.25 228.35 238.83 229.08 619.77 470.99 783.61 711.47 851.78 1114.75 608.59 821.07 854.79 946.00 616.10 656.29

R0 0.29 0.30 0.29 0.53 0.19 0.08 0.04 0.09 0.06 0.07 0.03 0.03 0.02 0.03 0.39 0.33 0.37 0.13 0.13 0.20 0.25 0.11 0.03 0.01 0.04 0.01 0.01 1.00 1.00 0.88 0.17 0.03 0.02 0.37 0.43 0.34 0.56 0.06 0.06 0.74 0.59 0.53 0.54 0.50 0.60 0.05 0.04 0.02 0.00 0.18 0.07 0.11 0.02 0.33 0.20

C1 815.78 693.09 775.19 701.38 762.87 858.78 990.74 1208.11 1137.69 1452.36 1105.31 1769.05 1274.47 1923.30 695.57 992.01 579.60 884.28 767.74 761.54 774.01 755.90 1686.40 929.61 1121.49 1381.21 1333.02 375.49 505.01 844.08 548.63 873.81 1087.01 700.55 1200.91 729.97 247.29 866.16 1140.95 695.92 222.37 232.49 229.08 589.41 466.92 736.71 655.67 785.64 1032.11 582.94 790.12 835.04 868.20 616.07 651.26

OBS k1 0.92 0.95 0.95 0.93 0.93 0.92 0.94 0.93 0.94 0.93 0.92 0.93 0.96 0.89 0.91 0.89 0.94 0.92 0.90 0.89 0.89 0.89 0.91 0.92 0.93 0.95 0.91 1.00 1.00 0.99 0.97 0.92 0.93 0.96 0.97 0.96 0.99 0.98 0.96 0.96 0.94 0.97 1.00 0.98 0.96 0.92 0.92 0.96 0.93 0.93 0.92 0.95 0.93 0.98 0.96

K1 5 5 7 5 6 6 7 7 10 10 10 10 10 11 5 5 5 7 6 7 7 7 9 10 11 9 11 4 4 4 6 11 16 7 7 5 4 10 8 4 3 2 2 9 5 11 9 11 18 7 11 10 11 5 6

R1 0.86 0.81 0.89 0.79 0.93 0.92 0.78 0.72 0.70 0.74 0.77 0.70 0.62 0.91 0.93 0.97 0.78 0.86 0.94 0.94 0.97 0.94 0.84 0.81 0.85 0.44 0.85 1.00 1.00 0.99 0.60 0.89 0.64 0.80 1.00 1.00 1.00 0.62 0.60 0.95 0.96 0.63 0.54 1.00 0.90 0.82 0.89 0.39 0.55 0.90 0.93 0.42 0.90 0.69 0.82

C2 812.35 689.80 766.79 691.36 753.48 850.76 988.79 1215.24 1137.90 1454.83 1096.94 1781.07 1268.44 1940.02 690.45 999.71 572.39 891.41 764.68 759.25 771.54 747.66 1711.55 926.23 1115.18 1396.91 1329.79 375.06 503.00 839.01 544.45 865.65 1091.48 691.54 1194.11 729.97 247.57 829.58 1128.28 682.55 208.97 223.00 225.86 590.00 463.23 734.01 650.47 786.72 1028.16 578.95 792.43 834.45 858.38 606.76 649.20

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k2 0.80 0.98 0.97 0.94 0.90 0.92 0.99 0.97 0.91 0.96 0.86 0.95 0.87 0.94 0.99 0.85 0.89 0.94 0.94 0.89 0.89 0.95 0.97 0.95 0.94 0.93 0.96 0.94 0.96 0.82 0.89 0.98 0.85 0.98 0.99 0.97 0.99 0.99 0.97 0.99 0.94 0.90 0.97 0.87 0.91 0.82 0.85 0.97 0.96 0.98 0.96 0.93 0.85 0.84 0.99

SAM RS K2 R2 5 1.00 6 1.00 6 0.94 6 1.00 6 1.00 7 1.00 7 1.00 8 1.00 11 1.00 10 0.96 11 1.00 10 0.98 11 1.00 11 1.00 5 0.65 5 1.00 5 1.00 7 1.00 6 0.99 7 1.00 7 1.00 8 1.00 9 0.93 10 0.98 11 1.00 10 1.00 11 1.00 5 1.00 4 1.00 5 1.00 6 1.00 11 0.85 18 1.00 7 1.00 7 0.98 5 1.00 5 1.00 9 1.00 8 1.00 5 1.00 3 1.00 3 1.00 3 1.00 9 1.00 6 0.96 12 0.97 10 1.00 12 1.00 18 1.00 7 1.00 11 0.97 11 1.00 12 0.97 5 1.00 6 0.99

C2 814.64 690.90 772.59 695.86 759.50 858.54 991.99 1220.51 1145.68 1466.67 1105.48 1794.66 1287.42 1967.24 691.87 1004.72 573.01 903.97 766.08 760.31 774.09 752.47 1725.40 933.57 1121.42 1416.50 1334.93 375.08 503.50 840.13 546.56 880.70 1106.59 697.06 1208.27 729.98 251.57 845.72 1149.91 689.62 209.20 223.17 226.00 591.01 463.44 745.36 655.37 790.58 1035.99 583.85 796.17 844.83 873.34 613.93 652.85

R2 0.97 0.93 0.89 0.91 0.95 0.97 0.97 0.90 0.93 0.90 0.97 0.90 0.95 0.92 0.85 0.93 0.99 0.99 0.95 0.92 0.94 0.93 0.97 1.00 0.97 0.94 0.93 1.00 1.00 0.99 0.95 0.93 0.94 0.95 0.99 1.00 1.00 0.93 0.87 0.97 0.91 0.91 0.93 1.00 0.98 0.98 0.97 0.94 0.98 0.96 0.95 0.91 0.96 1.00 0.96

Table 2. Results of the 55 VRPSD instances in a medium-variance scenario α = 0.25 (The best performing algorithm of the respective VRPSD instance is highlighted in Bold)

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# 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

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BKS 787.08 662.11 742.69 672.47 733.95 833.21 944.88 1146.77 1074.46 1355.80 1039.08 1622.15 1181.69 1766.50 676.09 956.29 553.16 834.92 754.22 744.23 749.96 712.92 1602.28 868.31 1039.36 1276.20 1227.90 375.28 505.01 837.67 524.61 835.26 1024.40 687.60 1164.73 723.54 241.97 819.56 1043.88 691.29 212.66 217.42 217.85 588.79 461.73 699.56 632.69 741.50 944.56 570.27 748.07 795.66 829.93 598.19 633.32

CV RP C0 889.99 751.24 814.24 732.29 820.56 969.20 1142.27 1411.09 1259.62 1631.51 1223.96 2047.45 1424.04 2207.69 775.84 1145.13 655.32 1008.46 847.67 853.48 881.49 886.93 2059.75 1103.81 1287.63 1660.42 1582.81 378.28 505.01 883.62 561.25 970.33 1221.11 743.71 1280.75 772.90 252.15 943.69 1230.53 703.87 236.63 244.14 232.30 620.24 477.88 806.07 733.49 876.30 1156.71 625.80 869.55 904.90 959.11 622.71 660.43

R0 0.23 0.20 0.18 0.36 0.13 0.07 0.03 0.06 0.04 0.04 0.01 0.01 0.01 0.02 0.32 0.22 0.27 0.09 0.09 0.14 0.16 0.08 0.02 0.01 0.02 0.01 0.01 0.93 1.00 0.73 0.12 0.01 0.00 0.09 0.40 0.30 0.45 0.06 0.03 0.53 0.46 0.44 0.47 0.49 0.47 0.02 0.02 0.01 0.00 0.09 0.02 0.05 0.01 0.18 0.12

C1 853.63 715.18 803.96 719.09 777.18 890.16 1026.08 1279.20 1189.56 1539.37 1154.93 1867.97 1321.27 1994.10 713.90 1036.03 602.72 929.01 790.20 790.54 794.33 774.66 1793.16 968.50 1170.91 1453.30 1383.07 378.25 505.01 844.13 555.26 894.96 1127.04 704.48 1208.99 731.01 247.51 922.16 1218.10 699.87 227.41 239.34 232.30 591.05 474.21 763.24 675.00 822.43 1087.04 593.15 809.56 861.92 889.26 621.75 659.67

OBS k1 0.92 0.89 0.95 0.85 0.90 0.93 0.95 0.93 0.87 0.94 0.93 0.89 0.91 0.89 0.89 0.91 0.88 0.88 0.92 0.89 0.89 0.88 0.91 0.84 0.88 0.90 0.86 0.99 0.88 0.98 0.94 0.92 0.89 0.93 0.97 0.96 0.99 0.96 0.98 0.96 0.95 0.99 1.00 0.98 0.95 0.91 0.93 0.96 0.87 0.92 0.92 0.93 0.93 0.89 0.90

K1 5 6 7 5 6 6 7 7 10 10 10 10 10 11 5 5 6 7 6 7 7 8 9 11 11 10 12 4 4 4 6 11 16 7 7 5 4 10 8 4 3 2 2 9 5 11 9 11 19 7 11 11 11 5 6

R1 0.55 0.77 0.58 0.91 0.86 0.60 0.44 0.35 0.71 0.30 0.34 0.47 0.43 0.53 0.86 0.66 0.78 0.86 0.61 0.61 0.66 0.80 0.38 0.85 0.63 0.46 0.69 0.93 1.00 1.00 0.73 0.44 0.43 0.79 0.91 0.98 0.99 0.30 0.14 0.78 0.85 0.48 0.47 0.97 0.64 0.44 0.47 0.14 0.52 0.79 0.54 0.33 0.47 0.99 0.96

C2 850.47 706.32 791.32 707.05 770.99 883.49 1022.54 1270.77 1186.39 1545.51 1146.64 1865.45 1336.18 2031.81 709.74 1040.93 591.86 925.95 781.24 788.15 794.55 772.63 1829.80 961.11 1164.81 1487.98 1379.10 376.10 503.00 840.80 551.13 900.90 1136.53 698.73 1216.47 730.80 247.51 854.43 1204.51 686.26 211.31 225.25 228.94 591.39 470.29 762.67 671.29 819.07 1076.66 589.22 815.36 864.46 884.89 612.04 654.28

k2 0.81 0.95 0.94 0.82 0.93 0.91 0.97 0.91 0.93 0.91 0.89 0.91 0.92 0.92 0.81 0.90 0.95 0.98 0.94 0.96 0.98 0.92 0.87 0.94 0.88 0.97 0.88 0.90 0.93 0.81 0.91 0.98 0.96 0.96 0.98 0.91 0.91 0.88 0.86 0.99 0.95 0.83 0.86 0.97 0.93 0.90 0.98 1.00 0.91 0.94 0.95 0.90 0.97 0.96 0.99

SAM RS K2 R2 5 0.96 5 0.88 6 0.86 5 1.00 6 1.00 7 1.00 7 1.00 8 1.00 10 1.00 10 1.00 11 1.00 10 0.87 12 1.00 11 1.00 5 0.80 5 1.00 5 0.87 8 1.00 6 1.00 8 1.00 7 1.00 8 0.97 10 1.00 11 1.00 12 1.00 11 1.00 11 0.91 5 1.00 4 1.00 5 1.00 6 0.95 11 1.00 16 0.92 7 1.00 8 1.00 5 1.00 5 1.00 9 0.94 8 1.00 4 0.94 3 1.00 3 1.00 3 1.00 9 1.00 5 1.00 11 0.93 9 0.84 11 1.00 19 1.00 8 1.00 11 1.00 12 1.00 11 0.91 5 1.00 6 1.00

C2 852.24 708.92 800.28 711.89 773.33 890.50 1027.54 1299.60 1201.57 1554.61 1153.28 1895.08 1341.17 2050.43 716.91 1048.46 596.15 929.06 783.51 791.95 798.92 778.14 1849.53 965.55 1178.39 1505.38 1393.62 376.18 504.00 842.43 554.54 912.63 1153.35 707.16 1226.02 730.92 249.67 858.66 1232.85 694.14 211.52 225.47 229.17 591.57 475.38 774.63 677.02 822.86 1087.28 599.56 824.16 874.14 904.10 617.80 658.00

R2 0.81 0.79 0.80 0.97 0.95 0.95 0.91 0.93 0.93 0.86 0.87 0.88 0.87 0.85 0.80 0.93 0.84 0.95 0.91 0.96 0.83 0.81 0.86 0.96 0.87 0.94 0.84 0.99 1.00 0.96 0.94 0.89 0.93 0.97 0.96 0.98 1.00 0.92 0.84 0.92 0.83 0.77 0.78 0.99 0.93 0.92 0.85 0.89 0.91 0.89 0.89 0.95 0.85 0.86 0.93

Table 3. Results of the 55 VRPSD instances in a high-variance scenario α = 0.75 (The best performing algorithm of the respective VRPSD instance is highlighted in Bold)

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The route reliability level refers to the probability (i.e., R1 and R2 ) that solution routes do not suffer from failure (i.e., capacity constraint violation) in the actual distribution phase. This is an important measure besides the expected cost, since the managerial control over frequency of route failure and expected cost penalty is dependent on the route reliability level. Monte Carlo simulation for computation of expected cost and reliability level is configured in the study at 10,000 trials. Three different statistical customer demand distributions are considered, with the variance level α configured to 0.05, 0.25, and 0.75, which simulates low, medium and high demand variances, respectively. 16 SAMRS-avg SAMRS-best OBS-best CVRP-best

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Figure 7. Gaps for solution at different variance levels (CVRP-best ≡ Gap0 ; OBS-best ≡ Gap1 ; SAMRS-best ≡ Gap2 ; SAMRS-avg ≡ Gap2 )

1 SAMRS-avg SAMRS-best OBS-best CVRP-best

0.9 0.8 0.7 Reliability

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0.6 0.5 0.4 0.3 0.2 0.1

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Figure 8. Reliability indices for solution at different variance levels (CVRP-best ≡ R0 ; OBS-best ≡ R1 ; SAMRS-best ≡ R2 ; SAMRS-avg ≡ R2 )

4.2.1.

Solution Quality

On solution quality, the relative performances of the algorithms in terms of the percentage gap of C0 , C1 , and C2 are depicted in Fig. 7. As observed from the figure, SAM RS consistently produces the lowest gap Gap2 when compared to Gap0 and Gap1 attained by CV RP and OBS, respectively. This suggests that SAM RS obtained the best solution with lowest expected cost for different customer demand uncertainty levels, indicating that its superiority, even at highvariance of customer demands.

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It is notable from the values of K1 and K2 in Tables 1, 2, 3 that the best solution obtained by OBS and SAM RS differ in terms of the number of routes, i.e., the best found solutions obtained by SAM RS have larger number of routes on approximately half of the problem instances considered. Typically, a larger number of routes than those of OBS, although indicating higher reliability, may also lead to higher expected costs. The results presented in Tables 1, 2, 3, however showed otherwise. Even for problem instances in which the best found solutions obtained by SAM RS have larger number of routes, the corresponding expected cost of the SAM RS solution are observed to remain lower than those arrived by OBS. Notably, the superior solution quality obtained by SAM RS in terms of expected cost, is contributed by the enhanced search performance of SAM RS, particularly a result of the adaptive search intensification brought about by the self-adaptive memeplex based individual learning provided in the SAM RS. To further analyse the performances of SAM RS in terms of search efficiency and solution quality, the results obtained from Tables 1, 2, 3 in terms of the expected cost and solution reliability level of best found solutions and average converged solutions are pitted against the best solutions obtained by OBS and CV RP at different variance levels in customer demands, as summarized in Figs. 7 and 8. 4.2.2.

Solution Reliability

On solution reliability, the reliability indices of the best found solutions at different variance levels are depicted in Fig. 8. The figure indicates that in terms of solution reliability, the best found solution obtained by SAM RS is consistently superior since it exhibits a much lower probability of route failure for the different levels of variance in customer demands. It is worth noting that even at high variance of customer demands (α = 0.75), the best solution obtained by SAM RS remains to maintain a high average reliability level of R2 = 0.975, while the reliability level of the best found solution obtained by OBS drops to R1 = 0.627. The higher reliability levels obtained over OBS, can also be partially attributed to the way the safety stock level k is chosen. In OBS, the optimal solution, in terms of expected cost and reliability, is picked from alternative solutions obtained via a set of pre-defined values of k. This restricts the possible choice of k value, and thus limits the possibility of finding improved solution for which the k value exists outside the pre-defined set. In contrast, SAM RS models k as a continuous variable that is self-adapted along with the evolutionary search so as to enhance convergence to optimal solution. 4.2.3.

Search Robustness

To investigate the robustness of the proposed SAM RS, we also compute the C2 and R2 for each VRPSD instance. C2 , R2 denote the sampled means on expected cost and reliability of the solutions obtained by SAM RS across N R = 30 independent runs, respectively. The results in terms of the C2 and R2 are then pitted against those obtained in OBS and CV RP , at different uncertainty levels of customer demands. The relative performances of the algorithms in terms of percentage gaps of C0 , C1 , C2 are depicted in Fig. 7. As illustrated by the histogram of average gaps, there is virtually no significant difference in values between the sample mean C2 obtained by SAM RS and the best solution C1 obtained by OBS, for the different uncertainty levels. This thus verifies the robustness of SAM RS in terms of solution quality. The relative performances of the algorithms in terms of R0 , R1 , R2 are depicted in Fig. 8. As illustrated by the histogram of average gaps, while CV RP and OBS drops significantly in reliability as the variance level increase, SAM RS has been able to maintain highly reliable solutions across all the independent runs at low probability of route failure for the different variance levels of customer demands. Even at high variance of customer demands (α = 0.75), the solutions obtained by SAM RS maintain high average reliability level of R2 = 0.899, while the reliability level of the best solution obtained by OBS slides to R1 = 0.627.

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As illustrated in Figs. 7 and 8, SAM RS also exhibits high robustness in terms of the expected cost and reliability on the best solutions over 30 independent runs, with an average expected cost that matches well the expected cost of the best solution obtained by OBS, and an average reliability that is superior to that of the best solution obtained by OBS. 5.

Conclusion

In this paper, a novel Self-Adaptive Memeplex Robust Search(SAM RS) has been introduced. SAM RS is composed of a self-adaptive memeplex individual learning, a robust solution search scheme , and a gene-meme co-evolution model. The memeplex representation bridges the two adaptive mechanisms in SAM RS, namely the meme coordination within the self-adaptive memeplex individual learning and the meme evolution within the gene-meme co-evolution model, thus efficiently managing lifetime learning phase for robust solution search. The robust solution search scheme in SAM RS directs the MA search towards robust solutions via the Baldwinian mode of inheritance by inducing perturbations into the phenotypic features of the solution, and also reduces the computational complexity involved in the fitness evaluation by serving as an approximation to the computationally intensive Monte Carlo simulation via its implicit averaging effect. Empirical results obtained on the VRPSD benchmarks demonstrated that SAM RS produced higher quality solutions in terms of not only the expected cost but also the solution reliability. In addition, SAM RS achieved robustness in algorithm performance under different customer demand variance level. In particular, while the compared algorithm OBS drops its solution reliability to 0.627 at high demand variance level, solutions obtained by SAM RS maintain a high average reliability of around 0.9 over the 30 independent runs. In summary, the SAM RS is shown to be an efficable means of generating high-quality robust and reliable solutions in safety stock based VRPSD.

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Xianshun Chen received his Bachelors degree in Electrical and Electronics Engineering from Nanyang Technological University. He is currently pursuing his doctorate degree in the field of competent Memetic Algorithms at School of Computer Engineering, Nanyang Technological University. His research interests include the design and development of competent and innovative memetic computing frameworks and various soft computing techniques as well as their applications.

Liang Feng received the B.Eng. degree in School of Telecommunication and Information Engineering from Nanyang Nanjing University of Posts and Telecommunications, China, in 2009. Currently, he is working toward the Ph.D. degree in computer engineering at the Center for Computational Intelligence, School of Computer Engineering, NTU. His primary research interests include Evolutionary Computation, Memetic Computing and Data mining, etc.

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Yew-Soon Ong received the BS and MS degrees in electrical and electronics engineering from Nanyang Technological University (NTU), Singapore, in 1998 and 1999, respectively. He completed the PhD degree on artificial intelligence in complex design from the Computational Engineering and Design Center, University of Southampton, UK in 2002. He is currently an Associate Professor and Director of the Center for Computational Intelligence at the School of Computer Engineering, NTU. Dr. Ong is the founding Technical Editor-in-Chief of Memetic Computing Journal, Chief Editor of the Springer book series on studies in adaptation, learning, and optimization, Associate Editor of IEEE Computational Intelligence Magazine, the IEEE Transactions on Systems, Man and Cybernetics - Part B and many others. He is currently also Chair of the IEEE Computational Intelligence Society Emergent Technology Technical Committee and has served as Guest Editors of several journals. His research interest in computational intelligence spans across Memetic Computing, Evolutionary Design, Machine Learning, Agent-based systems and Cloud computing.

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