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Modeling and algorithmic development of a staff scheduling problem
European Journal of Operational Research 128 (2001) 558±569
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Theory and Methodology
Modeling and algorithmic development of a sta scheduling problem Subhash C. Sarin *, Sanjay Aggarwal Department of Industrial and Systems Engineering, College of Engineering Institute, Virginia Polytechnic and State University, 250 New Engineering Building, Blacksburg, VA 24061, USA Received 30 September 1997; accepted 2 September 1999
Abstract The problem of scheduling workers at a hub of a trucking system for the stripping and loading of the trucks and the determination of the optimal number of workers is a dicult problem. The trucks arrive at the facility at dierent (but known) times and may have their own scheduled departure times. This problem is like a set partitioning problem but with a side constraint pertaining to the dynamic arrival of the trucks. We develop a procedure to solve this problem that is based on the column generation technique, and the solution of a set covering problem to obtain the integer solution. The performance of the procedure is demonstrated by applying it to the real-life data obtained from a trucking company. Its eectiveness is illustrated by comparing it with a lower bound and a well-known heuristic procedure. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Sta; Mathematical programming
1. Problem statement The problem addressed in this paper is that of scheduling workers at a hub of a trucking system. The trucks from dierent places arrive at the hub. The items on these trucks are then stripped and loaded onto the trucks going to intended destinations. The trucks arrive at the hub dynamically and information about the expected arrival time of each truck and its expected unloading and loading time is a known priori. The stripping and loading of the trucks is done manually. The problem is to schedule the trucks arrived during a shift to the minimum number of workers. Only those arrivals are considered that can be attended to within a shift.
*
Corresponding author. Tel.: +1-540-231-6656; fax: +1-540-231-3322. E-mail address: [email protected] (S.C. Sarin).
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 4 2 1 - X
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2. Model formulation Let there be m trucks to be stripped during a shift. Let aj be a vector of size m and be de®ned such that aij 1, if truck i is included in an assignment j (to a worker), and aij 0, otherwise. Thus, aj represents an assignment of the trucks to a worker. Let yj 1, if assignment j is selected for allocation to a worker during the shift, and yj 0, otherwise. The stang problem can thus be modelled as follows. Determine yj ; j 1; . . . ; N , so as to P1
N X yj Minimize j1
subject to N X j1
1 aj yj e;
yj 0 or 1; j 1; . . . ; N : Here, N represents all possible assignments of trucks for allocation to workers to perform the stripping and loading operations during a shift and e is a vector of 1Õs. Note that the number of assignments selected is equal to the number of workers as each selected assignment is allocated to only one worker. The constraints in (1) represent the fact that each truck is served by only one worker. The trucks are to be assigned in such a fashion that the total number of workers is minimized. This is a set partitioning problem where the trucks have to be partitioned into mutually exclusive sets so that a truck is served only by one worker while a worker could work on several trucks during the shift, albeit one-at-a-time. A key feature of this formulation is the use of vectors aj . These can be generated taking into consideration the prevailing conditions or constraints on the assignment of the workers to the trucks. One important consideration in this respect, for the problem on hand, is the dynamic arrival of the trucks and their scheduled departure (which is typically the end of the shift for all the trucks). These constitute the feasibility requirements for the generation of aj . An arbitrary assignment of trucks to the workers need not be feasible and thus this requirement restricts the number of aj Õs to be generated. As an illustration of this formulation, consider the case where four trucks (designated 1, 2, 3 and 4) arrive at a terminal. The arrival times of these four trucks are (0, 0, 1, 1.5) while the required processing times are (4, 5, 2, 2.5), respectively. Examples of several possible aj columns are:
In a1 , trucks 1 and 3 will be assigned to a worker while under a5 , the worker will work on trucks 3 and 4. The feasibility condition pertains to the fact that the worker assigned to an aj can only process the trucks after they become available and also the total assigned work can be ®nished within the shift time. Thus in a5 , the worker can begin work only at time 1, when the third truck becomes available. Also, if the shift is 8 h long, then a9 is not feasible. In the implementation of P1, only the feasible aj 's will be generated.
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The set partitioning problem has been studied in great detail since the mid 1960s. Inspite of its special form, the set partitioning problem has many applications. Some of these applications described in the literature include: airline ¯eet scheduling (Levin, 1969), truck routing (Balinski and Quandt, 1964; Clarke and Wright, 1964; Dantzig and Ramser, 1959; Gar®nkel and Nemhauser, 1969; Pierce, 1968), airline crew scheduling (Arabeyere et al., 1969; Kolner, 1966; Spitzer, 1961; Thiriez, 1969), information retrieval (Day, 1965), switching circuit design (Balinski, 1965; Cobham, 1962; Paul and Unger, 1959; Root, 1964), stock cutting (Pierce, 1970), assembly line balancing (Salveson, 1955), capital equipment decisions (Valenta, 1969), location of oshore drilling platforms (Daly and Spierer, 1969), facilities location problems (Revelle et al., 1970), political districting (Gar®nkel and Nemhauser, 1970; Wanger, 1968), coloring problems (Busacker and Saaty, 1965), symbolic logic (Cobham et al., 1961), and PERT-CPM (Cobham, 1962). The dierent procedures presented in the literature for the set partitioning problem include: heuristics (Cassidy and Bennett, 1975; Christo®des, 1974; Golden, 1975; Baker et al., 1979; Rubin, 1973), implicit enumeration (Balas, 1965; Pierce, 1968; Gar®nkel and Nemhauser, 1969; Pierce and Lasky, 1973; Marsten, 1974), simplex based cutting plane methods (Gomory, 1963), column generating algorithm which is different in concept from the column generation based method presented in this paper (Balas and Padberg, 1975), hybrid primal cutting plane/implicit enumeration method (Balas, 1975), set partitioning via node covering (Balas and Samuelson, 1974), and network ¯ow model (Moreland, 1966). The given problem can also be viewed as a bin packing problem in which the workers are like the bins and the trucks are like the items to be assigned to the bins. If we de®ne yj 1, if a worker j is used, and yj 0 otherwise; xij 1, if a truck i is assigned to the jth worker and xij 0, otherwise; then the bin packing formulation of the given problem is as follows: P2
min z
n X
yj
j
subject to m X wi xij 6 T :yj ; i1 n X
xij 1;
j 1; 2; . . . ; n;
i 1; . . . ; m;
i1
xij ; yj 0; 1;
i 1; . . . ; m; j 1; . . . ; n;
where n is the maximum possible number of workers, m the number of trucks, wi the time required to strip a truck i and to load its contents on another truck, and T is the total available time of a worker on a given day (same for all). In view of the example mentioned earlier, this formulation considers the available shift time a priori. However, the dynamic arrival of the trucks still must be considered during the implementation of this formulation on the problem at hand. The work on bin packing problems has been devoted mainly to the development of approximate algorithms. The most common ones are known as the next-®t (NF), ®rst-®t (FF) and best-®t (BF) algorithms. They generate solutions that are close to optimum. Worst case performance bounds for these are presented by Johnson (1974). Eilon and Christo®des (1971) presented a simple depth-®rst enumerative algorithm based on the ``best-®t decreasing'' branching strategy. On the other hand, Hung and Brown (1978) developed a branch-and-bound algorithm for the generalization of the bin packing problem in which the bins are allowed to have dierent capacities. Martello and Toth (1989) have proposed an algorithm based on a ``®rst-®t decreasing'' branching scheme.
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3. Development of a solution approach The proposed solution approach is based on the column generation method. In this method, the (0±1) requirements of the variables in P1 are relaxed and the resultant problem is solved as a linear program. The columns of this linear program are generated as and when they are needed. Recall that a column of the set partitioning problem on hand corresponds to an assignment of the trucks to a worker. If we represent the simplex multipliers of the relaxed P1 by p p1 ; . . . ; pm , then the column generation problem is to P3
maximize
m X
pi aij ÿ cj
i1
subject to m X
wi aij 6 T ;
j 1; . . . ; n;
i1
aij 0; 1;
i 1; . . . ; n;
where aij 's are the (0,1) entries in a column j, wi is the time required to unload and load truck i, and cj (the coecient of yj in the objective function of P1) 1. Not all trucks are assumed to be available at the start of a shift and can arrive at the hub at dierent times during the shift. This dynamic availability of the trucks must be taken into consideration while generating columns that satisfy the time availability constraint as mentioned earlier. Note that the problem (P3) is the knapsack problem with a side constraint corresponding to the dynamic availability of the trucks. Once the optimal solution to the relaxed version of (P1) is obtained, the optimal basis is used to obtain an integer solution. This is done by setting up another problem using the columns of the ®nal basis as follows. If a^j ; j 1; . . . ; m, are the columns corresponding to the ®nal basis, then the problem can be stated as follows: P4
minimize
m X
yj
j1
subject to m X a^j yj P e; j1
yj 0; 1;
j 1; . . . ; m:
The idea of this formulation is to select the minimum integer number of columns (workers) such that every truck is attended to at least once. Note that (P4) is the set covering problem and is solved as an integer problem. It is much smaller in size as compared to (P1). The overall scheme can now be summarized and is presented in Fig. 1. A LP code was developed to solve problem (P1) that also incorporated the procedure for generating columns (problem P3). Problem (P4) was solved using CPLEX. A key step of the overall scheme is the solution of the knapsack problem (P3) with a side constraint as mentioned earlier. The branch and bound based procedure due to Martello and Toth (1988), to solve the knapsack problem, was modi®ed to incorporate the truck availability (also referred to in the discussion as feasibility) constraint. The procedure begins by building an integer solution from the greedy solution of the continuous version of the knapsack problem as follows:
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Fig. 1. Solving the linear relaxation of the given problem.
0 6 aij 6 1;
i 1; . . . ; m:
If ` is the greatest index for which ai;j 1;
i 1; . . . ; `;
ai;j 0;
i ` 1; . . . ; m:
P`
i1
wi 6 T then
and
This solution is tested for truck availability. If the generated column is a feasible integer solution to the knapsack problem and satis®es the side constraint corresponding to the dynamic availability of the trucks, then it is pivoted into the simplex basis. If this solution is infeasible, a depth-®rst branch and bound search is then performed. A forward move is performed where an attempt is made to introduce the largest possible set of new elements into the current solution. The solution is updated by setting to 1 the aij values of the
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elements found by the forward move. The solution at this stage is tested for feasibility. If infeasible, a backtracking move is performed. This move consists of updating the current solution by setting to 0 the values of akj with k maxfqjaqj 1g. Then, if at least one of the elements following the kth element has a value of wj small enough to allow its introduction into the current solution after backtracking, the forward move is performed. The algorithm continues with feasibility checks at each stage. As soon as a column with feasible assignment is found, it is entered into the basis. A ¯ow diagram depicting the computer implementation of the proposed algorithm is shown in Fig. 2.
4. Results of implementation The algorithm presented above was coded in the C programming language and applied to the real-life data received from the hubs at two locations of a transportation company. The hub at location 1 is a small operation involving up to 25 trucks a day. The stripping and loading of the trucks is done during a single shift lasting 8±9 hours. The hub at location 2 handles on an average 200 trucks a day and works 24 hours a day divided into three shifts of 8 hours each. A sample data of one week (10/3/94±10/7/94) is presented in Table 1 for the hub at location 1. Note that the number of trucks handled in one day vary from 15 to 25 during the week. The arrival times of the trucks are given assuming the shift starting time to be time ``0''. The proposed algorithm was applied to two weeksÕ data. The results are presented in Table 2. The third column of this table presents the LP lower bound values for the number of workers, while the fourth and the ®fth columns present, respectively, the solution values obtained by the proposed method and the ®rst-®t bin packing heuristic. The proposed method obtains solutions equivalent to the lower bound in 8 out of the 10 cases. The problems were solved to optimality using an integer programming routine as well. The results are shown in column seven. The solutions obtained by the proposed method are found to be optimal in 9 out of the 10 cases. The computational times of the proposed method are presented in the sixth column. The results for the hub at location 2 are presented in Table 3. A work day at this hub is divided into three shifts of 8 hours each. The ®rst shift begins at 12:00 midnight. For the analysis, trucks to be processed during each shift are considered according to the following criteria: · trucks that arrive before the beginning of the ®rst shift are processed in the ®rst shift only, · trucks that arrive during any shift and require processing time more than the time remaining in that shift are processed in the next shift. These trucks are assumed to have arrived at time 0.00 for the next shift. Approximately three weeks of data were available for this hub. The proposed method generates solutions that are signi®cantly better than the bin packing heuristic. The integer optimal solutions were not obtained for the data from the hub at location 2 in view of the excessive computational time required. However, the percentage dierences between the LP lower bound and the solutions obtained by the proposed method are found to be in the range of 0±22%. Given that the lower bound can be signi®cantly away from the optimal solution (33%, according to the data in Table 2), the solutions obtained by the proposed method are close to optimum. This is further illustrated by the fact that the mean deviation from the lower bound is only 8.6% with a standard deviation of 5.6%. The median is at 8.3%. The last column depicts the computational times. The computer runs were made on an IBM PS/277 486DX2 machine. The computational time requirement generally increases with the size of the problem. However, the run times for various instances of the same problem size tend to deviate due to interactions between the availabilities of trucks, loading/unloading times and the value of T. The algorithm was also run on an IBM 750-P90 machine with Pentium processor and the run times in all the instances tried were approximately 1/3 of the times presented in Table 3.
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Fig. 2. Computer implementation of the algorithm.
a
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.42 0.00 0.00 2.42 0.00 0.00 0.00
0.75 0.75 1.17 1.42 1.75 1.08 1.17 0.75 0.83 0.58 1.58 1.50 2.08 1.17 0.67 1.92 1.75 0.92 1.42 2.17 2.00 0.67 3.00 2.17 2.25
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.93 2.43 2.43 3.06 4.15 4.42 4.67 4.35 4.67 5.75 4.07 5.75 5.00 5.50 4.15 4.35 4.42
Arrival time 0.42 0.75 1.75 0.83 1.25 2.83 0.42 1.17 0.42 1.17 0.67 1.25 0.50 1.33 0.47 0.92 1.00 0.67 1.83 1.00 0.50 0.17 1.92 1.25 0.92
Operating time
Day 2 (10/4/94)
Arrival time
Operating timea
Day 1 (10/3/94)
Operating time is given in hours.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Truck no.
Table 1 One weekÕs data from the hub at location 1
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Arrival time 0.92 1.42 0.58 1.17 0.75 0.50 0.92 1.00 1.67 1.42 1.17 0.17 0.75 0.50 1.58
Operating time
Day 3 (10/5/94)
0.00 0.00 0.00 0.00 0.50 0.50 2.17 2.85 2.85 4.08 4.08 4.48 4.48 4.75 4.75 4.95 4.15 6.00 4.65 4.65 7.48
Arrival time 3.83 4.00 0.83 0.58 1.25 0.67 1.08 0.67 1.50 1.08 1.17 1.17 1.50 0.58 0.58 0.83 2.08 0.50 0.92 1.83 0.50
Operating time
Day 4 (10/6/94)
0.00 0.00 0.00 1.25 0.00 2.50 2.57 1.25 4.50 2.50 2.50 0.00 2.33 4.50 2.57 1.92
Arrival time
0.33 1.17 2.75 0.92 3.00 0.58 1.17 0.75 1.50 1.42 1.83 4.37 2.25 1.17 2.83 1.92
Operating time
Day 5 (10/7/94)
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Table 2 Comparison of solution values (number of workers) obtained using LP lower bound, the proposed method and a bin packing heuristic method for the data from the hub at location number 1 Date
Trucks/shift
LP lower bound
Solution by the proposed method
Solution by the bin packing heuristic
Run time (min) of the proposed method
IP optimal solution
10/3/94 10/4/94 10/5/94 10/6/94 10/7/94 10/10/94 10/12/94 10/13/94 10/14/94 10/17/94
25 25 15 21 16 26 16 19 20 24
5 4 2 4 4 5 4 3 5 6
5 4 2 4 4 5 4 4 5 7
6 5 3 6 5 6 4 5 7 7
0:43 1:45 0:25 0:76 0:28 0:42 0:55 0:43 0:60 0:35
5 4 2 4 4 5 4 4 5 6
Table 3 Comparison of solution values (number of workers) obtained using LP lower bound, the proposed method and a bin packing heuristic method for the data from the hub at location number 2 Date
Shift
Truck/shift
LP lower bound
Solution by the proposed method
Solution by the bin packing heuristic
% Dierence between LB and the proposed method
10/1/94 10/1/94 10/1/94 10/3/94 10/3/94 10/3/94 10/4/94 10/4/94 10/4/94 10/5/94 10/5/94 10/5/94 10/6/94 10/6/94 10/6/94 10/7/94 10/7/94 10/7/94 10/8/94 10/8/94 10/8/94 10/10/94 10/10/94 10/10/94 10/11/94 10/11/94 10/11/94 10/12/94 10/12/94
a b c a b c a b c a b c a b c a b c a b c a b c a b c a b
53 72 37 26 24 42 67 90 45 80 86 51 71 91 32 77 70 37 55 57 10 32 37 48 74 84 42 91 70
18 21 14 8 9 8 22 28 13 29 28 11 23 31 8 27 26 9 18 16 4 8 10 13 9 23 24 32 25
19 24 15 8 9 9 23 33 13 30 31 13 25 32 9 27 29 9 19 18 4 9 11 14 11 24 26 34 28
26 25 16 11 10 11 32 36 13 41 32 13 29 36 10 35 36 10 24 23 5 13 13 17 11 34 26 44 28
6 14 7 0 0 13 5 18 0 3 11 18 9 3 13 0 12 0 6 13 0 13 10 8 22 4 8 6 12
Run time (min)
5:35 9:68 0:60 0:53 0:35 2:90 0:36 36:96 0:61 27:05 17:36 2:46 22:96 15:38 0:71 21:23 3:86 2:50 3:50 2:00 0:21 1:31 1:70 3:01 24:65 19:48 1:96 33:03 6:86
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Table 3 (Continued) Date
Shift
Truck/shift
LP lower bound
Solution by the proposed method
Solution by the bin packing heuristic
% Dierence between LB and the proposed method
10/12/94 10/13/94 10/13/94 10/13/94 10/14/94 10/14/94 10/14/94 10/15/94 10/15/94 10/15/94 10/17/94 10/17/94 10/17/94 10/18/94 10/18/94 10/18/94 10/19/94 10/19/94 10/19/94 10/20/94 10/20/94 10/20/94 10/21/94 10/21/94 10/21/94 10/22/94 10/22/94 10/22/94
c a b c a b c a b c a b c a b c a b c a b c a b c a b c
54 78 89 53 83 92 46 69 67 26 25 23 50 61 99 46 77 93 42 75 85 56 77 76 45 64 55 28
14 30 25 12 30 33 15 29 18 6 9 6 12 19 31 9 27 29 10 30 28 12 26 27 11 26 17 11
16 31 26 14 33 34 17 30 20 7 9 7 13 20 34 11 29 32 11 32 30 14 29 28 12 28 18 12
17 34 28 17 39 38 18 37 26 10 11 8 16 27 37 11 33 34 13 37 35 17 33 36 13 32 21 13
14 3 4 17 10 3 13 3 11 17 0 17 8 5 10 22 7 10 10 7 7 17 12 4 9 8 6 9
Run time (min)
7:83 9:30 34:13 7:83 18:10 23:56 0:83 7:86 4:91 1:03 13:43 0:41 7:16 12:33 10:00 2:23 20:13 49:66 1:28 13:90 5:93 9:60 13:53 3:78 1:28 4:01 3:83 0:38
5. Concluding remarks In this paper, we develop an algorithm to solve a sta scheduling problem and apply it to the real-life data obtained from a transportation company serving up to 200 trucks a day. This problem can be viewed as a set partitioning problem with a side constraint. The proposed algorithm is based on the column generation method and solution of a set covering problem de®ned over the LP optimal solution of the relaxed problem. The solutions obtained by the proposed method are compared to the LP lower bound and those obtained by a bin packing heuristic procedure, and are shown to be signi®cantly better than those obtained using the bin-packing heuristic and are close to optimum. The computational time required by the proposed procedure is also quite reasonable. The proposed solution procedure also suggests an alternate way to solve the set partitioning problem.
Acknowledgements The authors thank the referees for several insightful comments which helped in further improving the paperÕs exposition.
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www.elsevier.com/locate/dsw
Theory and Methodology
Modeling and algorithmic development of a sta scheduling problem Subhash C. Sarin *, Sanjay Aggarwal Department of Industrial and Systems Engineering, College of Engineering Institute, Virginia Polytechnic and State University, 250 New Engineering Building, Blacksburg, VA 24061, USA Received 30 September 1997; accepted 2 September 1999
Abstract The problem of scheduling workers at a hub of a trucking system for the stripping and loading of the trucks and the determination of the optimal number of workers is a dicult problem. The trucks arrive at the facility at dierent (but known) times and may have their own scheduled departure times. This problem is like a set partitioning problem but with a side constraint pertaining to the dynamic arrival of the trucks. We develop a procedure to solve this problem that is based on the column generation technique, and the solution of a set covering problem to obtain the integer solution. The performance of the procedure is demonstrated by applying it to the real-life data obtained from a trucking company. Its eectiveness is illustrated by comparing it with a lower bound and a well-known heuristic procedure. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Sta; Mathematical programming
1. Problem statement The problem addressed in this paper is that of scheduling workers at a hub of a trucking system. The trucks from dierent places arrive at the hub. The items on these trucks are then stripped and loaded onto the trucks going to intended destinations. The trucks arrive at the hub dynamically and information about the expected arrival time of each truck and its expected unloading and loading time is a known priori. The stripping and loading of the trucks is done manually. The problem is to schedule the trucks arrived during a shift to the minimum number of workers. Only those arrivals are considered that can be attended to within a shift.
*
Corresponding author. Tel.: +1-540-231-6656; fax: +1-540-231-3322. E-mail address: [email protected] (S.C. Sarin).
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 4 2 1 - X
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2. Model formulation Let there be m trucks to be stripped during a shift. Let aj be a vector of size m and be de®ned such that aij 1, if truck i is included in an assignment j (to a worker), and aij 0, otherwise. Thus, aj represents an assignment of the trucks to a worker. Let yj 1, if assignment j is selected for allocation to a worker during the shift, and yj 0, otherwise. The stang problem can thus be modelled as follows. Determine yj ; j 1; . . . ; N , so as to P1
N X yj Minimize j1
subject to N X j1
1 aj yj e;
yj 0 or 1; j 1; . . . ; N : Here, N represents all possible assignments of trucks for allocation to workers to perform the stripping and loading operations during a shift and e is a vector of 1Õs. Note that the number of assignments selected is equal to the number of workers as each selected assignment is allocated to only one worker. The constraints in (1) represent the fact that each truck is served by only one worker. The trucks are to be assigned in such a fashion that the total number of workers is minimized. This is a set partitioning problem where the trucks have to be partitioned into mutually exclusive sets so that a truck is served only by one worker while a worker could work on several trucks during the shift, albeit one-at-a-time. A key feature of this formulation is the use of vectors aj . These can be generated taking into consideration the prevailing conditions or constraints on the assignment of the workers to the trucks. One important consideration in this respect, for the problem on hand, is the dynamic arrival of the trucks and their scheduled departure (which is typically the end of the shift for all the trucks). These constitute the feasibility requirements for the generation of aj . An arbitrary assignment of trucks to the workers need not be feasible and thus this requirement restricts the number of aj Õs to be generated. As an illustration of this formulation, consider the case where four trucks (designated 1, 2, 3 and 4) arrive at a terminal. The arrival times of these four trucks are (0, 0, 1, 1.5) while the required processing times are (4, 5, 2, 2.5), respectively. Examples of several possible aj columns are:
In a1 , trucks 1 and 3 will be assigned to a worker while under a5 , the worker will work on trucks 3 and 4. The feasibility condition pertains to the fact that the worker assigned to an aj can only process the trucks after they become available and also the total assigned work can be ®nished within the shift time. Thus in a5 , the worker can begin work only at time 1, when the third truck becomes available. Also, if the shift is 8 h long, then a9 is not feasible. In the implementation of P1, only the feasible aj 's will be generated.
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The set partitioning problem has been studied in great detail since the mid 1960s. Inspite of its special form, the set partitioning problem has many applications. Some of these applications described in the literature include: airline ¯eet scheduling (Levin, 1969), truck routing (Balinski and Quandt, 1964; Clarke and Wright, 1964; Dantzig and Ramser, 1959; Gar®nkel and Nemhauser, 1969; Pierce, 1968), airline crew scheduling (Arabeyere et al., 1969; Kolner, 1966; Spitzer, 1961; Thiriez, 1969), information retrieval (Day, 1965), switching circuit design (Balinski, 1965; Cobham, 1962; Paul and Unger, 1959; Root, 1964), stock cutting (Pierce, 1970), assembly line balancing (Salveson, 1955), capital equipment decisions (Valenta, 1969), location of oshore drilling platforms (Daly and Spierer, 1969), facilities location problems (Revelle et al., 1970), political districting (Gar®nkel and Nemhauser, 1970; Wanger, 1968), coloring problems (Busacker and Saaty, 1965), symbolic logic (Cobham et al., 1961), and PERT-CPM (Cobham, 1962). The dierent procedures presented in the literature for the set partitioning problem include: heuristics (Cassidy and Bennett, 1975; Christo®des, 1974; Golden, 1975; Baker et al., 1979; Rubin, 1973), implicit enumeration (Balas, 1965; Pierce, 1968; Gar®nkel and Nemhauser, 1969; Pierce and Lasky, 1973; Marsten, 1974), simplex based cutting plane methods (Gomory, 1963), column generating algorithm which is different in concept from the column generation based method presented in this paper (Balas and Padberg, 1975), hybrid primal cutting plane/implicit enumeration method (Balas, 1975), set partitioning via node covering (Balas and Samuelson, 1974), and network ¯ow model (Moreland, 1966). The given problem can also be viewed as a bin packing problem in which the workers are like the bins and the trucks are like the items to be assigned to the bins. If we de®ne yj 1, if a worker j is used, and yj 0 otherwise; xij 1, if a truck i is assigned to the jth worker and xij 0, otherwise; then the bin packing formulation of the given problem is as follows: P2
min z
n X
yj
j
subject to m X wi xij 6 T :yj ; i1 n X
xij 1;
j 1; 2; . . . ; n;
i 1; . . . ; m;
i1
xij ; yj 0; 1;
i 1; . . . ; m; j 1; . . . ; n;
where n is the maximum possible number of workers, m the number of trucks, wi the time required to strip a truck i and to load its contents on another truck, and T is the total available time of a worker on a given day (same for all). In view of the example mentioned earlier, this formulation considers the available shift time a priori. However, the dynamic arrival of the trucks still must be considered during the implementation of this formulation on the problem at hand. The work on bin packing problems has been devoted mainly to the development of approximate algorithms. The most common ones are known as the next-®t (NF), ®rst-®t (FF) and best-®t (BF) algorithms. They generate solutions that are close to optimum. Worst case performance bounds for these are presented by Johnson (1974). Eilon and Christo®des (1971) presented a simple depth-®rst enumerative algorithm based on the ``best-®t decreasing'' branching strategy. On the other hand, Hung and Brown (1978) developed a branch-and-bound algorithm for the generalization of the bin packing problem in which the bins are allowed to have dierent capacities. Martello and Toth (1989) have proposed an algorithm based on a ``®rst-®t decreasing'' branching scheme.
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3. Development of a solution approach The proposed solution approach is based on the column generation method. In this method, the (0±1) requirements of the variables in P1 are relaxed and the resultant problem is solved as a linear program. The columns of this linear program are generated as and when they are needed. Recall that a column of the set partitioning problem on hand corresponds to an assignment of the trucks to a worker. If we represent the simplex multipliers of the relaxed P1 by p p1 ; . . . ; pm , then the column generation problem is to P3
maximize
m X
pi aij ÿ cj
i1
subject to m X
wi aij 6 T ;
j 1; . . . ; n;
i1
aij 0; 1;
i 1; . . . ; n;
where aij 's are the (0,1) entries in a column j, wi is the time required to unload and load truck i, and cj (the coecient of yj in the objective function of P1) 1. Not all trucks are assumed to be available at the start of a shift and can arrive at the hub at dierent times during the shift. This dynamic availability of the trucks must be taken into consideration while generating columns that satisfy the time availability constraint as mentioned earlier. Note that the problem (P3) is the knapsack problem with a side constraint corresponding to the dynamic availability of the trucks. Once the optimal solution to the relaxed version of (P1) is obtained, the optimal basis is used to obtain an integer solution. This is done by setting up another problem using the columns of the ®nal basis as follows. If a^j ; j 1; . . . ; m, are the columns corresponding to the ®nal basis, then the problem can be stated as follows: P4
minimize
m X
yj
j1
subject to m X a^j yj P e; j1
yj 0; 1;
j 1; . . . ; m:
The idea of this formulation is to select the minimum integer number of columns (workers) such that every truck is attended to at least once. Note that (P4) is the set covering problem and is solved as an integer problem. It is much smaller in size as compared to (P1). The overall scheme can now be summarized and is presented in Fig. 1. A LP code was developed to solve problem (P1) that also incorporated the procedure for generating columns (problem P3). Problem (P4) was solved using CPLEX. A key step of the overall scheme is the solution of the knapsack problem (P3) with a side constraint as mentioned earlier. The branch and bound based procedure due to Martello and Toth (1988), to solve the knapsack problem, was modi®ed to incorporate the truck availability (also referred to in the discussion as feasibility) constraint. The procedure begins by building an integer solution from the greedy solution of the continuous version of the knapsack problem as follows:
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Fig. 1. Solving the linear relaxation of the given problem.
0 6 aij 6 1;
i 1; . . . ; m:
If ` is the greatest index for which ai;j 1;
i 1; . . . ; `;
ai;j 0;
i ` 1; . . . ; m:
P`
i1
wi 6 T then
and
This solution is tested for truck availability. If the generated column is a feasible integer solution to the knapsack problem and satis®es the side constraint corresponding to the dynamic availability of the trucks, then it is pivoted into the simplex basis. If this solution is infeasible, a depth-®rst branch and bound search is then performed. A forward move is performed where an attempt is made to introduce the largest possible set of new elements into the current solution. The solution is updated by setting to 1 the aij values of the
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elements found by the forward move. The solution at this stage is tested for feasibility. If infeasible, a backtracking move is performed. This move consists of updating the current solution by setting to 0 the values of akj with k maxfqjaqj 1g. Then, if at least one of the elements following the kth element has a value of wj small enough to allow its introduction into the current solution after backtracking, the forward move is performed. The algorithm continues with feasibility checks at each stage. As soon as a column with feasible assignment is found, it is entered into the basis. A ¯ow diagram depicting the computer implementation of the proposed algorithm is shown in Fig. 2.
4. Results of implementation The algorithm presented above was coded in the C programming language and applied to the real-life data received from the hubs at two locations of a transportation company. The hub at location 1 is a small operation involving up to 25 trucks a day. The stripping and loading of the trucks is done during a single shift lasting 8±9 hours. The hub at location 2 handles on an average 200 trucks a day and works 24 hours a day divided into three shifts of 8 hours each. A sample data of one week (10/3/94±10/7/94) is presented in Table 1 for the hub at location 1. Note that the number of trucks handled in one day vary from 15 to 25 during the week. The arrival times of the trucks are given assuming the shift starting time to be time ``0''. The proposed algorithm was applied to two weeksÕ data. The results are presented in Table 2. The third column of this table presents the LP lower bound values for the number of workers, while the fourth and the ®fth columns present, respectively, the solution values obtained by the proposed method and the ®rst-®t bin packing heuristic. The proposed method obtains solutions equivalent to the lower bound in 8 out of the 10 cases. The problems were solved to optimality using an integer programming routine as well. The results are shown in column seven. The solutions obtained by the proposed method are found to be optimal in 9 out of the 10 cases. The computational times of the proposed method are presented in the sixth column. The results for the hub at location 2 are presented in Table 3. A work day at this hub is divided into three shifts of 8 hours each. The ®rst shift begins at 12:00 midnight. For the analysis, trucks to be processed during each shift are considered according to the following criteria: · trucks that arrive before the beginning of the ®rst shift are processed in the ®rst shift only, · trucks that arrive during any shift and require processing time more than the time remaining in that shift are processed in the next shift. These trucks are assumed to have arrived at time 0.00 for the next shift. Approximately three weeks of data were available for this hub. The proposed method generates solutions that are signi®cantly better than the bin packing heuristic. The integer optimal solutions were not obtained for the data from the hub at location 2 in view of the excessive computational time required. However, the percentage dierences between the LP lower bound and the solutions obtained by the proposed method are found to be in the range of 0±22%. Given that the lower bound can be signi®cantly away from the optimal solution (33%, according to the data in Table 2), the solutions obtained by the proposed method are close to optimum. This is further illustrated by the fact that the mean deviation from the lower bound is only 8.6% with a standard deviation of 5.6%. The median is at 8.3%. The last column depicts the computational times. The computer runs were made on an IBM PS/277 486DX2 machine. The computational time requirement generally increases with the size of the problem. However, the run times for various instances of the same problem size tend to deviate due to interactions between the availabilities of trucks, loading/unloading times and the value of T. The algorithm was also run on an IBM 750-P90 machine with Pentium processor and the run times in all the instances tried were approximately 1/3 of the times presented in Table 3.
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Fig. 2. Computer implementation of the algorithm.
a
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.42 0.00 0.00 2.42 0.00 0.00 0.00
0.75 0.75 1.17 1.42 1.75 1.08 1.17 0.75 0.83 0.58 1.58 1.50 2.08 1.17 0.67 1.92 1.75 0.92 1.42 2.17 2.00 0.67 3.00 2.17 2.25
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.93 2.43 2.43 3.06 4.15 4.42 4.67 4.35 4.67 5.75 4.07 5.75 5.00 5.50 4.15 4.35 4.42
Arrival time 0.42 0.75 1.75 0.83 1.25 2.83 0.42 1.17 0.42 1.17 0.67 1.25 0.50 1.33 0.47 0.92 1.00 0.67 1.83 1.00 0.50 0.17 1.92 1.25 0.92
Operating time
Day 2 (10/4/94)
Arrival time
Operating timea
Day 1 (10/3/94)
Operating time is given in hours.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Truck no.
Table 1 One weekÕs data from the hub at location 1
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Arrival time 0.92 1.42 0.58 1.17 0.75 0.50 0.92 1.00 1.67 1.42 1.17 0.17 0.75 0.50 1.58
Operating time
Day 3 (10/5/94)
0.00 0.00 0.00 0.00 0.50 0.50 2.17 2.85 2.85 4.08 4.08 4.48 4.48 4.75 4.75 4.95 4.15 6.00 4.65 4.65 7.48
Arrival time 3.83 4.00 0.83 0.58 1.25 0.67 1.08 0.67 1.50 1.08 1.17 1.17 1.50 0.58 0.58 0.83 2.08 0.50 0.92 1.83 0.50
Operating time
Day 4 (10/6/94)
0.00 0.00 0.00 1.25 0.00 2.50 2.57 1.25 4.50 2.50 2.50 0.00 2.33 4.50 2.57 1.92
Arrival time
0.33 1.17 2.75 0.92 3.00 0.58 1.17 0.75 1.50 1.42 1.83 4.37 2.25 1.17 2.83 1.92
Operating time
Day 5 (10/7/94)
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Table 2 Comparison of solution values (number of workers) obtained using LP lower bound, the proposed method and a bin packing heuristic method for the data from the hub at location number 1 Date
Trucks/shift
LP lower bound
Solution by the proposed method
Solution by the bin packing heuristic
Run time (min) of the proposed method
IP optimal solution
10/3/94 10/4/94 10/5/94 10/6/94 10/7/94 10/10/94 10/12/94 10/13/94 10/14/94 10/17/94
25 25 15 21 16 26 16 19 20 24
5 4 2 4 4 5 4 3 5 6
5 4 2 4 4 5 4 4 5 7
6 5 3 6 5 6 4 5 7 7
0:43 1:45 0:25 0:76 0:28 0:42 0:55 0:43 0:60 0:35
5 4 2 4 4 5 4 4 5 6
Table 3 Comparison of solution values (number of workers) obtained using LP lower bound, the proposed method and a bin packing heuristic method for the data from the hub at location number 2 Date
Shift
Truck/shift
LP lower bound
Solution by the proposed method
Solution by the bin packing heuristic
% Dierence between LB and the proposed method
10/1/94 10/1/94 10/1/94 10/3/94 10/3/94 10/3/94 10/4/94 10/4/94 10/4/94 10/5/94 10/5/94 10/5/94 10/6/94 10/6/94 10/6/94 10/7/94 10/7/94 10/7/94 10/8/94 10/8/94 10/8/94 10/10/94 10/10/94 10/10/94 10/11/94 10/11/94 10/11/94 10/12/94 10/12/94
a b c a b c a b c a b c a b c a b c a b c a b c a b c a b
53 72 37 26 24 42 67 90 45 80 86 51 71 91 32 77 70 37 55 57 10 32 37 48 74 84 42 91 70
18 21 14 8 9 8 22 28 13 29 28 11 23 31 8 27 26 9 18 16 4 8 10 13 9 23 24 32 25
19 24 15 8 9 9 23 33 13 30 31 13 25 32 9 27 29 9 19 18 4 9 11 14 11 24 26 34 28
26 25 16 11 10 11 32 36 13 41 32 13 29 36 10 35 36 10 24 23 5 13 13 17 11 34 26 44 28
6 14 7 0 0 13 5 18 0 3 11 18 9 3 13 0 12 0 6 13 0 13 10 8 22 4 8 6 12
Run time (min)
5:35 9:68 0:60 0:53 0:35 2:90 0:36 36:96 0:61 27:05 17:36 2:46 22:96 15:38 0:71 21:23 3:86 2:50 3:50 2:00 0:21 1:31 1:70 3:01 24:65 19:48 1:96 33:03 6:86
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Table 3 (Continued) Date
Shift
Truck/shift
LP lower bound
Solution by the proposed method
Solution by the bin packing heuristic
% Dierence between LB and the proposed method
10/12/94 10/13/94 10/13/94 10/13/94 10/14/94 10/14/94 10/14/94 10/15/94 10/15/94 10/15/94 10/17/94 10/17/94 10/17/94 10/18/94 10/18/94 10/18/94 10/19/94 10/19/94 10/19/94 10/20/94 10/20/94 10/20/94 10/21/94 10/21/94 10/21/94 10/22/94 10/22/94 10/22/94
c a b c a b c a b c a b c a b c a b c a b c a b c a b c
54 78 89 53 83 92 46 69 67 26 25 23 50 61 99 46 77 93 42 75 85 56 77 76 45 64 55 28
14 30 25 12 30 33 15 29 18 6 9 6 12 19 31 9 27 29 10 30 28 12 26 27 11 26 17 11
16 31 26 14 33 34 17 30 20 7 9 7 13 20 34 11 29 32 11 32 30 14 29 28 12 28 18 12
17 34 28 17 39 38 18 37 26 10 11 8 16 27 37 11 33 34 13 37 35 17 33 36 13 32 21 13
14 3 4 17 10 3 13 3 11 17 0 17 8 5 10 22 7 10 10 7 7 17 12 4 9 8 6 9
Run time (min)
7:83 9:30 34:13 7:83 18:10 23:56 0:83 7:86 4:91 1:03 13:43 0:41 7:16 12:33 10:00 2:23 20:13 49:66 1:28 13:90 5:93 9:60 13:53 3:78 1:28 4:01 3:83 0:38
5. Concluding remarks In this paper, we develop an algorithm to solve a sta scheduling problem and apply it to the real-life data obtained from a transportation company serving up to 200 trucks a day. This problem can be viewed as a set partitioning problem with a side constraint. The proposed algorithm is based on the column generation method and solution of a set covering problem de®ned over the LP optimal solution of the relaxed problem. The solutions obtained by the proposed method are compared to the LP lower bound and those obtained by a bin packing heuristic procedure, and are shown to be signi®cantly better than those obtained using the bin-packing heuristic and are close to optimum. The computational time required by the proposed procedure is also quite reasonable. The proposed solution procedure also suggests an alternate way to solve the set partitioning problem.
Acknowledgements The authors thank the referees for several insightful comments which helped in further improving the paperÕs exposition.
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