The GAP was defined by Ross and Soland [118], and is inspired by real-life prob- lems such as assigning jobs to computer networks (see Balachandran [5]) and fixed charge plant location where customer requirements must be satisfied by a single plant (see Geoffrion and Graves [58]). Other applications that have been studied are the p-median location problem (see Ross and Soland [119]), the maximal covering loca- tion problem (see Klastorin [76]), routing problems (see Fisher and Jaikumar [47]), R & D planning problems (see Zimokha and Rubinshtein [140]), and the loading problem in flexible manufacturing systems (see Kuhn [82]). Various approaches can be found to solve this problem, most of which were summarized by Cattrysse and Van Wassenhove [26] and Osman [101].
The Single Sourcing Problem (hereafter SSP) is a particular case of the GAP
where the requirements are agent-independent, i.e. aij = dj for each i = 1, . . . , m.
This problem was in fact introduced before the GAP by De Maio and Roveda [34]. They interpret the SSP as a special transportation problem where each demand point must be supplied by exactly one source. Allocating the items necessary for production and maintenance operations in a set of warehouses in order to minimize delivery costs originated the SSP. Srinivasan and Thompson [126] propose agent-dependent requirements as an extension of the SSP, i.e., what is now known as the GAP. In Part III we study another extension of the SSP, the Multi-Period Single-Sourcing Problem, where the demand and the capacities are time-varying and capacity can be transferred to future periods.
Due to its interest, this problem has been studied from an algorithmic point of view extensively. Different exact algorithms and heuristics have been proposed in the literature. Nevertheless, all approaches suffer from the N P-Hardness of the GAP (see Fisher, Jaikumar and Van Wassenhove [48]). Moreover, the decision problem associated with the feasibility of the GAP is an N P-Complete problem (see Martello and Toth [89]). (See Garey and Johnson [54] for a definition of N P-Hardness and N P-Completeness.) Therefore, even to test whether a problem instance has at least one feasible solution is computationally hard. In their proofs, Fisher, Jaikumar and Van Wassenhove [48] and Martello and Toth [89] use problem instances of the GAP with agent-independent requirements. Hence, the complexity results also hold for the SSP.
3.3
Existing literature and solution methods
In this section we summarize the research devoted to the GAP. We first concentrate on algorithmic developments. In this respect, we present bounding techniques which has been incorporated in branch and bound schemes to solve to optimality the problem. We then describe heuristic and meta-heuristic approaches for the GAP. Finally, we present other studies of the GAP.
Different bounds for the GAP have been proposed to be embedded in a branch and bound scheme. Ross and Soland [118] relax the capacity constraints (3.1) to
aijxij ≤ bi. This turns into a simple cost minimization problem where each task
is assigned to its cheapest (feasible) agent in the optimal solution. In general, this solution violates the capacity constraints of some of the agents. Therefore, some
tasks must be reassigned yielding a penalty of the objective value. The bound is improved by adding the minimal penalty incurred to avoid this violation. Martello and Toth [88] show that the algorithm of Ross and Soland is not fast enough when the capacity constraints are tight. Instead, they propose to calculate bounds by removing the semi-assignment constraints. The relaxed problem decomposes into m knapsack problems. Again, the corresponding bound is improved by adding a lower bound on the penalty to be paid to satisfy the violated semi-assignment constraints. Fisher, Jaikumar and Van Wassenhove [48] obtain bounds using Lagrangean relaxation in the semi-assignment constraints. A multiplier adjustment method (see Fisher [45]) is used to find good multipliers. Guignard and Rosenwein [66] observe that the largest test-problems reported in the literature contained at most 100 variables. They propose some enhancements and additions to the approach of Fisher, Jaikumar and Van Wassenhove [48] to be able to solve larger problems. First, they enlarge the set of possible directions used by the multiplier adjustment method. Second, if the obtained
solution violates Pm
i=1
Pn
j=1xij = n, then the corresponding surrogate constraint
(Pm i=1 Pn j=1xij ≤ n or P m i=1 Pn
j=1xij ≥ n) is added to improve the bound given
by the heuristic. They were able to solve problems with 500 variables. Karabakal, Bean and Lohmann [74] argue that the multiplier adjustment methods proposed by Fisher, Jaikumar and Van Wassenhove [48] and Guignard and Rosenwein [66] move along the first descent direction (for a maximization formulation of the GAP) found with nonzero step size. They propose to use the steepest descent direction. Unfortunately, no comparisons are shown with the two previous methods. Linear
programming bounds have been used by J¨ornsten and V¨arbrand [72], improved by
valid inequalities from the knapsack constraints. They also consider new branching rules. Numerical results are only shown for problem instances of size m = 4 and n = 25. Cattrysse, Degraeve and Tistaert [23] also strength the linear relaxation of the GAP by valid inequalities. Savelsbergh [121] proposes a branch and price algorithm for the set partitioning formulation of the GAP (see Section 2.5).
Due to the hardness of the GAP, a significant number of heuristic procedures have been proposed. First, we describe those ones based on the LP-relaxation of the GAP. Benders and Van Nunen [14] prove that the number of infeasible tasks, i.e. the ones assigned to more than one agent, in the optimal solution for the LP-relaxation is at most the number of agents used to full capacity. They propose a heuristic that assigns the infeasible tasks of the optimal solution for the LP-relaxation of the GAP. Cattrysse [24] proposes to fix the feasible tasks of the LP-relaxation of the GAP and solve the reduced problem by a branch and bound technique. He observes that adding cuts to the LP-relaxation increases the number of fractional variables, so that less tasks are fixed. Numerical experiments show that this increases the success of his primal heuristic. Trick [131] uses another property of the LP-relaxation of the GAP to propose his LR-heuristic. He defines the variable associated with the assignment of task j to agent i useless if the requirement of task j on agent i is larger than the capacity of agent i, which means that without loss of optimality useless variables can be fixed to zero. The basic idea is to solve the LP-relaxation of the GAP and fix all the feasible tasks. We then obtain a new GAP with the same number of agents which has at most m tasks (see Benders and Van Nunen [14]). Moreover, useless variables
3.3. Existing literature and solution methods 61
are fixed to zero. This procedure is successively repeated.
A considerable number of the heuristic approaches for the GAP are based on Lagrangean relaxations. Chalmet and Gelders [27] use subgradient optimization for the two possible Lagrangean relaxations. They notice that the constraint matrix when relaxing the capacity constraints is totally unimodular. Thus, this bound co- incides with the optimal value of the LP-relaxation of the GAP, see Geoffrion [57]. Nevertheless, they claim that the former one can be calculated more efficiently for large problem instances. By relaxing the semi-assignment constraints, better bounds are expected since the unimodularity property does not hold. Klastorin [75] uses a subgradient method for the relaxation of the capacity constraints. A branch and bound scheme was implemented to search in the neighborhood of the current solution.
J¨ornsten and N¨asberg [71] apply a Lagrangean decomposition approach (see Guig-
nard and Kim [65]). This approach enables to combine the two structures obtained by Lagrangean relaxation. Moreover, this bound is at least as good as these two Lagrangean relaxations. A subgradient method is used to find bounds, and heuristic procedures try to find primal solutions. There is no description of the way the test-
problem instances were generated. Barcia and J¨ornsten [8] use the bound improving
technique (see Barcia [7]) to tighten the bound obtained by Lagrangean decompo- sition. Lorena and Narciso [85] propose a subgradient method for the Lagrangean relaxation of the capacity constraints and the surrogate relaxation of those ones. Primal solutions are searched for by a greedy heuristic from the class of Martello and Toth [88] and a constructive heuristic. In a later work, Narciso and Lorena [97] pro- pose a Lagrangean/surrogate relaxation. The numerical results are averaged output for all the problem instances tested.
Cattrysse, Salomon and Van Wassenhove [25] propose a heuristic based on the set partitioning formulation of the GAP. They solve its LP-relaxation by a multiplier adjustment method combined with a subgradient method. They look for primal solutions by reduction techniques.
Martello and Toth [88] propose one of the most widely used greedy heuristics for the GAP (see Section 2.3.1). They also add a local search phase where they try to improve the objective value of the current solution. Wilson [137] uses the solution where each task is assigned to its cheapest agent as the starting point for an exchange procedure where the violation of the capacity constraints is decreased in each step.
Meta-heuristics have also been proposed for the GAP. Cattrysse [24] implements a simulating annealing concluding that it is only competitive for small problem sizes. Racer and Amini [105] describe a variable depth search heuristic (where the main idea is to adaptively change the size of the neighborhood). They compare their results with the heuristic from Martello and Toth [88] on five classes of problem. At the expense of high computation times, the variable depth search heuristic finds better feasible solutions than the greedy heuristic for one of the problem classes. In order to decrease computation times, Amini and Racer [2] describe a hybrid heuristic where initial solutions are generated with the heuristic from Martello and Toth [88] and refined with a variable depth search heuristic. Osman [101] proposes a simulating annealing and a tabu search. Chu and Beasley [32] and Wilson [136] propose genetic algorithms for the GAP. In the first one, a family of potential solutions is generated, and steps are
made to improve feasibility and optimality. On the contrary, good starting solutions in term of objective value are assumed in the second one. Ramalhinho Louren¸co and Serra [106] propose two meta-heuristic approaches for the GAP. The first one is a greedy adaptive search heuristic (see Feo and Resende [42] for a general description of such GRASP heuristics), and the second one is a MAX-MIN ant system (see
St¨utzle and Hoos [127]). Both of them are combined with a local search and a tabu
search schemes. Yagiura, Yamaguchi and Ibaraki [139] notice that searching only in the feasible region may be too restrictive. Therefore, they propose a variable depth search heuristic where it is allowed to move to infeasible solutions of the problem. Yagiura, Ibaraki and Glover [138] propose an ejection chain approach combined with a tabu search.
Shmoys and Tardos [123] propose a polynomial-time algorithm that, given C ∈ R, either proves that there is no feasible solution for the GAP with cost C or find a feasible assignment of cost at most C with a consumption of the resource at agent i
of at most 2bifor all i. We can also mention an aggregation/disaggregation technique
for large scale GAP’s proposed by Hallefjord, J¨ornsten and V¨arbrand [69].
Apart from algorithms solving the GAP, there are papers devoted to other aspects. In this respect, Gottlieb and Rao [63, 64] perform a polyhedral study of the GAP. It is straightforward to see that any valid inequality for the Knapsack Problem is also valid for the GAP; they also prove that each facet of the Knapsack Problem is also a facet for the GAP. They found other valid inequalities based upon more than one knapsack constraint. Amini and Racer [1] present an experimental design for computational comparison of the greedy heuristic of Martello and Toth [88] and a variable depth search heuristic which is apparently the same one as the authors have proposed in Racer and Amini [105]. Stochastic models for the GAP have been proposed by Dyer and Frieze [38] and Romeijn and Piersma [110]. In the latter paper a probabilistic analysis of the optimal solution of the GAP under this stochastic model was performed, studying the asymptotic behaviour of the optimal solution value as the number of tasks n goes to infinity. Furthermore, a tight condition on the stochastic model under which the GAP is feasible with probability one when the number of tasks goes to infinity is derived.