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A Branch-And-Price Algorithm for the Generalized Assignment Problem

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A BranchandPrice Algorithm for the Generalized Assignment Problem Martin Savelsb ergh Georgia Institute of Technology School of Industrial and Systems Engineering Atlanta GA USA Abstract The generalized assignment problem examines the maximum prot assignmentof jobs to agents such that each job is assigned to precisely one agent sub ject to capacity restrictions on the agents A new algorithm for the generalized assignment problem is presented that employs b oth column generation and branchandb ound to obtain optimal integer solutions to a set partitioning formulation of the problem July Revised Novemb er Revised Octob er Intro duction The Generalized Assignment Problem GAP examines the maximum prot assignment of n jobs to m agents suchthateach job is assigned to precisely one agent sub ject to capacity restrictions on the agents Although interesting and useful in its own right its main imp ortance stems from the fact that it app ears as a substructure in manymodels develop ed to solve realworld problems in areas suchasvehicle routing plant lo cation resource scheduling and exible manufacturing systems The GAP is easily shown to b e NPhard and a considerable b o dy of literature exists on the search for eectiveenumeration algorithms to solve problems of a reasonable size to optimality Ross and Soland Martello and Toth Fisher Jaikumar and Van Wassenhove Guignard and Rosenwein Karabakal Bean and Lohmann A recent survey by Cattrysse and Van Wassenhove provides a comprehensive treatment of most of these metho ds In this pap er we present an algorithm for the GAP that employs b oth column generation and branchandb ound to obtain optimal integer solutions to a set partitioning formulation of the problem We discuss various branching strategies that allow column generation at any no de in the branchandb ound tree Therefore the algorithm can b e viewed as a branch andprice algorithm that is similar in spirit to the branchandcut algorithms that allowrow generation at any no de of the branchandb ound tree Manyvariations of the basic algorithm have b een implemented using MINTO a Mixed INTeger Optimizer Nemhauser Savelsb ergh and Sigismondi MINTO is a software system that solves mixedinteger linear programs by a branchandb ound algorithm with linear programming relaxations It also provides automatic constraint classication pre pro cessing primal heuristics and constraint generation Moreover the user can enrichthe basic algorithm byproviding a variety of sp ecialized application routines that can customize MINTO to achieve maximum eciency for a problem class This pap er is organized as follows Section intro duces b oth the standard and the set partitioning based formulation for the GAP Section presents the basic branchandprice algorithm and discusses issues related to column generation and branchandb ound Section examines the various branching strategies Section covers various implementation issues and Section describ es the computational exp eriments that have b een conducted Finally Section examines approximation algorithms derived from the branchandprice algorithm Formulations In the GAP the ob jective is to nd a maximum prot assignmentofn jobs to m agents such that each job is assigned to precisely one agent sub ject to capacity restrictions on the agents The standard integer programming formulation is the following X max p x ij ij imj n sub ject to X x j fng ij im X w x c i f mg ij ij i j n x f g i fmgjf ng ij where p ZZ is the prot asso ciated with assigning job j to agent i w ZZ the ij ij claim on the capacity of agent i byjobj if it is assigned to agent i c ZZ the capacity i of agent i and x avariable indicating whether job j is assigned to agent i x ij ij or not x ij The formulation underlying the branchandprice algorithm discussed in this has an exp onential number of variables and can b e viewed as a disaggregated version of the ab ove formulation i i i Let K fx x x g b e the set of all p ossible feasible assignments of jobs to agent i k i i i i i is a feasible solution to x x x i ie x nk k k k X i w x c ij i jk j n i x f g j fng jk i Let y for i f mg and k K b e a binary variable indicating whether a feasible i k i i i assignment x is selected for agent i y or not y The GAP can nowbe k k k formulated as X X i i max p x y ij jk k j n imk k i sub ject to X i i x y j f ng jk k imk k i X i i f mg y k k k i i f g i fmgk K y i k where the rst set of constraints enforces that each job is assigned to precisely one agent and the second set of constraints enforces that at most one feasible assignment is selected for each agent This set partitioning formulation has b een used by Cattrysse Salomon and Van Wassenhove to develop an approximation algorithm for the GAP The knapsack problem asso ciated with agent i in the standard formulation ie X p x max ij ij j n sub ject to X w x c ij ij i j n x f g j f ng ij has b een replaced in the disaggregated formulation by X X i i max p x y ij jk k j n k k i sub ject to X i y k k k i i i where x x are the integral solutions to the knapsack problem Because the linear k i programming relaxation of a knapsack problem contains the convex hull of the integer solutions the LP relaxation of the disaggregated formulation provides a b ound that is at least as tight as the b ound provided by the LP relaxation of the standard formulation Observe that the disaggregated formulation is essentially obtained by applying Dantzig Wolfe decomp osition to the standard formulation where the knapsack constraints have b een placed in the subproblem Consequently the value of the b ound provided by the LP relaxation of the disaggregated formulation is equal to the value of the Lagrangean dual obtained by dualizing the semiassignment constraints ie X X X min max p x x ij ij j ij imj n j n im sub ject to X w x c j f ng ij ij i j n x f g i fmgjf ng ij See for example Nemhauser and Wolsey Section I I for an exp osition of the relation b etween Lagrangean relaxation and DantzigWolfe decomp osition The algorithms of Fisher Jaikumar and Van Wassenhove Guignard and Rosenwein and Karabakal Bean and Lohmann are based on b ounds obtained by solving the ab ove Lagrangean dual Our computational exp eriments will show that the branchandprice algorithm discussed in this pap er although in theory using the same b ounds outp erforms the optimization algorithms of Fisher Jaikumar and Van Wassenhove Guignard and Rosenwein and Karabakal Bean and Lohmann A plausible explanation for this phenomenon is the fact that the use of the simplex metho d provides much b etter convergence prop erties than the use of subgradient and dual ascent metho ds for the solution of the Lagrangean dual Let y be any feasible solution to the LPrelaxation of the disaggregated formulation and P i i let z x y then z constitutes a feasible solution to the LPrelaxation of the ij k k jk k i standard formulation Furthermore wehave the following i Prop osition If y is fractional then there must b e a j such that z is fractional ij k i Pro of Supp ose there is no job j such that z is fractional Let F fk K j y g ij i k b e the set of fractional variables asso ciated with agent iWemay assume that jF j P i i i Note is fractional for every j with x y x b ecause if F fpg then z ij k k jp k jk i P i y Therefore that the convexity constraint asso ciated with agent i implies that k F k P P P i i i i i is either or for y forj n Consequently x y y x k F k F k F k jk k k jk P P i i i i i i j nIf y then x forall k F if x y then x x k F k F jk k jk jk k jk for all k F But that means that wehave duplicate columns a contradiction Branchandprice algorithms Column generation is a pricing scheme for solving largescale linear programs LPs Instead of pricing out nonbasic variables byenumeration in a column generation approach the most negative or p ositive reduced price is found by solving an optimization problem Gilmore and Gomory intro duced the column generation approach in the context of cutting sto ck problems In their case as in many other cases the linear program is a relaxation of an integer program IP However when an LP relaxation is solved by column generation the solution is not necessarily integral and it is not clear how to obtain an optimal or even feasible integer solution to the IP since standard branchandb ound techniques can interfere with the column generation algorithm Recentlyvarious researchers have started to develop customized branching strategies to handle these diculties eg Desro chers Desrosiers and Solomon for vehicle routing problems Desro chers and Soumis and Anbil Tanga and Johnson for crew scheduling problems and Vance Barnhart Johnson and Nemhauser for cutting sto ck problems Consider the linear programming relaxation of the disaggregated formulation for the GAP This master problem cannot b e solved directly due to the exp onential number of columns However a restricted master problem that considers only a subset of the columns can b e solved directly using for instance the simplex metho d Additional columns for the restricted master problem can b e generated as needed by solving the pricing problem max fz KP v g i i im where v is the optimal dual price from the solution to the restricted master problem as i so ciated with the convexity constraintofagent i and z KP isthevalue of the optimal i solution to the following knapsack problem X i max p u x ij j j j n sub ject to X i w x c ij i j j n i x f g j f ng j with u b eing the optimal dual
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