NEW METHODS USING COLUMN GENERATION: AN EXAMPLE - ASSIGNING JOBS TO IDENTICAL MACHINES

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1 NEW METHODS USING COLUMN GENERATION: AN EXAMPLE - ASSIGNING JOBS TO IDENTICAL MACHINES Fran B Parng, Illinois Institute of Technology 58 W Boardwal Drive, Palatine IL 667 fparng@alcatel-lucent.com ABSTRACT This paper describes two new column generation methods and illustrates them by solving the problem of assigning obs to identical machines. The obective is to minimize the number of machines used to perform the obs. Two new methods have been developed and compared to the standard column generation methods. The paper compares using the new, Combined method vs. Standard method for the Master Problem. In addition, we also compare the standard subproblem (called Method A) with a new subproblem (called Method B). The paper presents computational results using Lingo on eight sets of data using two different initialization methods. Key Words: Column Generation, Branch-and-Price INTRODUCTION The standard integer programming formulation of the general column generation model being considered is demonstrated below: Minimize cx Subect to Ax b x & integer where the constraints are partitioned into a class of global or lining constraints ( Ax b ) and a class of specific constraints referred to as subproblem constraints (, x & integer). The cost vector c is a x n, x is an n x unnown solution vector, A is a m x n coefficient matrix, b is a m x vector of right hand side constants, B is a m x n coefficient matrix, b is a m x vector of right hand side constants. With identical machines, there is only one subproblem to solve. In Section 2, we introduce the general problem of using column generation techniques; explain why column generation is a unique method and useful approach for identical subproblems. We introduce a variety of problems that are amenable to column generation as well as the basic idea associated with branch-and-price In Section 3, we describe a new formulation of the Master Problem and contrast it to the Standard Master Problem The Standard Master Problem uses the traditional Phase I method. The new method, called the Combined method, combines the subproblem constraints into the Master Problem and does not require solution of a subproblem to generate attractive subproblem solutions)

2 Also, we present a new subproblem, called Method B, and contrast it with the usual formulation that we call Method A. Method B limits the dual prices to the range in which they are nown to be correct. Also, Method B includes the lining constraints in the subproblem formulation. In Section 4, we present numerical results on examples where the obs must be done multiple times. We discussed the benefit and advantage of using the new methods, the Combined method and Method B. Column Generation LITERATURE REVIEW Dychoff, 98 [] is the first to propose an integer linear programming formulation for the problem of getting a guaranteed global optimum to cutting stoc problems under the integrality requirement. Degraeve and Schrage, 999 [2] described a method that embedded the column generation procedure within a branch-and-bound scheme to find optimal integer solutions for cutting stoc problems. Vanderbec and Wolsey, 996 [3] described an exact algorithm for integer programs with a large number of columns. They found no satisfactory general branching scheme has been proposed for identical subproblems and a corresponding method to modify the subproblems after branching, and verify that it can be implemented. They developed a combined branching & subproblem modification scheme that applied to general column generation problems and described the use of lower bounds to reduce the tailing-off effect. Branch-and-price In recent years, branch-and-price has been applied successfully to airline crew scheduling. Vance, Barnhart et al., 997 [4] presented a new formulation for the airline crew scheduling problem such that its LP relaxation provides a tighter bound on the optimal IP solution than the traditional set partitioning formulation. Savelsbergh and Sol, 998 [5] described a branch-andprice algorithm in the transportation system for the general picup and delivery problem. Branch-and-price algorithms have solved many large-scale set partitioning problems successfully. Routing and scheduling has been a particularly good application area of branchand-price; see Desrosiers et al., 995 [6] for a survey of these results. Alternate Formulations for the Master Problem The Standard Method FORMULATIONS OF PROBLEMS Say we have visible solutions to the subproblem. Then the Standard Master Problem is Minimize α ( ) m cx y + α a 2 = i= i

3 Ax y + Ia b y, ai = Subect to ( ) & continuous If α = andα 2 =, it is a pure Phase I. Ifα = andα 2 =, a are forced to be, it is a pure Phase II. If both are positive, we call this is a composite obective function and both feasibility and optimality are considered at the same time. The Combined Method The second way of generating feasible solution is to combine the subproblem variables into the Master Problem with integer variables for the sought subproblem solution. To start Phase I, no integer feasible subproblem solutions are nown, and then the first Combined Master Problem solved is: Minimize α ( cx) Subect to m + α 2 i= a i Ax + Ia b x & integer α α, are constants, a & continuous, 2 Note: We are using a composite obective function. If α = and α 2 =, we have the Pure Phase I. The solution is x. Say we now one subproblem solution for the Combined Master Problem x. The second Combined Master Problem is: { } Minimize ( ) Subect to ( ) m α cx + c x y + α ai 2 i= Ax + Ax y + Ia b x & integer y, a & continuous Say this yields an optimal integer feasible subproblem solution x, call this x 2. In general, if there are integer feasible subproblem solutions, the Combined Master Problem is: Minimize α cx + ( cx ) y + α2 ai = i= Subect to Ax + ( Ax ) y + Ia b = x & integer y, a & continuous

4 =, = and no new x is found and a i >, then no feasible solution exists to the If α α2 original problem. As soon as the artificial variables are all, we have a feasible solution to the original problem. The Combined Master is used until the original obective function value does not decrease in two subsequent problems. Then, we revert to the Standard Master. Alternate formulations of the Subproblem Method A Method B Minimize ( c π ) A x Subect to x & integer Minimize cx π ( r s) Subect to Ax ( r s) r, s & continuous r Down limit, r is the down limit. Method A s obective function calculates the reduced cost of the most attractive subproblem feasible solution to add to the Master problem. Method A assumes π will remain the same, even if we go past the point where the set of binding constraints changes. When we apply Method B, we use both π and down limit r obtained from the Master Problem solution. Method B assumes that the impact on the Master Problem only continues until there is a basis change. We don t now how π will change after the basis change. In Method B, we want to associate this value π with the down limit. We solve the problem first using down limit, and we assume its new value will be after the basis change. Because this is conservative and may not be true, we must use Method A to prove optimality of the continuous Master Problem after Method B no longer gives attractive subproblem solutions. NUMERICAL RESULTS A set of test problems was generated to evaluate the new methods Combined for the Master and Method B for the subproblem. For these test problems, the machines were identical but the number of times each ob needed to be done was larger than one. Two initialization methods were used. Initialization (called the Big M method) started with one column corresponding to doing every ob once. This ensures a feasible solution to the continuous Master Problem, but since all the obs could not be done on one machine, it was given an obective function value of Big M. This very large value would then cause the initial column to leave the solution, somewhat lie Phase I. Initialization 2 (called single ob-single machine) introduced one master column for each ob, doing it only once. This assured there would be a feasible solution to the Master Problem, but would use one machine for each ob and every repetition of the ob. These initial variables were given an obective function coefficient of one because using one machine for each

5 ob and repetition is not attractive. Using either initialization eliminated the necessity of the explicit inclusion of the traditional Phase I artificial variables, one for each ob type to be done (the lining constraints).. Summary Of The Test Results And Conclusions Table : Summary of Results 8/29/28 Summary of the results Methods used Single ob-single machine initialization Machines used Big M initialization Machines used Method B, Method A Combined Master, Method A Method B, Method A Combined Master, Method A Bound 728* 728* 728* 728* * 685* * 8* 8* 8* * 546* * 45* 45* 45* * 782* * 752* * 728* 728* 728* * * * 8* 8* 8* * 45* 45* 45* * * Note: If there is an asteris, we now that it is an optimal solution based on the bound. Several points need to be observed in the numerical tests. First, when Combined was used, it was always followed by the Standard method. Combined is an alternate method of searching for attractive subproblem solutions. Second, Method A always followed Method B (using downlimits) when it was used. This assures that the optimal obective value to the final continuous master is a lower bound on the optimal obective function value for the master solved with integer variables (assuming all columns are available). Third, after establishing the lower bound, the Master was solved as an integer programming model. Note that the number of machines used must be integer so if the obective function lower bound is rounded up (say to T), that is a valid lower bound for the integer master with all subproblem columns present. If the

6 Combined master solved with integer variables yields an obective function value of T, we are assured that we have an optimal solution. Table shows the number of machines used to do all the obs. It loos lie the Combined Master method generally did at least as good as and sometimes better than Standard Master. That is, it often required fewer machines. Compare column 3 to column and column 4 to column 2. So Combined loos lie it actually helps. If you loo at using with the downlimit vs. followed by no downlimit vs. no downlimit at all, sometimes resulted exactly the same, but using the downlimit first actually resulted a better solution. Compare column 2 to column and column 4 to column 3. So using the downlimit at least for the example we looed at, never made things worse, but sometimes made things better. Also, we observe that Big M is inferior to single obsingle machine initialization. From the experiment datasets results, we conclude that Combined is better than Standard in terms of total machines used. Method B using downlimit is better than Method A not using down limit. Method A assumes the dual price remains the same forever when you decrease that right hand side; Method B says the dual price is outside of the limits, and that maes sure that the subproblem solution improvement potential doesn t get overestimated. BIBLIOGRAPHY [] H. Dychoff, A new linear programming approach to the cutting stoc problem, Operational Research [29] [98,pp. 92-4] [2] Zeger Degraeve and Linus Schrage, Optimal integer solutions to industrial cutting stoc problems, INFORMS Journal on Computing Vol. [] No. [4], Fall [999] [3] Francois Vanderbec and Laurence A. Wolsey, An exact algorithm for IP column generation, Operations Research Letters [9] [996, pp.5-59] [4] Pamela H. Vance, Cynthia Barnhart, Ellis L. Johnson and George L. Nemhauser, Airline Crew Scheduling: A new formulation and decomposition algorithm, Operations Research Vol. [45], No. [2], March-April, [997] [5] Martin Savelsberg and Marc Sol, Drive: Dynamic routing of independent vehicles, Operations Research Vol. [46], No. [4], July-August [998] [6] J. Desrosiers, Y. Dumas, N.M. Solomon, and F. Soumis [995]. Time constrained routing and scheduling, M.E. Ball, T.L. Magnanti, C. Monma, and G.L. Nemhauser (eds.) Handboos in Operations Research and Management Science, Volumn [8]: Networ Routing, Elsevier, Amsterdam, [ pp. 35-4]

The successful solution of large-scale mixed integer

The successful solution of large-scale mixed integer BRANCH-AND-PRICE: COLUMN GENERATION FOR SOLVING HUGE INTEGER PROGRAMS CYNTHIA BARNHART*, ELLIS L. JOHNSON, GEORGE L. NEMHAUSER, MARTIN W. P. SAVELSBERGH, and PAMELA H. VANCE** Georgia Institute of Technology,