PAPER

An Algorithm for Crew Route Scheduling Focusing on Manpower Efficiency Takashi SAKAGUCHI Satoshi KATO Laboratory Head, Researcher, Transport Operation Systems Laboratory, Signalling and Transport Information Technology Division Naoto FUKUMURA Senior Researcher, Signalling and Transport Information Technology Division Ideally, railway crew planning rosters should draw as little as possible on available crew resources. Methods of crew route scheduling with the evaluation criterion of the total number of required days have been studied because we could not figure the number of required crews out exactly at the time of setting it up. However this method for compiling crew route schedules has made roster scheduling more difficult. This paper therefore proposes a crew route scheduling method with the evaluation criterion of the estimated number of required crews and shows its effectivity for this problem. Keywords: crew route scheduling, mathematical programming, column generation technique 1. Introduction

2. Crew Operation Schedule

Crew operation schedules are just one of a number of schedules required in railway transport scheduling alongside timetabling and vehicle operation schedule. Crew operation schedules are composed of crew route schedules and crew roster scheduling to ensure that a crew is assigned each train scheduled on the timetable. Efficient crew route/roster schedules are difficult for planners to prepare, given the many different conditions which must be taken into account. Efficient schedule in this case means one which uses fewer crew resources, without imposing an excessive workload. The difficulty in the task of ensuring scheduling efficiency arises from the fact that not only does a single shift have to be efficient in terms of allocated staff, successive shifts must also be efficient in terms of overall staff rosters, i.e. the required number of staff should not vary too wildly between shifts, there should not be overly long breaks, and dead-heading should be kept at a minimum. In other words, it is necessary to not only focus on local efficiency, but also on efficiency of the final outcome (crew roster schedule). To date crew route scheduling algorithms including criteria for evaluating the number of shift-days (i.e. total number of the required days for each shift) have been studied [1] and refined to such a level which could solve certain practical aspects thanks to advances in the mathematical optimization technology and computer performance. However almost no studies have been made for schedules evaluated with a criterion to ensure optimum (minimum) number of crew members. This paper therefore proposes a crew scheduling method including a criterion for estimating the required number of crew members based on a mathematical planning method. This method was then validated as an effective manpower scheduling tool.

This paper first describes conditions underlying the crew route schedule and crew roster schedule in order to clarify their definitions. In the following passages we deal with the drivers as the kind of crews, which scheduling conditions are more complex and versatile.

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2.1 Crew Route Schedule A crew shift identified by the shift number consists of a certain number of trips and additional works along with breaks between them, and contents of all shifts are different from each other. Figure 1 is an example of the following crew shift: one member of the crew in the shift travels from Station A to Station B as a passenger. This person then works as the driver in the crew on Train 2 travelling from Station B to Station C. The same crew continues working on the same vehicle (now called“train 3”) travelling from Station C back to Station A. The same crew then accompanies the vehicle to the parking yard in Station A. The crew route schedule is necessary to satisfy the following conditions. (1) The crew member in the shift should begin and end his/her shift at the nearest station to their home base. In many cases, crew route schedule includes a number of crew bases. (2) A shift is composed of one travel period, i.e. a“trip” (either as a driver, i.e. driving trip or as a passenger, i.e. a passenger trip) and a“break”. A driving trip also includes shunting operations. (3) Each“trip”starts from the station where the previous one ended. (4) Crew change-over can only occur at certain stations. We call such a station“crew-change station”, and we refer to the part of the train time table separated by crew-change stations as a“leg”. (5) The whole of the legs covered by the crew route schedule are fixed and given by the diagram, and the legs 197

Sta. B

Sta. A

Sta. C

trip as passenger

(parking of train)

6.00

12.00

Shift 1

3 day Shift 2 4th day

18.00

24.00

Shift 2

rd

Train 1 (crew change)

0.00 1st day 2nd day

Train 2

Train 3

trip as driver

Day off

Shift 3

5th day 6th day Shift 3 7th day 8th day

Day off Day off

Fig. 2 Example of a Roster

Fig. 1 Example of a Shift that are able to constitute shifts depend on the home bases. (6) For any leg, only one shift that includes it as driving trip is exist. (7) There is a regulation on the limit of real working hours for each shift, especially that for midnight driving shifts which include several trips between 10 p.m. and 5 a.m. more than 2 hours in total. (8) There is a regulation on the limit of working hours (= period from the start time to the finish time) for each shift. (9) There is a regulation on the limits of time and distance for each driving trip. (10) There is a regulation on the lower limit of the period between trips. 2.2 Crew Roster Schedule

hours. (11) The ratio of days on midnight driving shifts to the whole days in the roster does not exceed a specified value. 3. Mathematical Programming Based Formulae and Solution Method 3.1 Crew Route Scheduling formulae A crew route schedule is defined as a problem that finds a subset of the set of potential shifts satisfying condition 2.1(6) with the minimum total cost. This problem can be formulated based on the mathematical programming model known as“set covering problem”as follows: minimize. s.t.

The crew roster schedule represents a set of shift patterns covering several weeks. Figure 2 is an example of an 8 day rotating shift pattern. This type of shift pattern is known as a roster. Each crew works along certain roster repeatedly. The number of days of a roster is equal to the number of crews required to carry out the roster. A crew roster schedule must satisfy the following conditions. (1) The roster is composed of the sequences of shifts and day-offs for one or a couple of days. (2) Each shift is included in only one roster and appears only once in it. (3) The average of real working hours of the roster does not exceed the specified limit. (4) The roster includes a certain number of day-offs for every fixed periods. (5) There is a regulation on the lower bound of the length of the break period between shifts, which depends on the former shift. We call such break“inter-shift break.” (6) There is a regulation on the lower bound of the length of the break period between sequences of shifts. We call such break“inter-sequence break.” (7) The finish time of the shift just before the day-off is not later than specified time. (8) The start time of the shift just after the day-off is not earlier than specified time. (9) The number of consecutive midnight driving shifts does not exceed the specified limit. (10) An inter-shift break period next to a couple of consecutive midnight driving shifts is not less than specified 198

∑ ∑ c x + ∑t u

(1)

∑∑a x

(2)

k∈K j∈J k

k∈K j∈J k

k j

k j

i i

i∈I

k ij

k j

− ui = 1, ∀i ∈ I

x kj ∈ {0, 1} ,

∀j ∈ J k , ∀k ∈ K

(3)

ui ≥ 0,

∀i ∈ I

(4)

where I = set of legs in a diagram; K = set of crew bases; Jk = set of potential shifts from crew base k (a potential shift j that belongs to crew base k is described as“shift (j, k)” in the text); aijk = 1 (it means that the potential shift (j, k) includes the leg i) or = 0 (It does not includes the leg i); cjk = the estimated cost of the potential shift (j, k); ti = the estimated cost of the leg i; xjk = a variable that is 1 if the potential shift (j, k) is adopted in the schedule, and is 0 if not adopted; and ui = number of passenger trips during leg i. The first term of (1) represents the total cost for all the shifts in the crew route schedule. The second term of (1) represents the total cost of passenger trip hours. Equation (2) represents the condition 2.1(6). Expression (3) and (4) represent domains of the variables. 3.2 Solution Method for Crew Route Scheduling The crew route scheduling problems are formulated on the assumption that all the potential shifts are given. However the enumeration of all of them is impossible in practice. Therefore, for the sake of efficiency only useful QR of RTRI, Vol. 54, No. 4, Nov. 2013

shifts are listed, using an optimization method called the “column generation method [2].”In the column generation method, a network whose nodes correspond to legs and whose arcs correspond to connections between legs satisfying conditions 2.1(2), (3), (4), (5) and (10) as part of a shift are set. As a result, the problem of finding useful potential shifts is reduced to the problem of searching shortest paths corresponding to the useful potential to satisfy conditions 2.1(1), (7), (8), and (9) on the network. Column generation gives a set of poptential shifts by performed the network construction and the search for useful potential shifts on it by turns. Finally a mixed integer programming problem represented at section 3.1 is calculated by using any commercial mathematical programming solver. 3.3 Formulation of the Crew Roster Scheduling The aim of this study is to make it possible to draw up efficient crew route schedules. However, the crew roster schedules corresponding to them have to be checked to estimate their efficiency. In order to carry out this estimation, a particular kind of crew roster schedule was defined where each crew base only had one associated roster (or shift pattern). In addition, it was assumed that condition 2.2(6) is satisfied whenever both conditions 2.2(7) and 2.2(8) were met which meant that the order of shift sequences on the formulation. The Crew Roster Scheduling redefined above is a problem that finds a subset of the set of potential shift sequences satisfying condition 2.2(2) with the minimum total number of days of the sequences taking account of the number of additional days-off. This problem can be formulated based on the mathematical programming model named set partitioning the problem as follows: minimize. s.t.

∑d

n∈N k

k n

∑b

n∈N k

(5)

ynk

k mn

ynk = 1,

∑ (q r − v

n∈N k

∑ (d

n∈N k

k n 1

k n

∀m ∈ M k

(6)

) ynk ≥ 0

(7)

r − snk ) ynk ≥ 0

(8)

k n 2

ynk ∈ {0, 1} ,

∀n ∈ N k

(9)

k where M = number of shifts associated to crew base k (a shift m that belongs to crew base k is described as“shift (m, k)”in the text); Nk = number of shift sequences in Mk (a shift sequence n that belongs to crew base k is described as“sequence (n, k)” in the text); k bmn = 1 (means that the sequence (n, k) includes the shift (m) or = 0 (otherwise); dnk = number of days in sequences (n, k), including statutory rest days.; qnk = number of days of in sequences (n, k), excluding statutory rest days; k n n = total real working hours of shifts included in the sequence (n, k); snk = number of midnight driving shifts in the sequence (n, k); r1 = maximum average real working hours; r 2 = maximum ratio of the number of midnight driving

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shifts to the roster; and ynk = 1 if the sequence (n, k) is adopted as an element of the crew roster schedule, or 0 if otherwise. Expression (5) represents an objective function to reduce to a minimum the required number of crew. Equation (6) represents the constraint of set-partitioning corresponding to the condition 2.2(1). Each of (7) and (8) corresponds to each of conditions 2.2(3) and 2.2(11). Expression (9) defines the domain of the variables of the problem. 3.4 Solution Method for the Crew Roster Scheduling Just like the crew route scheduling, for crew roster scheduling formulae it is assumed that all potential shift sequences are given. And the column generation method is applied too. A network whose nodes represent shifts in the crew route schedule is set up. The arcs in this network represent connections between shifts satisfying conditions 2.2(5), (6), (7), and (8). In order to identify the useful shift sequences by the shortest path search, a search is made for paths satisfying conditions 2.2(4), (9), and (10). Finally an integer programming problem represented in Section 3.3 is calculated for the set of shift sequences thus given. 4. Manpower Efficiency Algorithm 4.1 Manpower Efficiency Efficiency of crew route schedules (i.e. index for the value of the expression (5)) is difficult to determine directly. However some of the constraints linked to crew shifts which are used in the crew roster scheduling such as real working hours will affect the result of crew roster scheduling. For example, excessively long working hours will make it more difficult to ensure breaks between shifts. Such attributes are therefore referred to as efficiency factors. Efficiency factors in the crew route scheduling were therefore controlled in order to ensure manpower efficiency. Because it is not possible to decide suitable values for each efficiency factor, and because they appear to depend on crew bases, another method was developed which integrates efficiency factors into the evaluation function along with adjustable parameters. In fact, we examined the effect of setting the cost of shifts in (1) as follows: c kj = w1α kj + w2 β jk

(10)

k j

where a = the number of shift-days in the shift (j, k) (this is the conventional definition of the cost for crew route scheduling); k b j = the working hours of the shift (j, k); and each of w1, w2 = the adjustable parameter of the efficiency k k factors a j , b j . Three cases were examined: (a) w1=1, w2=0 (i.e. conventional definition); (b) w1=0,w2=1; and (c) w1=1, w2=0.06. In each case, two crew route scheduling problems were solved for different actual lines, and then for each result, a crew roster scheduling problem was solved by using the solution method in Section 3.4. The last case (c) could minimize the number of required crew in the result of both problems. This result shows that it is possible to reduce the manpower of the corresponding crew roster schedule 199

using the evaluation function of the crew route scheduling problem into which efficiency factors have been integrated. However it is not possible to decide suitable values of the adjustable parameters in advance. Therefore an algorithm was proposed for solving the crew route schedule repeatedly by adjusting the parameters according to the result of the crew roster scheduling based on the idea by Caprara [3]. 4.2 The Framework of the Algorithm Figure 3 represents the framework of the algorithm. Adjustable parameters are initialized according to the conventional definition of the cost for crew route scheduling at Step 1. Here, the number of shift-days, working hours, real working hours, inter alia were used as efficiency factors making up the cost of shifts similar to (10). A solution for the crew route schedule was created using the method shown in Section 3.2 on the current setting of adjustable parameters in Step 2. A roster schedule for each crew base corresponding to the solution was solved by the method shown in Section 3.4 in Step 3, in the result, the minimum total number of required crews corresponding to the parameters was fixed. In Step 4, the algorithm stops iterating schedules when the number of the scheduling iterations reaches a specified value, and the best crew route schedule between iterations is output. Before the crew route scheduling is executed again, adjustable parameters of the efficiency factors are adjusted corresponding to the variations of some values such as the number of required crew and/or values of the efficiency factors in Step 5. Start Step 1

Initialize Adjustable Parameters Step 2

Crew Route Scheduling (Shown in Section 3.2) Step 5

Step 3

Calculate Minimum Number of Required Crews by Crew Roster Scheduling (Shown in Section 3.4) Step 4

Adjust Parameters by the Result of the Step 3

No

Specified Iterations were Done?

Yes End

Fig. 3 Overview of the Crew Route Scheduling Algorithm

4.3 Examination Results The algorithm was examined on a general PC for 4 railway areas in Japan and results were compared to the actual operated schedules and solution results using the conventional cost definition. In the examination, the limit of iterations was set at around 30 counts and the time limit for optimization in Step 2 and Step 3 was set at 600 seconds in both cases, therefore the maximum calculation time for the whole scheduling of an area was set at 5 hours. Table 1 shows the comparison result. It shows that in all areas crew route schedules using the proposed algorithm could reduce manpower for their crew operation. It is also possible to see that the proposed algorithm increases the number of shift-days compared to solutions found using the conventional method, and decreases the required number of crews in all cases. One of the possible reasons for this is that reducing only the number of shift-days tends to make shifts longer and consequently has a negative influence on some efficiency factors, while actually controlling efficiency factors can avoid this problem. 5. Comparison of Distribution of Efficiency Factors The distributions of the 4 efficiency factors were compared, namely, working hours, real working hours, start time, and finish time of each shift between three scheduling results shown in Table 1 in order to look into the effectiveness and the validity of the algorithm. Figure 4 shows the distributions of the shifts of area B. In Fig. 4, each bar corresponds to each efficiency factor. Scales on it mean ranges of the efficiency factor, and the color between adjacent scales means the number of shifts corresponding to the range of an hour of the crew route schedule. The warmer colors correspond to a larger number of shifts, the cooler colors corresponds to a lower number of shifts, while white means no shift. Comparing schedules produced using the proposed algorithm and the original (actual) schedule both shift distributions are similar in terms of start and finish time. However, in the former schedule shifts with over 23 working hours increased while shifts with around 14 working hours decreased in comparison with the original schedule. The same tendency appeared for real working hours, that is, long shifts increased while shorter shifts decreased. Based on this tendency, shift sequences as a result of crew roster scheduling were examined, and it was found that

Table 1 Comparison between Examination Results and Others Area

200

Proposed Algorithm

Conventional Method

Actual Schedules

Shift-days

Required Crews

Shift-days

Required Crews

Shift-days

Required Crews

A

256

348

249

380

330

390

B

220

294

214

333

272

303

C

268

362

261

389

303

373

D

387

520

373

594

436

550

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(hours)

(time)

(time)

24

(hours) 16

24

24

24

16

24

24

14

10

12

12

14

10

12

12

4

4

rs

ou

H ing

ork

W

al

Re

rs

ou

H ing

ork

W

0

e

rt Sta

Tim

0

sh ini

(hours)

4

e

Tim

F

ork

W

rs

ou

H ing

ork

W

al Re

Middle Color

(time)

4

rs

ou

H ing

(a) An Actual Schedule Warmer Color

(hours)

0

e

rt Sta

method has neither shifts with short working hours nor shifts with short real working hours. This characteristic found with the conventional method is considered to be one of the obstacles to improving manpower efficiency in roster scheduling. It could therefore be said that maintaining wide range in shift distribution in terms of working hours and real working hours wide when creating crew route scheduling is an effective means to improve manpower efficiency.

(time)

Tim

0

e

ish

Tim

Fin

(b) A Schedule Made by Proposed Algorithm Colder Color

The Number :Greater of Shifts

Fewer

None of Shifts

Fig. 4 Shift Distribution in Area B for 4 Efficiency Factors there were significantly fewer consecutive night shifts. This therefore indicates that the reason the algorithm improves manpower efficiency is that the algorithm creates longer shifts helping to avoid an increase in consecutive night shifts with a number of critical restraints on the roster scheduling. Figure 5 shows the comparison between three scheduling results for Area A. The shift distribution of working hours and real working hours for both the actual schedule and the schedule made by the proposed algorithm are similar. However the schedule produced using the conventional (hours)

(hours)

24

24

(hours) 24

14

14

14

4

4

4

de ) al Ma ule ion nd ched vent d a H lS o n a Co Meth ctu (A

sed po m Pro orith Alg

(hours)

(hours)

16

16

(hours) 16

10

10

10

4

4

4

de ) nal Ma ule tio n nd ched e a od H lS nv Co Meth tua c (A

sed po m Pro orith Alg

6. Conclusions This paper proposes an algorithm for crew route scheduling with a view to increasing efficiency of manpower resource allocation in crew operation scheduling. After application of the proposed method to real railway scenarios, confirmation was obtained of the algorithm’s high performance in comparison with conventional methods for minimizing the number of shift-days. As a result of an investigation of the mechanism giving such high performance, it was shown that there is a possibility that wide distribution ranges of working hours and real working hours to the same extent as the actual schedule could contribute to improving the manpower efficiency. It is hoped that following on the algorithm will be applied for scheduling on other lines with the different conditions to verify the effect of our method and to improve the algorithm. References [1] Tomii, N., Fukumura, N., Sakaguchi, T., and Hirai, C., Scheduling Algorithms in Railways. NTS Publishing Inc., Tokyo, Japan, pp. 106-114, 2005(in Japanese). [2] Desaulniers, G., Desrosiers, J., and Solomon, M.M. (Eds.), Column Generation. Springer Publishing Inc., New York, U.S.A., pp. 1-32, 2005. [3] Caprara, A., Monaci, M., and Toth, P.,“A global method for crew planning in railway applications,”In: Daduna, J.R., and Voß, S. (Eds.), Computer-Aided Scheduling of Public Transport, Lecture Notes in Economics and Mathematical Systems, Vol. 505, Springer Publishing Inc., Berlin, Germany, pp. 17-36, 2001.

Fig. 5 Comparison between Scheduling Methods

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201