SCHOOL OF COMPUTER STUDIES RESEARCH REPORT SERIES

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1 University of Leeds SCHOOL OF COMPUTER STUDIES RESEARCH REPORT SERIES Report A Column Generation Approach to Bus Driver Scheduling by Sarah Fores, Les Proll & Anthony Wren Division of Operational Research & Information Systems August 996 To be presented at the 4th Meeting of the EURO Working Group on Transportation, Newcastle, 9 - September, 996

2 INTRODUCTION The problem of scheduling public transport vehicles and their drivers has been the subect of six international workshops (see, e.g., Desrochers and Rousseau, 992; Daduna et al., 995). Research has progressed to a point where there are several computer scheduling systems in widespread commercial use, e.g. HASTUS (Rousseau and Blais, 985) and IMPACS (Smith and Wren, 988), which were originally developed for use in the bus industry. This paper outlines an alternative solution method which has been incorporated into a system which originated from IMPACS. Improved results on a selection of real bus driver problems are presented. THE DRIVER SCHEDULING PROBLEM Although some systems attempt to schedule vehicles and drivers simultaneously, or even to schedule drivers first, it is generally the case that a vehicle schedule is built initially to cover a set of predetermined ourneys. At various stages during the work of each vehicle there are convenient driver changeover locations with associated times and these are known as relief opportunities. An indivisible period between any two relief opportunities is known as a piece of work. A shift usually consists of two or three spells of work which each cover several consecutive pieces of work on the same vehicle. The formation of shifts is governed by a set of labour agreement rules to ensure that there is adequate provision for mealbreaks and acceptable working hours etc. The driver scheduling problem is then to define a set of shifts in such a way that all the vehicle work is covered and some measure of the number of drivers and the costs of shifts is minimised. Usually this requires minimisation of the number of shifts as a priority, and with that minimum finding a good compromise between cost and some subective measure of quality. Mathematical programming solution methods applied to driver scheduling have become more dominant and successful with improvements in computer technology but the problem is still too large to be able to guarantee an optimal schedule in most cases. Thus mathematical programming is frequently combined with heuristic approaches to provide a viable solution method. THE TRACS II MODEL TRACS II is the driver scheduling system which was developed at the University of Leeds and retained for research purposes. This system uses the same general approach as the commercially available IMPACS system (Smith and Wren, 988). TRACS II first generates a set of valid shifts, then reduces the set to a manageable size and finally selects from it a subset of shifts which cover the bus work. This final stage in the solution process involves solving a set covering model which ensures that each piece of work is covered by at least one driver and that the overall cost is minimal. As the introduction of each driver incurs a large cost the primary obective is to minimise the number of shifts. The remaining shift cost involves the actual wage cost of the shift plus some subective cost reflecting any penalty for shifts

3 2 containing undesirable features. In order properly to combine the two obectives a Sherali strategy (Sherali, 982) was adopted by Willers et al. (995), viz : N N Minimise W x + C x = = where x is if the th generated shift is used, and 0 otherwise, C is the combined wage and penalty cost of shift, and W is the weight that prioritises the reduction of shifts in the solution. The Sherali weight W is calculated as : N W = + UB C x = where UB[] is an upper bound on the sum of the shift costs. The upper bound can be taken to be the sum of the S largest C values, where S is itself an upper bound to the number of shifts which might be used in a sensible solution and can be set to be the number of shifts in the initial solution. We can now define a new obective : Minimise N = D x where D represents the combined wage and penalty cost of shift plus the Sherali weighting. It should be noted that in a practical problem there are generally many million potential shifts, too many to be treated by the model. For this reason restrictions are placed on the form of the shifts to be generated and retained. These restrictions are governed by parameters designed to ensure that the most likely shifts are available to the above model, and that a good choice of shifts is available for each piece of work. Nevertheless, it is quite possible that some apparently inefficient shifts may be required in a good solution, and that these will have been excluded by the parameters. For this reason TRACS II uses a set covering approach which allows pieces of work to be overcovered, i.e. covered by more than one shift. As overcover involves a wage cost for more than one shift the optimisation will deter overcover from being formed. Where overcover remains in the final schedule, the relevant shifts can be edited to form shorter shifts excluded from the generation process. There are instances where the existence of overcover would not be sensible, e.g. on the first and last few pieces of work on any bus, and so these corresponding constraints can be defined as equality constraints to be covered by only one

4 3 driver. The equality constraints cut down the problem size when forming an initial solution and are necessary in one of the branching strategies of the branch and bound algorithm. The TRACS II constraints can thus be defined as follows : N N A x = for i =,K E = i A x for i = E +, K, M = i where E is the number of equality constraints and A i = if shift covers piece of work i. It is then possible for the user to add side constraints which typically are used to limit the number of shifts of a particular type or the total number of shifts. Such constraints are of the form: N δ N x U or δ x L = = where δ = if the corresponding x is of the type that is to be constrained, = 0 otherwise. Method of Solution of the Model ) Create an initial solution by selecting shifts to cover pieces of work in increasing order of the number of shifts available to cover them; each successive shift is selected so as to minimise the cost per currently uncovered workpiece covered by this shift. 2) Solve the linear programming relaxation of the TRACS II model using a primal steepest edge algorithm. 3) If the minimum number of duties is fractional, add a constraint which increases the number of duties to the next highest integral value and re-solve if necessary using a dual approach. 4) Find an integer solution using specialised Branch and Bound techniques. COLUMN GENERATION Column generation is an approach which can be used to solve mathematical programming problems involving many variables, which would otherwise have to be decomposed or reduced before being solved. Scheduling problems are an important class of combinatorial optimisation problems which frequently contain many variables. TRACS II uses heuristics to reduce the number of shifts in order to enable conventional mathematical programming

5 4 solvers to produce a solution. Further investigation has taken place into incorporating column generation techniques within TRACS II. It is shown that column generation methods overcome many of the difficulties which arise in solving large problems, and also improve upon the solution and/or speed of the process. Column generation systems normally use the Revised Simplex Method to solve the problem over a subset of the columns. The process then repeatedly adds further columns which will potentially improve the solution to the subset and re-solves over the new set. This process continues until no further columns can be found which will improve the obective. Assuming that all the columns are available in the matrix of constraints, the Revised Simplex Method begins with an initial solution and iteratively improves the obective value by swapping a basic variable for a non-basic variable with favorable reduced cost, until no further improvement can be made. In the case of column generation only a subset of the columns is available at the outset. The Revised Simplex Method is used to find the solution which is optimal over the subset. One then generates new columns or searches through columns not previously considered. If there are any new columns which have favorable reduced costs then some, or all, of them are added to the subset as non-basic variables so that the current solution is no longer optimal. Using the current basic solution, the Revised Simplex Method continues to swap columns where necessary to reoptimise over the current larger subset. For any LP relaxation which is optimal over its available subset the overall optimal solution is attained when no more columns which would improve the obective can be added to the set. Iterations of the Revised Simplex Method within a particular shift subset are known as minor iterations; augmentations of the shift subset are known as maor iterations. COLUMN GENERATION AND DRIVER SCHEDULING The maor disadvantage of using most computerised scheduling systems, including TRACS II, is that the search for optimality is limited to an heuristically reduced shift set, and so for some problems the ILP fails to find a feasible solution. In TRACS II the constraint introduced in step 3 of the solution process must then be relaxed, thereby allowing more shifts in the solution. Large problems can be decomposed (Wren and Smith, 988) but this is usually deemed undesirable by users. Heuristics often limit the formation of shifts to include only efficient shifts, but because it may be the case that some inefficient shifts are required to link with them to produce the optimal solution, ideally we would wish to consider allowing all possible valid shifts which could be formed to enter the set covering model. In the context of driver scheduling a column generation method implicitly considers many more valid shifts than standard linear programming approaches whilst retaining a much smaller working subset of shifts which are available to the mathematical program solver. The HASTUS scheduling system (Rousseau and Blais, 985) contains a module, Crew-Opt (Desrochers et al., 992), which uses column generation techniques to form bus driver schedules and has produced encouraging results (Rousseau, 995). A subset of valid shifts is identified and further shifts are added to the subset based upon utilising a shortest path algorithm to identify shifts which will improve the current solution. Once no new shift can be

6 5 found which will reduce the schedule cost the current solution is optimal over the relaxed model. A branch and bound technique which also incorporates column generation is then applied to find a good driver schedule. The model works well on smaller problems but takes substantial time to perform the shift additions and the optimality can be compromised in the branch and bound phase as the search often stops at a good solution. A COLUMN GENERATION MODEL WITHIN TRACS II A column generation approach to solving the driver scheduling problem requires strategies for resolving the following three issues : How to choose the initial shift subset. How to generate or search for further shifts. How to produce an integer solution from the LP solution. These issues will be discussed in an attempt to find the most appropriate technique to be incorporated into the TRACS II system, and an outline of the model will follow. Shift Generation Incorporating column generation techniques into the solution strategy of the set covering model potentially allows all valid shifts to be considered. However, to guarantee the overall continuous optimum one of the following two methods would have to be used: I. All valid shifts would have to be formed initially and their reduced costs calculated during the column generation process to ensure optimality. II. A subproblem must be available to generate any remaining shifts with favorable reduced costs once optimality has been found over the current shift subset. Crew-Opt uses the second method, with a subproblem defining a constrained shortest path technique to generate further shifts as paths through a network. In the current TRACS II system the shift costing not only depends upon the duration of the shift, but may also include a subective weighting to deter some shifts from being included in a schedule. This is because many organisations seek to minimise unpopular or administratively difficult shifts. The penalty costs are currently defined in a subroutine specific to each organisation and typical requirements include: penalties added to three-part shifts penalties added to shifts whose meal breaks are not close to the middle of the shift penalties added to early shifts which start after the earliest time for split shifts.

7 6 Although these penalties can be added cumulatively as a shift is being constructed, they are not added as resource constraints and cannot be built into a shortest path network. If the shortest path through the network corresponds to a shift with undesirable features its reduced cost may not be the most favorable once the penalty cost has been added to the shift cost. Also, although Crew-Opt hopes to be able to handle larger problems, published work (Rousseau, 995) states that it is restricted. Indeed, larger problems will require large networks and for each set of labour agreement rules there will be a certain amount of time required to convert them into network constraints. In order to speed up the process of column generation it is usual for methods to add more than one column at each iteration, and for larger problems it will become complex and time consuming to traverse a network to generate such shifts. Decomposing a larger problem will compromise optimality. For the reasons given above, the technique which has been explored is that of removing the heuristics which reduce the size of the generated shift set in TRACS II, hence allowing more shifts to be considered by the set covering model. A network formulation is not used, and so shifts from this larger set are selected to enter a working subset by means of a less sophisticated enumeration method. An improvement in continuous LP optimum may be achieved by using this larger set, and so for each bus company both a larger shift set and an heuristically reduced shift set are used in the experiments reported here. As the current system produces an LP solution, the column generation method on the same data set will not improve the cost, as both will be optimal. However, the column generation method may prove to be faster, and the larger data set may produce a lower LP cost than the smaller data set, which would normally have been used in TRACS II. Integer Solution If the overall continuous optimum solution has been found, then in order to guarantee an optimal integer solution it would be inadequate to terminate the branch and bound phase as soon as a good solution had been found. Crew-Opt uses a column generation technique within a branch and bound structure in order to ensure that the integer solution is not limited to the shifts formed in finding the optimal LP solution. Continuing through the search tree in this way would guarantee an overall optimal integer schedule, although in practice Crew-Opt pauses at the first integer solution and uses the continuous optimum to analyse whether the expected improvement in integer solution cost ustifies the probable extra computational cost of further searching. At each node the column generation method used in the branch and bound phase limits the shifts which are formed to those relevant. Results given for Crew-Opt showed that a large number of new columns are added overall in the branch and bound phase even though only one path through the tree will provide the integer schedule. In TRACS II, because the shift costs include subective weightings, many integer solutions may be equally as good as or possibly preferable to the apparently optimal schedule with respect to shift content, and so spending time searching for the optimal schedule may be wasted. Whilst the solution time of computerised systems is acceptable compared to previous manual methods, bus companies would tend to give execution time reduction a higher priority

8 7 than improving a good integer solution. Faster solutions would allow a user more opportunity to analyse the effect of relaxing certain conditions or altering parameters. Since a method of producing schedules from a larger shift set cannot guarantee the overall optimal LP solution, the optimal schedule cannot be guaranteed either. For this reason the branch and bound phase used in TRACS II is adequate, and although it considers only some of the shifts retained in the model, this has not detracted from its ability to find a good solution. Initial Shift Set Crew-Opt uses an initial set consisting of short shifts in order to provide a feasible solution quickly. In the work reported here a large set of potential shifts known as a superset is generated initially and a subset of these is used in such a way as to provide an initial subset of good shifts, and hence a good feasible initial solution. Details of the method of selecting a shift subset are given in the next section. A Column Generation Model There are many possible ways of implementing a column generation strategy, but the method of solution which will be adopted is outlined as follows: Step 0 Generate a shift superset. Step Create an initial solution and form an initial shift subset. Step 2 Solve the LP over the current shift subset. Step 3 Add a set of shifts to the current subset which will improve the solution. Step 4 If no favorable shifts can be found then the LP solution is optimal, otherwise go to Step 2. Step 5 Find an integer solution using branch and bound. IMPLEMENTATION WITHIN TRACS II Having outlined the column generation model and decided that a larger set is to be generated at the outset, it is now necessary to define which shifts are to be included in the initial subset and the factors which determine how many further shifts are added at any intermediate solution stage. Full details of the implementation strategies and results are given by Fores (996). Initial Solution and Initial Shift Subset Given that all reasonable shifts are available at the outset, the method chosen to form an initial solution is the same as that currently used by TRACS II. As pieces of work are being considered in increasing order of the number of available shifts covering them, shifts are added to an initial subset if there are fewer than a specified number of other shifts, currently in

9 8 the subset, which cover that piece of work. Research has shown that a value of 0 provides a sufficiently large and varied initial subset to reduce the number of maor iterations required. A larger set would take more time to form and contain many more inefficient shifts, which would slow down the mathematical program solver. Addition of Further Shifts Once an initial shift subset has been formed, the Revised Simplex Method is used to optimise the model. This produces simplex multipliers which are used to calculate the reduced cost of any potential shift which exists in the superset. The reduced cost of a shift k is defined to be : D k M π a i= i ik where : M is the number of constraints D k is the cost of shift k π i is the simplex multiplier for constraint i a ik is the coefficient of shift k in constraint i A shift which will improve the current obective will have a negative reduced cost. Experiments show that the simplex multipliers can vary significantly more than the D k due to the presence of the prioritising weight. It is likely therefore that shifts which cover the pieces of work associated with the highest simplex multipliers may have the lowest reduced costs. The simplex multipliers are therefore considered in decreasing order and shifts covering their corresponding pieces of work are added to the subset if they have a negative reduced cost and if they are allowed to be added based upon the following rules, which limit the number of shifts which can be added during any maor iteration. Total Shifts Added Per Maor Iteration As it is time consuming to consider every simplex multiplier at each iteration, and the simplex multipliers have been ordered so that the later values are less likely to produce further good shifts, a limit has been imposed on the percentage of simplex multipliers used to produce further shifts, viz: P% of the total number of pieces of work for which additions can be made For many simplex multipliers no further shifts will be added but there is a limit on the number of remaining simplex multipliers which can be considered in any maor iteration. Experimentation has shown that a value of 20 for P adds sufficient further efficient shifts. Total Number of Shifts Added There is also a parameter limiting the total number of shifts which can be added to the subset defined by :

10 9 M = MAX{50,X% of the total number of shifts remaining} The constant value 50 is used to ensure that if the number of remaining shifts decreases to a very low value, a reasonably large number of shifts can still be added in order to avoid extra maor iterations. The parameter X reflects the need for fewer shifts to be added towards the end of the process but it is observed that P is normally a tighter limit than M and so X is set to 0 to allow it to be useful in very large data sets. Total Shifts Added Per Piece of Work Each piece of work is potentially covered by a very large number of shifts and so a limit is placed upon the number of shifts which can be added which cover a particular piece of work : I = MAX{5,Y% of the total shifts remaining/total no. pieces of work} The parameter Y again reflects the need to consider fewer shifts as the process continues and the constant value 5 deters more maor iterations from having to be performed as the process nears its conclusion. Setting Y to 50 provides a sufficient number of shifts per piece of work without compromising the time taken to evaluate the reduced costs of the shifts involved. COMPUTATIONAL RESULTS Seven problems were tested, each with two sizes of shift superset. The smaller is a subset of the larger and is of a size which would normally be used within TRACS II. At the time of this research, TRACS II was limited to deal with problems containing fewer than shifts but this limit was increased in order to compare results and times with those produced by the column generation approach on all problems, the largest of which contains over 90,000 shifts and over 400 pieces of work. Details of these problems are given in Table. Data Set #Pieces of Work Smaller Subset(S) Superset Size Final Subset Size #Pieces of Work Larger Subset(L) Superset Size Final Subset Size AUC CTJ CTR GMB RI STK SYD Table The aims of the research were twofold: to investigate whether better solutions could be found by making more potential shifts available to the optimisation process;

11 0 to determine whether for a given number of presented shifts, the optimisation process could be made faster. Results of the experiments on the test problems are given in Tables 2 and 3. The experiments were performed on a Silicon Graphics Iris Indigo workstation with a 33MHz R300 MIPS processor. In Table 2, * means an attempt was made to find a solution with the given number of shifts and means that the maximum node limit was reached in the branch and bound phase before an integer solution was found. Time to solve LP (mins) Total Solution Time(mins) Data Set Continuous # Shifts Final # Shifts Maor Iterations TRACS II Column Generation TRACS II Column Generation AUC(S) AUC(L) CTJ(S) CTJ(L) CTR(S) * CTR(L) * GMB(S) GMB(L) RI2(S) RI2(L) STK(S) STK(L) SYD(S) SYD(L) Table 2 Table 2 shows that for AUC, CTR, and STK, the continuous solution produced by the larger shift set indicated too few shifts to enable a feasible schedule to be produced. However, the addition of a side constraint which specified a shift total at the next highest integer enabled a schedule to be found which was better than that produced by the smaller shift set. STK found a solution with two fewer shifts, AUC produced a solution with the same number of shifts but at lower cost, and CTR produced a solution where none had been found either with the smaller data set or with TRACS II on the larger set. Schedules with fewer shifts were also found by using the larger shift set on three out of the remaining four problem instances. Table 2 also presents the computer time necessary to solve the relaxed LP and the total time to the ILP solution for each of the runs. Overall, results show that the larger shift sets used in the column generation approach can yield better schedules in terms of the minimum number of shifts necessary to cover the work. In addition, the results show a reduction in execution time using column generation for 0 of the 2 data sets for which a comparison is meaningful.

12 As a maor aim of the research has been to determine whether better solutions could be obtained by using column generation on a larger subset than used by TRACS II, we compare in Table 3 the column generation solutions to the larger problems with the TRACS II solutions to the corresponding smaller ones. The small increases in computing time are ustified by the potential savings. Where there has been a large increase (AUC), all but eight minutes of this is attributable to a longer route having been taken, by chance, through the branch and bound tree. TRACS II (S) Column generation (L) Data #shifts run time (mins) #shifts run time (mins) set relaxed LP total to ILP relaxed LP total to ILP AUC CTJ CTR GMB RI STK SYD Table 3 It was surprising that the new approach should, on three occasions, achieve a significantly better result than TRACS II (saving a single shift is very significant to a bus company). TRACS II consistently achieves better results than those obtained by conventional methods, and when tested against other computer systems has proved more efficient. Users of TRACS II are very happy with the quality of solution obtained. We have to conclude from the above that there are opportunities for significantly better savings than are currently being realised. CONCLUSION Although optimal solutions still cannot be guaranteed using the column generation approach described, results show that the technique can produce better solutions more quickly than a successful and widely used driver scheduling system. Increased solution speed opens up the possibility of solving larger problems without decomposition which, in turn, may lead to better solutions. TRACS II is now being used successfully to solve large train driver scheduling problems (Kwan et al., 996a, 996b), using a form of decomposition where necessary. The new column generation approach is currently being adapted for the production version of the system, and is expected to enable TRACS II to yield better and faster solutions to such problems.

13 2 REFERENCES Daduna, J. R., I. Branco and J. M. P. Paixáo (995). Computer-Aided Transit Scheduling - Proceedings of the Sixth International Workshop on Computer-Aided Scheduling of Public Transport. Springer-Verlag, Berlin. Desrochers, M. and J.-M. Rousseau (eds.) (992). Computer-Aided Transit Scheduling - Proceedings of the Fifth International Workshop on Computer-Aided Scheduling of Public Transport. Springer-Verlag, Berlin. Desrochers, M., J. Gilbert, M. Sauvé and F. Soumis (992). CREW-OPT: Subproblem modelling in a column generation approach to urban crew scheduling. In: Computer Aided Transit Scheduling (M. Desrochers and J.-M. Rousseau, eds.), pp Springer-Verlag, Berlin. Fores, S (996). Column Generation Approaches to Bus Driver Scheduling. Ph.D. Thesis, University of Leeds. Kwan, A. S. K., R. S. K. Kwan, M. E. Parker and A, Wren (996a). Producing train driver shifts by computer. In: Computers in Railways V - Volume : Railway Systems and Management (J. Allan, C. A. Brebbia, R. J. Hill, G. Sciutto and S. Sone, eds.), pp Computational Mechanics Publications, Southampton. Kwan, A. S. K., R. S. K. Kwan, M. E. Parker and A, Wren (996b) Scheduling train drivers and investigating alternative scenarios by computer. Presented at 4th Meeting of the EURO Working Group on Transportation, 9- September, University of Newcastle. Rousseau, J.-M. (995). Results obtained with Crew-Opt, a column generation method for transit crew scheduling. In: Computer-Aided Transit Scheduling (J. R. Daduna, I. Branco and J. M. P. Paixáo eds.), pp Springer-Verlag, Berlin. Rousseau, J.-M. and J.-Y. Blais (985). HASTUS: An interactive system for buses and crew scheduling. In: Computer Scheduling of Public Transport 2 (J.-M. Rousseau ed.), pp North-Holland, Amsterdam. Sherali, H. D. (982). Equivalent weights for lexicographic multi-obective programs. European Journal of Operations Research, 8, pp Smith, B. M. and A. Wren (988). A bus crew scheduling system using a set covering formulation. Transportation Research, 22A, pp Willers, W. P., L. G. Proll and A. Wren (995). A dual strategy for solving the linear programming relaxation of a driver scheduling system. Annals of Operations Research, 58, pp Wren, A. and B. M. Smith (988). Experiences with a crew scheduling system based on set covering. In: Computer-Aided Transit Scheduling (J. R. Daduna and A. Wren eds.), pp Springer-Verlag, Berlin.