2 5 Reference Model for Competition Routing Problems
V- A (OM)-A
V-V OM-V
Figure 2.2: Entity-Relationship Model for CRP
(RM-CRP) (relationships) D M -D M Decision-maker-decision-maker (addresses) / (vehicles) / (decision makers)/ (relationships) / (objectives) (A-A restrictions) (V-A restrictions) (V- (DM)-A restrictions) (V -V restrictions) (DM-A restrictions) (DM-V restrictions) (DM-DM restrictions) OM-OM (2.7) (2.8)
V-(DM)-A Vehicle-address modified by decision maker, in which we may specify the relation ship between a subset of vehicles and a subset of addresses for a particular subset of decision makers.
We replace Sentences (2.2) and (2.5) by Sentences (2.7) and (2.8) , in which we have renamed (problem characteristics) with (relationships) . Note that there are now five fields to the
classification instead of four. Each of these five fields is called a component of the RM-CRP.
To put some flesh on the bones of this model, we must consider the fundamental characteristics of each decision maker
(i) . Interaction with other decision makers.
(ii) . Individual objectives.
(iii). Interactions with addresses and vehicles (or teams) .
2.5. Reference Model for Competition Routing Problems 31
2.5.3 Decision Maker Component
We can now expand the components corresponding to the decision makers.
2.5.3.1 Decision Maker Attributes
Number. How many decision makers are there? Any decision makers who are completely de pendent actually constitute a single decision maker.
Computational Resource. We must distinguish between on-line (real time or near to real time) computation and off-line decision making, subject to some kind of computational budget. This is where each player is allocated a computational budget, e.g., an amount of CPU time proportional to the duration of the game.
Observational Ability. Perfect observation may be unrealistic in the real world since the ter rain will be either partially unsighted, like an inner city street network, or much too ex pensive to obtain, e.g., requiring satellite imagery.
2.5.3.2 Decision Maker-Decision Maker Relationships
Communication. Is it possible communicate between decision makers? Is this communication secure between parties or may others eavesdrop on conversations? What knowledge held by decision makers is private or public knowledge?
Coalitions. We assume that decision makers are independent. However, with or without com munication, players may enter into permanent or temporary coalitions. This involves non cooperative game theory considerations. What negotiations are possible and what side payments may be exchanged? How will contracts be enforced when there may be oppor tunity for a double-cross? What restrictions are placed on coalitions, e.g., permanence, binding rules, duration, some players not being permitted to cooperate, some possibly having to cooperate for the first part of the competition?
Piracy and Pursuit. Is it possible to steal from opponents if associated vehicle locations coin cide? Is there any element of incentive to pursue or evade an opponent?
Simultaneous or Sequential Decisions. Is the underlying game a simultaneous or sequential move game, i.e., do all decision makers determine their actions concurrently or one after another?
Supply. There are problems in which there is some degree of autonomy and some degree of one
(or more) DM having a great effect upon the outcome. For example, consider a problem in which there may be many salesman competing to deliver to a set of customers but in which there is also some sort of roving depot supplier who is also an independent decision maker with his own objectives to sell wholesale to the salesmen.
2.5.3.3 Decision Maker-Vehicle Relationships
Teams. A team is a fixed association of a number of vehicles to a single decision maker. We distinguish between a team and a coalition where a commitment to a team is binding and permanent (corresponds to a single DM) , but a coalition is transient and not binding (corresponds to a DM for each member). We assume that there is at least one vehicle associated with a DM and that an objective is realised by some kind of vehicle routing activity.
Mercenaries. It may be possible to hire vehicles from an outside pool of vehicles, although if there is any competition for a scarce number of hire vehicles, then the rental agency would need to be modelled as a decision maker.
Communications. Can vehicles communicate with their associated decision maker, e.g., a cen tral dispatcher? Does this occur frequently, infrequently or on demand? Are communica tions secure?
2.5.3.4 Decision Maker-Address Relationships and Decision Maker modified
Vehicle-Address Relationships
Unit Task Completion. What are the rules governing servicing a customer, delivery to or collection from a customer or general execution of some task? This is essential since the task is the basic unit of competition and we must have a well-defined understanding of when any rewards or penalties accrue.
2.5.4 Objectives Component
The next component to consider is that of objectives.
2.5.4.1 Public and Private Objectives
A public objective is common knowledge between all players. A private objective is known only by the player who holds that objective. No assumption is made about the private objective of the opponent. However if an opponent's objective is public then we assume the opponent is rational with respect to that public objective. Note also that a public objective may include some kind of overall constraint such as the time limit since these may be hard or soft. A player's
objectives may be conflicting, partially conflicting or cooperative. Of these, the first two
may evolve direct competition, incidental competition or cooperation for mutual benefit.
2.5.4.2 Types of Objective
There are three fundamental components to a player's private objective:
1 . value collected (by each player)
2. constraints on the overall time deadline for collection of prizes
2.5. Reference Model for Competition Routing Problems 33
Several pure private objectives are possible, assuming that the objective involves prize values:
Win. Claim more in sum of prize values claimed than any opponent. This is compatible with
the traditional notion of a game with a winner and the concept of trying to dominate the opponent.
Max. Maximize the sum of prize values claimed, Le., gaining as much value in prizes as possible but not caring about how much the opponents may claim. This is compatible with the economic interpretation in a market place of trying to improve market share or increase your own income.
Restrict . Minimise the sum of prize values claimed by the opponents.
Goal. Achieve some goal value in sum of prize values claimed in minimal time. However, what
happens once that goal is actually achieved, or even before that, when it is guaranteed that is can be achieved by the player providing a verifiable path? Do we continue on until the other player does or cannot achieve his goal value, and in case we do, then what strategy does the first player play? What bonuses may be available for exceeding a goal or from preventing the opponent from achieving his goal?
Spread. Maximize the magnitude of a win or minimise the magnitude of a loss by maximizing
the sum of prizes claimed less the sum of prize claimed by the most successful opponent.
Harvest. Maximize the harvest rate, i.e., the prize value claimed per unit time interval of har vesting, over a minimum/maximum time period.
Iterated. Maximize the best, worst or average sum of prize values claimed over a number of iterations of the problem instance.
Although these are the basic pure objectives, many combinations could be constructed to suit the application. In particular is the idea of achieving certain private or public goals throughout the duration of the game that may involve, e.g., cooperation, harvesting and winning at different stages, similar to a cycling points race.
Other useful objectives include minimising the number of vehicles required to satisfy all the constraints and time dependent objectives, such as the Travelling Repairman Problem (TRP) in which we are concerned with the time of visitation to each and every customer, that is
their service time (Lucena [149]). This may profitably be combined with a prize value as in the MCPTDR of Brideau and Cavalier [28] (see Section 2.1.5).
Note. An important property of objectives for a CRP is whether the objective is memoryless.
For example, the MAX objective remains as trying to claim as much in prize value as possible
during the remaining time available. However, the WIN objective varies in consequence depending
on how well a player does with respect to its opponent. 2.5.4.3 Constraints and Penalties
Suppose we have a time duration for a game that cannot be exceeded. We associate with each prize i E V a penalty 7ri � O. If prize i is unclaimed at the end of the game then both players have
where T; C; (objective) (function) P; ( (vehicle constraints» ) Cj Pj ( (address constraints») Vj o V (operator) (function) T; V C; V P; « (vehicle constraints» ) V CiV Pi ( (address constraints» ) [route duration] [vehicle costs] [vehicle penalty] [address costs] [address penalty] [path prize value]
(2.9) (2. 10)
7r; deducted from their score. Hence objective maximizing players may need to truly cooperate in ensuring that they can claim the badly penalising prizes between them so that they can both be better off.
Note that in a TSSP, there rewards contribute to the subtour value but penalties contribute to the subtour costs. For the CRP objective component, Sentences (2.9)-(2.10) summarise the major reward, cost and penalty contributors.
2.5.5 Address and Vehicle Components
The last components to consider are those corresponding to addresses and vehicles. The major of these attributes and relationships are inherited from RM-VRSP. Hence we consider those additions arising from the presence of competition.
2.5.5.1 Address Attributes
What are the attributes of the (address) component?
Tasks. To keep some of combinatorial structure to the CRP, we adopt the task as the basic unit of competition and assume that at least some tasks are contestable. A task may be a service, pickup, delivery or pickup-and-delivery. Mosheiov [163] considers the Travelling Salesman Problem with Pick-up and Delivery (TSPD) in which delivery customers
are served by delivery of goods from a central warehouse and pick-up customers need to deliver goods from their locations to the warehouse. General pickup and delivery problems (e.g. dial-a-ride) involve a unit task of picking up a load from some point and completing a delivery of that load. However, we allow that not every task need be undertaken, but rather permit the tactical selection of a subset of the tasks.
Prizes. We also assume that there must be some incentive for servicing a particular task, such as a reward, or some disincentive for not servicing a particular task, such as a penalty. Prize values may be time dependent and prizes may require some nonzero time to service.
2.5. Reference Model for Competition Routing Problems 35
Environment. The environment is the structure in which addresses are embedded, e.g. , the Euclidean plane, a graph or network, street map, Geographic Information Systems (GIS) coverage, Digital Terrain Model (DTM). The environment may model obstacles or a physi cal terrain and may either be completely known in advance or have some element of explo ration required, hence requiring a balance between accumulating prizes, exploring terrain and observation of opponent or prizes, i.e., between gaming tactics and exploration tactics. Also the environment may have dynamic elements such as wind speed and direction. Selection and Covering. Not all prizes or tasks need to be selected, as in the TSSP. In addition
we may consider allocation of off-tour prizes to on-tour prizes as in the SVRAP of Beasley and Nascimento [15] (see Section 2.1).
Calling Process. Tasks or prizes may become available for selection, i.e., arrival, according to some stochastic arrival process of calls from customers, as in the DTSP and DVRSP.
2.5.5.2 Address-Address Constraints
What are the relationships between (addresses)? Location-routing problems, like those de
scribed by Laporte [133], can be modelled as relationships between tiers of addresses such that a load must be delivered to a depot address before it can be picked up from the depot address and delivered to the customer address.
2.5.5.3 Vehicle Attributes
What are the attributes of the (vehicle) component?
Movement. Relative speed, possibly a function of the location on the terrain. Energy con strained by vehicle load and surface traversal, i.e., a tiredness budget.
Capacitated. If prizes are considered to have some weight or volume then capacity of vehicles becomes an important consideration. Also, vehicles may be able to drop off prizes collected either at some depot or just drop them somewhere in the environment, henceforth able to be claimed by another vehicle, either from the same team, coalition or an opponent. If a vehicle must return to the depot for some reason then we might have depots; for example, if the vehicle has a capacity and the prizes have some mass, then the vehicle may first collect prizes and then return to the depot to deposit them, and then continue prize collection. We may wish to exchange prizes at a location if we can get a better one in exchange or miss out on it, especially if a vehicle can only go out once (the knapsack version).
Redistribution. On-route redistribution of payload. Roving depots. On-route redistribution of payload may be permitted, especially between vehicles in a team, or between members of a coalition. There is no "currency" so that negotiation of giving away or exchanging a prize is between players.
2.5. 5.4 Vehicle-Vehicle Constraints
2.5.5.5 Vehicle-Address Constraints
What are the relationships between (vehicles) and (addresses)? What was previously (type
of strategy) in the classification scheme of Desrochers, Lenstra and Savelsbergh [54] is now
incorporated in (V-A restrictions). Competition related constraints would be included in (V (DM)-A restriction) of Section 2.5.3 since they must necessarily involve a decision maker. Non-competition related constraints would be inherited from the RM-VRSP.
2.5.6 Illustrative Examples
Having described the components of the RM-CRP, we can now develop a few complete illustrative examples. Some familiar problems can be described succinctly by the new classification scheme: TSP is 1/1/1/ /T, OP is 1/1/1, dur/ /MAX, and TOP is 1/I/m, dur/ /MAX.
2.5.6.1 Household Milk Delivery
Up until a few years ago, the delivery of milk in glass bottles to homes was common in New Zealand. However, many people now purchase their milk from the supermarket in plastic con tainers, although some people still support the milk vendor. Customers would place the exact number of required milk bottles at the gate of their home, together with prepaid plastic tokens. Milk vendors usually employed teenagers, pushing milk trolleys loaded with full bottles, to service the customers. Historically, milk vendors have cooperated with each other in defining districts within which each vendor will operate. Presumably this is because of the low economic margins within which these businesses operate and because demand is low compared with potential supply.
If however, demand was high in comparison with supply, a CRP would be an appropriate formulation of the problem since customers are indifferent between suppliers. Vehicles correspond to the milk trolley and milk truck. Decision makers are the vendors. Vehicles are capacitated but it is also possible to transfer (or replenish) bottles between vehicles.
2.5.6.2 Bicycle Couriers
Bicycle couriers are common in the central business district (CBD) of many large cities. These couriers are directed to pickup and deliver small packages among offices by the dispatcher in the courier company. Many small courier businesses operate within the CBD. At present, customers requiring transit of packages contact the courier dispatcher of a particular business who arranges
for a bicycle courier to pickup the package. Tight requirements of delivery service are often required by customers.
2.5.6.3 Terrain Navigation
In the presence of an unknown terrain or obstacles even a single prize TSSP may be non-trivial. Taylor [197] considered a variation of the OP that employed a digital elevation model of a phys ical terrain. Simulations involved both exploring the terrain to discover the prize locations and efficiently sequencing those prizes whose locations were observed from being in the viewshed of some previous location.
2.5. Reference Model for Competition Routing Problems 37
A generalization of this problem to a CRP involves a terrain and a number of competitors, much like a true Orienteering event, except that prizes (control points) may only be claimed by one orienteer. An autonomous vehicle must construct a map of the terrain on which it is navigating; it should try to exploit as much knowledge of the terrain as possible in order to make the best decisions about its path (Mitchell [160]) but must also balance considerations of exploration and prize targeting.
2.5.6.4 Airport Shuttles
Boyd, Clarke, Gemmell and Miller [26]model an airport shuttle bus problem as a CRP. Two competing airport shuttle companies each have four vehicles to pickup passengers from the airport and delivery then to their homes or pickup from homes and deliver to the airport. Whenever a new customer arrives in the system, via a stochastic arrival process, both companies determine the best way they can serve the customer, based on their current vehicle routes, and submit a bid price for servicing that customer. The customer chooses a company by considering the bid prices and the reputation of each company; reputation is based on service satisfaction and prize satisfaction. Having determined which company to choose, the customer is assigned to that company and is served accordingly. The goal for each company is to make a profit. Each company's behaviour is dictated by an operating policy, which defines how vehicle routes are selected, the vehicle routing strategy, and how bid prices are determined, the pricing strategy.
A central idea is that of reputation as a common measure of each company's perceived level of service. Each carrier is committed to a set of customers and also knows the commitment of each of the other carriers to customers. Also, each customer is assigned to a carrier when the call is placed; no trading or poaching of customers is permitted. The nature of this problem indicates that information about routing and forecasting customers are the two key components. Note that in this model it is sensitive to much information we have about the other carrier's vehicle's locations since then positioning ourselves relative to our opponents becomes very important.
2.5.6.5 Treasure and Pirates
A pirate has buried his lifetime's collection of treasure over a number of rugged terrain islands in the South Pacific. At his death, his crew, each looking after their own interests, descend upon the island. Each of them, having served the pirate over his treasure-burying career, had detailed records of the dig spots. Suppose each pirate had a GPS navigation system and could track the