tion of a best solution from a finite set of feasible" solutions according to some objective function
which gives a value to each feasible solution. A feasible solution is usually some combinatorial arrangement of elements of some set, e.g., a sequence of customers in the case of the TSP. Al though simply stated, the TSP is a prototype of a difficult COP since, in the worst case, a solution method may need to consider all feasible solutions to find the best solution.l However, solution concepts from the TSP are often applied to other COPs, e.g., in a VRSP solution method as a subroutine to determine the routes of the individual vehicles.
The Travelling Salesman Subset-tour Problem (TSSP) differs from the TSP in that the salesman is not required to visit every city (Mittenthal and Noon [161)). A salesman collects a
prize (or reward) , Vj , in every city j that he visits and pays a fixed penalty (or isolation cost), 'Irk , for every city k that he fails to visit. The salesman travels between cities i and j at cost Cij . Such a subset-tour (or subtour) may vary from visiting no cities to visiting all the cities. The salesman must determine both which subset of cities to visit and also in which order to visit those cities; these are two interdependent decisions.
The Travelling Salesman Subset-tour Problem with One Additional Constraint (TSSP+ 1) is just that, a TSSP with a single constraint. Noon, Mittenthal and Pillai [168] and Bowers, Noon and Thomas [25] have applied a TSSP + 1 as a subproblem within a decomposition scheme for solving the classical capacity-constrained VRP. Within a capacity-constrained VRP
2 . 1 . Combinatorial Subset Selection Routing Problems 15
the driver's problem is modelled as a TSSP+1 . The TSSP+ l decomposition is based on a
Lagrangian relaxation and is capable of producing valid lower bounds for the VRP. The best bound attainable is shown to be at least as good as the bound obtained by solving the linear programming relaxation of the classic set partitioning formulation of the VRSP. Pekny and Miller [175] define the Resource Constrained Travelling Salesman Problem (RCTSP) which is equivalent to the TSSP+ 1. They describe an application to scheduling sequence dependent transition costs with respect to an aggregate due date and design a parallel algorithm for problem instances comprising up to 200 cities.
More generally, Beasley and Nascimento [15] consider the possibility of gaining reward by allocating some customers not directly visited by the salesman to some nearby customer who is on the salesman's tour. They provide a framework formulation, the Single Vehicle Routing Allocation Problem (SVRAP) for the TSSP class of problems that also incorporates various problems with a covering component. Three types of selection occur: on-route customers, who contribute a fixed reward; off-route (allocated) customers who contribute a benefit dependent upon the on-route customer it is allocated to; and isolated customers who contribute a fixed penalty cost. Not all customers need be visited by the vehicles but customers not visited are either allocated to some customer on one of the vehicle routes (covered) or left isolated. Also, customers may have a preallocated type. The SVRAP also generalizes such problems as the Covering Tour Problem (CTP) of Gendreau, Laporte and Semet [77], the Shortest Covering Path Problem (SCPP) of Current, Pirkul and Rolland [45] , and the Covering Salesman Problem (CSP) , Median Tour Problem (MTP), and Maximal Covering Tour Problem (MCTP), of Current and Schilling [46, 47] .
The TSSP i s characteristic of a class of problem which h as evolved i n the literature over the last decade or so, which we call Combinatorial Subset Selection Routing Problems. We now survey the growing literature for this problem class by categorising the problem models as unconstrained, reward constrained, cost constrained, multiple salesman and time dependent.
2 . 1 . 1 Unconstrained TSSP
Keller [126] and Keller and Goodchild [127J propose the Multiobjective Vending Problem
(MVP) : to find all TSSP solutions, 8, such that the multiple objective (v(8) , £(8)) is Pareto optimal with respect to maximizing v(8) and minimizing £(8), i.e., find all those solutions such that there is no solution which simultaneously has greater value and shorter length. The MVP also generalizes problems involving shortest paths where each path is associated with two objectives (see Henig [97]).
Another unconstrained TSSP, the salesman wishes to minimise the sum of his travel costs
and isolation costs. We call this the Bienstock variation of the Prize Collecting Travelling
Salesman Problem (Bienstock-PCTSP) considered by Bienstock, Goemans, Simchi-Levi and
Williamson [20], Williamson [206] and Goemans and Williamson [83] . Volgenant and Jonker [202]
study the same problem, calling it the Generalized Travelling Salesman Problem (GTSP),2
2There are many problems called the Generalized Travelling Salesman Problem (GTSP) in the liter ature, each a generalization of the TSP.
and a special case called the Shortest Path Problem with Specified Nodes (SPPSN) in which a specified set of nodes must be visited exactly once, remaining nodes at most once and large penalties are incurred for not visiting the specified nodes. They give a transformation for the GTSP into an asymmetric TSP with twice the number of vertices and four times the number of edges.
Malandraki and Daskin [152] define a Maximum Benefit Travelling Salesman Problem (MBTSP) in which the requirement for a city to be visited at least once is further relaxed. A city may be visited any number of times or not at all, each visit accruing a diminishing additional reward. The objective is to maximize the net profit (reward value collected less travel costs) . Malandraki and Daskin also describe a similar variation of the Chinese Postman Problem (CPP) in which the requirement for each edge to be traversed at least once is relaxed and a benefit is derived from every traversal of an edge. This is called the Maximum Benefit Chinese Postman Problem (MBCPP) which finds a tour of maximum total net benefit. An edge may be traversed more than once or not at all, for decreasing benefit for each traversal. They also consider the multiojective MBCPP of maximizing the route's benefit and minimizing the route's length.
2 . 1 .2 . Reward Constrained TSSP
A Reward Constrained TSSP is a TSSP constrained by a single constraint on the sum of prize values on the subtour.
The salesman wishes to minimise the sum of his travel costs and isolation (penalty) costs whilst including in his tour enough cities to collect a prescribed amount Vmin of prize money.
The salesman departs from a given depot and returns to the depot at the end of the subtour. We will call this the Balas variation of the Prize Collecting Travelling Salesman Problem
(Balas-PCTSP) considered by Aneja and Punnen [4] , Awerbuch, Azar, Blum and Vempala [5], Balas [10, 1 1] , Dell'Amico, Maffioli and Varbrand [51] and Fischetti and Toth [68]. The Balas PCTSP was originally formulated for scheduling the daily operation of a steel rolling mill but has found numerous applications in routing and machine scheduling.
Hamacher and Moll [95] consider a special case of the Balas-PCTSP in which all the prize
values are equal, naming it the Travelling Salesman Selection Problem which we will call the
Hamacher-TSSP. Existing heuristics are based on approximations for the k-Minimal Spanning
Tree Problem to find the node cluster containing the shortest subtour satisfying the requirement to visit the given number of cities. The Hamacher-TSSP does not include a specific depot that has to be visited by the subtour.
2 . 1 . 3 Cost Constrained TSSP
A Cost Constrained TSSP is a TSSP constrained by a single constraint on the cost or length of the sub tour .
Gensch [78] studies an industrial problem in which the salesman wishes to maximize the net profit (reward value collected less travel costs) whilst not exceeding a prescribed travel cost
2 . 1 . Combinatorial Subset Selection Routing Problems 1 7
reserve the acronym (Gensch-TSSP) . Kataoka and Morito [125] comment that Gensch's algorithm is not strictly an algorithm (but a heuristic) because it does not necessarily find the optimal solution to one of the component subproblems.
Golden, Levy and Dahl [86] consider a generalization of Gensch-TSSP problem in which there is a net profit, Pij , associated with each edge and the salesman wishes to maximize the total net profit over the edges traversed whilst not exceeding a prescribed travel cost budget. This is called the Time Constrained Travelling Salesman Problem (TCTSP).3
The most popular TSSP+l studied in the literature is the Orienteering Problem (OP), in which the salesman wishes to maximize the reward value collected without exceeding a prescribed travel cost budget, visiting each city at most once. The motivation for this problem is the sport of Orienteering, concisely described by Chao, Golden and Wasil [35]:
"Orienteering is an outdoor sport usually played in a mountainous or heavily forested area. Armed with a compass and map, competitors start at a specified con trol point, try to visit as many other control points as possible within a prescribed time limit, and return to a specified control point. Each control point has an as sociated score, so that the objective of orienteering is to maximize the total score. Competitors who arrive at the finish point after time has expired are disqualified, and the eligible competitor with the highest score is declared the winner. Since time is limited, competitors may not be able to visit all control points. The competitors have to select a subset of control points to visit that will maximize their total score subject to the time restriction."
Hayes and Norman [96] model a real world orienteering event, the 1974 Lake District Mountain Trail, in England, as a dynamic program, to compare optimal paths against the actual routes selected by participants. They also consider the design of the course and the siting of the control points. They do not, however, formulate a combinatorial optimization problem.
Tsiligirides [200] appears to be the first to consider the combinatorial optimization problem formulation now known as the OP, although Tsiligirides called it a Generalised Travelling Salesman Problem, i.e., yet another GTSP. Golden, Levy and Vohra [89) were the first to coin the name Orienteering Problem. They compared a number of stochastic and deterministic subtour construction and improvement heuristics with those proposed by Tsiligirides. Golden, Wang and Liu [91] improved these heuristic ideas with a multifaceted heuristic including centre of gravity improvement, randomness, subgravity and a learning capability. Keller [126] adds a heuristic for the MVP (but restricted to the OP) to these computational comparisons. Ramesh and Brown [182] propose another heuristic for the OP employing local subtour operations including insertions, deletions and improvements. Wang, Sun, Golden and Jia [205) modify a continuous Hopfield neural network to find solutions for the OP. The neural network finds an initial feasible solution which is then improved using traditional two-exchanges and cheapest insertion. Chao, Golden and Wasil [35] develop a local search heuristic for the OP and compare their heuristic 3Note that Baker [9] and Kindervater, Lenstra and Savelsbergh [128] consider problems called the
Time Constrained Travelling Salesman Problem both of which are equivalent to a Travelling Salesman Problem with Time Windows (TSPTW), not a cost constrained TSSP.
with a number of these other authors' heuristics and also some algorithms, concluding that their heuristic is computationally efficient and outperforms most other heuristics on a set of 107 test problems.
Sokkappa [196} also studied the OP, but called it the Cost Constrained Travelling Sales man Problem (CCTSP). Golden, Levy and Vohra [89} shows that OP is NP-hard (see Garey and Johnson [73]) but Sokkappa proves that no K-approximation algorithm or fully polynomial approximation scheme exists for the OP, unless P = NP. Awerbuch, Azar, Blum and Vempala [5] propose the Bank Robber Problem, which turns out to be yet another name for the OP. More importantly they are able to provide a polylogarithmic performance guarantee for the OP and the Balas-PCTSP.
Algorithms have also been proposed to find optimal solutions to the OP. Kataoka and Morito [125} introduced the Maximum Collection Problem (MCP), which is equivalent to OP, and proposed a branch and bound algorithm with an Assignment Problem (AP) relaxation. Laporte and Martello [138} provide an integer linear programming formulation of the Selective Travelling Salesman Problem (STSP), also equivalent to the OP, and propose simple greedy heuristics, upper and lower bounding procedures and a branch and bound algorithm. Ramesh, Yoon and Karwan [183] design a branch and bound algorithm using a Lagrangean relaxation solved by a degree-constrained spanning tree procedure. Leifer and Rosenwein [143} contribute a number of strong linear programming relaxations for the OP by adding a sequence of valid in equalities. They determine upper bounds on the optimal value by solving three successive linear programs.
A capacitated TSSP is also possible. Diaby and Ramesh [55] consider the Distribution Problem with Carrier Service (DPCS). Each location has a certain demand, the distribution vehicle has a load capacity, and the entire operation should be completed within a certain time. An outside carrier is available for direct service of locations from the central facility. The problem is to determine a feasible tour for the company vehicle and the locations to be served by the outside carrier such that the total cost of the operations is minimised. The features of this problem are feasibility with respect to vehicle load as well as the travel time constraint, penalty costs for not visiting a customer and no rewards.
Finally, Millar [156] and Millar and Kiragu [157] present a novel application of the OP to a fisheries patrol problem in the Scotia-Fundy region of the Atlantic Coast of Canada. The OP serves as a static snapshot of a more dynamic problem; the prize values are used to approximate urgency and importance criteria.
2 . 1 .4 Multiple Salesman TSSP
In the Multiple Travelling Salesman Problem (MTSP) , m salesman must start from a depot, each visiting a number of prizes and returning to the depot, such that every prize is visited by at least one salesman and the sum of the distances travelled by the salesmen is minimised. The VRSP, then, is simply a capacitated MTSP. We can similarly define multiple salesman versions of the TSSP.
2 . 1 . Combinatorial Subset Selection Routing Problems 19
problem, there are m orienteers who compete as a team to maximize the sum of prize values
collected by the team members subject to a common time limit. Hence there are three interde pendent decisions: which prizes to visit, for each prize which team member to allocate to that prize, and for each team member what sequence of allocated prizes to follow. The paper modifies the local search heuristic developed for the OP in Chao, Golden and Wasil [35] and modifies the stochastic algorithm (heuristic) of Tsiligirides [200].
Butt and Cavalier [31] and Butt and Ryan [30] look at the Multiple Tour Maximum. Collection Problem (MTMCP), which is equivalent to TOP. Butt and Cavalier [31] provide an integer programming formulation and propose a local search heuristic, whereas Butt and Ryan [30] propose a branch and bound algorithm for finding optimal solutions.
Golden, Assad and Dabl [85] apply the TCTSP of Golden, Levy and Dabl [86] in the context of large scale vehicle routing with an inventory component. The goal of the distribution system is to maintain an adequate level of inventory for all customers. A profit is attached to every customer depending on the urgency of resupplying the customer, which is a nonlinear function of proportion of remaining tank level. The three interdependent decisions of TOP can easily be seen in this application. Customer selection involves identification of the customers to be serviced on a particular day and is intimately related to the inventory component of the problem. Customer vehicle assignment involves the assignment of customers for service on a particular day to one of the trucks. Finally, routing involves the construction of efficient routes for each truck over the set of its assigned customers.
2 . 1 .5 Time Dependent TSSP
Three forms of Time Dependent TSSP have been investigated in the literature: time depen dent rewards, time windows and time dependent travel times.
Brideau and Cavalier [28] and Erkut and Zhang [64] look at the Maximum Collection Problem with Time Dependent Rewards (MCPTDR) . In particular they include service times at the prizes and prize values of the form
(2.1)
for prize i at time t � 0 with decay rate S i � O. Brideau and Cavalier attempt to model a CRP
by using time-dependent rewards as a static snapshot.
"A simple example of the MCPTDR involves a salesman in a competitive envi ronment. Consider a salesman who wishes to visit a subset of cities in such a way as to maximize the number of sales made. All potential markets that fall within a restricted radius of travel are candidate cities to be visited by the salesman. From historical data, the salesman knows the potential reward at each city, that is, the number of potential sales that could be made in each market. However, as is common in a competitive market, there are other salesman working the same area, soliciting sales from potential customers. Thus, as the days progress, potential customers pur chase goods and services from the salesman's competitors, making the customers unavailable for solicitation. Depending on the number of initial customers and the
rate at which they make commitments, the number of potential sales decrease in each city until a time is reached when there are no potential sales left to be made. The salesman thus realizes that the longer the delay before visiting a city, the fewer the number of prospective customers remain. The salesman's objective is thus to chose a tour that would maximize the number of potential sales."
This problem indicates that the CRP has real world application for which researchers have pro posed models which attempt to capture the competition element without explicitly including dynamic competition in the model.
Problems involving restrictions on when you may visit locations are usually collected together as the class of Time Window Problems. For example the Travelling Salesman Problem with Time Windows (TSPTW) is the same as the TSP except that each customer may have specified an early time (which you cannot visit that location before) and a lateness time (which you must
visit that location before) . Kantor and Rosenwein [124] propose the Orienteering Problem with Time Windows (OPTW) in which three types of decision are coordinated: allocation of customers to be serviced; sequencing of customers that are to be serviced; and scheduling of customer deliveries with respect to the time windows at each customer.
Finally, Malandraki and Dial [153] and Malandraki and Daskin [151] consider time dependent travel times in their formulations.