Comparisons

Table 4.6compares solution values from Table 4.4, generated using evolutionary learning, against the same problem instances (with 3 depots or more) obtained using distributed game theoretic approach as per Table4.5. The proposed evolutionary procedure provides

Table 4.6: Passive learning vs. Active negotiation for MDVRP solution generation Problem [nodes/depots] (Max.Veh./Cap.) zh

∣zgame[gap=1-zgame/zh] P01 [50/4] (4 per depot/80) 608 ∣ 637 [-0.047] P02 [50/4] (2 per depot/160) 489 ∣ 514 [-0.051] P03 [75/5] (3 per depot/140) 663 ∣ 703 [-0.060] P06 [100/3] (6 per depot/100) 927 ∣ 1061 [-0.144] P07 [100/4] (4 per depot/100) 949 ∣ 1090 [-0.148]

better solutions. The quality of near-optimal solutions obtained from negotiation approach is restricted by the satisfaction of the game equilibrium. Once the game equilibrium is reached it is dicult to improve such a solution in the negotiation approach.

Figure 4.5: Comparative Study of Multi-Depot Vehicle Routing Solution Approaches

Figure 4.5 depicts a comparative study of various aforementioned solution nding approaches as discussed in relation to multi-depot vehicle routing problems. On the x-axis, we project time in seconds while, on the y axis, we evaluate solution cost for the modied- E016-03m problem of the case study. The comparison clearly shows that the distributed learning technique presents comparable solution with centralized heuristic/meta-heuristic solution search. However, in collaborative or cooperative setup, the distribution mechanism and the solution search together take longer time than centralized solution generation. As depicted in the gure, split-delivery is proven advantageous for this MDVRP instance since it lowers routing cost compared to the centralized solution approach.

4.5.3 Advantages and Limitations

We summarize next the advantages and limitations of two proposed solution generation approaches for MDVRP in distributed setting. First of all, both approaches oer task de- composition by splitting the problem of multi-depot vehicle routing in distributed setting. Such a decomposition allows distributed decision makers to handle sub-problems locally in a decentralized setup, to use their own input and search parameters (without sharing with others) and to enforce organizational policies. Thus, participants have a larger control in decision making as needed in various organizational setups. Furthermore, as the task de- composition divides the original problem instance into multiple smaller problem instances, each participant can locally produce high quality near-optimal solutions using less memory and computation time.

However, overall solution generation for the MDVRP instances takes longer time with the increase in number of participants since the result synthesis from divided sub-problems and proper task decomposition for the original problem turn more and more complex. Our benchmark results indicate that the solution quality for the MDVRP instances is less com- petitive for both approaches compared to the near-optimal solution found in centralized setting. This mainly relates to the diculty of nding the most appropriate partitioning of the customer nodes among participating decision makers. Moreover, these proposed dis- tributed approaches require further improvement to address shared delivery of commodities using vehicles from dierent depots.

4.6 Summary

In this chapter, we rst discussed a model of a chance constraint optimization problem to address MDVRP in distributed settings. We have presented two innovative distributed ap- proaches. First, the multi-round evolutionary learning (passive learning) approach enables collaborative decision makers to search near-optimal solutions for relevant size problem in- stances by locally computing sub-problems and interacting with other participants without explicitly collecting or sharing eet/capacity information. Second, the active negotiation approach leverages a game theoretic interaction among cooperative participants where

mechanisms are designed to generate distributed solution search for MDVRP instances. The proposed approach assigns customers to participating depots over transport networks with small-world characteristics using a VCG strategyproof mechanism while aiming to minimize total routing cost. The case studies and benchmark results for both approaches show that near-optimal solutions can be found in these distributed settings despite the lack of all information at each participant's side. In this regard, the learning approach nds better quality near-optimal solution. In future, for passive learning approach, it is possible to investigate more eective initial assignment policies for quicker convergence of the solution search.

Chapter 5

Collaborative Monitor Deployment

Problem

In this chapter, we investigate centralized and distributed models and ap- proaches to determine (near) optimal deployment locations of execution mon- itors over a well-tracked transport network under a xed budget. The goal of the optimization is minimizing the weighted average energy consumption for data communication between the sensors and the monitors. We illustrate a col- laboration strategy of monitor deployment when the total deployment budget is unknown and split among multiple decision makers. We also determine a satisfactory (fair) sharing of monitor deployment cost for every participant.

5.1 Introduction

Typically, SCN involves the ow of products from producers/distributors to customers. Such a network consists of physical locations and traversal paths among these locations. Formally, these locations can be represented as the vertices of a graph while the directed edges between vertices may stand for the traversal paths (arcs) among locations. Monitors can be deployed on the vertices [75] or the edges [104] of a network. In SCN, monitor deployment between two locations (e.g. road, railway, etc.) is costly from security and maintenance perspectives. Thus, this research and development eort focuses on deploying

monitors over a subset of vertices. In this regard, the focus of this chapter can be stated as follows:

ˆ A mathematical model for monitor deployment with multiple decision makers; ˆ Centralized and distributed approaches to minimize energy consumption; ˆ Conduct and analyze a case study and generate new benchmark results.

The other contribution of this chapter relates to an automated collaborative negotiation mechanism toward near-optimal monitor deployment with individual budgets. Also, a heuristic is proposed to locally compute solutions under the budget constraint. The optimal selection of monitor locations in SCN, where the total budget is split among participants, is a distributed problem derived from classical facility location [75] and p-median problems [141] which have N P-Hard computation complexity. This requires heuristic or meta- heuristic techniques to eciently solve large problems [67, 75, 141]. These problems are often addressed with known budget which simplies the formulation. If the budget is split among the participants, the formulation requires coupling through a joint chance constraint [174] to limit the probability of constraint violation by the participants.

The rest of the chapter is organized as follows. Section 5.2 describes the problem, its assumptions and models. First, a base model is formulated using a known budget constraint. Second, this model is extended for the distributed case where total budget is split among participants. Section5.3presents the proposed approach. An exact (optimal) solution algorithm is discussed followed by a faster heuristic technique. Afterward, a distributed approach is detailed whereby participants locally run the heuristic technique and collaborate toward a near-optimal solution. A deployment cost sharing mechanism is also proposed in this regard. Section 5.4 presents a case study. Section 5.5 reports on the results obtained for some Problem Instances (PI). Useful insights are shared on the obtained results and the heuristic performance. Section5.6draws concluding remarks and hints on the future work.

5.2 Problem Description and Modeling

Let G = (V, E) be a complete directed graph representing an SCN. Vertices are divided into monitor nodes (or monitor) and relay nodes (or node). Appropriate equipment can be deployed on a monitor i at a deployment cost ci, to collect task execution data. The

execution information is produced by agents (e.g. vehicles) who visit a subset of vertices in sequence (also called route) through a connecting paths (e.g. roads). A relay node sends collected information to single monitor. Each edge ⟨i, j⟩ ∈ E, is associated to a pair of integers: δij and wij. The proposed problem refers to deploying monitors on a subset

of vertices such that weighted average energy consumption in sending execution data is minimum. In the process of optimal solution generation, P participants (Decision Makers) will collaboratively determine monitor locations by individually allocating own budget Cp

on a subset of vertices. Every solution should respect the following: ˆ Each vertex is either a monitor or a relay node;

ˆ Deployment cost on each selected vertex is split among a subset of decision makers; ˆ Every relay node sends data to the monitor incurring least energy consumption. 5.2.2 Assumptions

Task planning (e.g. product delivery) and communication between any two vertices are considered independent of other vertices since G is a complete directed graph. Each vertex is assumed as a source of at least one execution path. Thus, wij is a positive integer.

A number of factors (e.g. communication radius, obstacles, electromagnetic interference, attenuation, environmental situation) aects the energy consumption value δij between

two locations. Young et al. [253] characterized these eects on radio signal strength over a log normal shadowing model. In contrast, this eort attempts to minimize the weighted average energy consumption based on predetermined values for δij specic to

every arc that serves as an input to the problem instance. We assume that each relay node always receives execution information from all the agents moving from it to their

next destinations. However, in this context, we ignore the energy consumption in data communication for each agent to its last departing vertex. All deployed monitors are considered to have innite capacity to receive execution data. No participant knows the exact budget of others. However, it is assumed that a feasible solution always exists and the total budget is sucient to deploy at least one monitor.

As discussed in Section2.1.3, in the model formulation, two types of decision variables are employed: control variables and state variables [174]. The monitor deployment problem with a known budget can be represented through a system of linear equations. The values assigned to the control variables (respecting the system of equations) represent a solution. However, monitor deployment with individual budgets is described as a dynamic system with multiple states where each state represents a system of equations with its own control variables. In this context, state variables describe the mathematical state of that dynamic system. A subsequent set of state variables typically depends on its previous corresponding set.