Bus transportation crews planning by goal programming

Socio-Econ. Plan. Sci. Vol. 18, No. Printed in the U.S.A.

3, pp. 207-210,

00384121/84 Pergamon

1984

$3.00 + .oO Pres Ltd.

BUS TRANSPORTATION CREWS PLANNING BY GOAL PROGRAMMING? JAI PRAKASH

and S. B. SINHA

Department of Mathematics, LIT., Kharagpur 721 302, India

and S. S. SAHAY Industrial Management Centre, I.I.T., Rharagpur 721 302, India (Received

27 August 1982; in revisedform

14 March 1983)

Abstract-In the present paper, an integer goal programming model formulation is presented for the bus transportation crews planning to find an optimal schedule in which each crew is assigned two days off (consecutive or nonconsecutive) per week and several goals like upper and lower limits on the total number of crews having consecutive and/or non-consecutive days off on each day, lower limit on the number of crews having consecutive days off on the particular pairs of consecutive days, and minimization of the number of crews having non-consecutive days off, are taken into consideration. The model is demonstrated by a numerical example.

1. INTRODUaION

Exorbitant hike in petroleum price has opened many challenging venues for the transportation management. Several measures are being evolved to face this increasing economic burden. One of the measures being commonly advocated is to have five working days a week in place of six by increasing the working hours per day. But there are some continuous processing industries and service organizations which must function on all days of the week. This situation is also valid for the essential public services like medical, police, fire brigade and transportation etc. However, the workers of these manufacturing and service organizations would also like to have five days working in a week necessiating a rescheduling of manpower. In a service organization like bus transportation, where the requirement of the crews for bus operation differ from day to day of the week due to varying passenger density; planning of bus crews scheduling for five working days a week is faced with deciding the number of crews to be given consecutive or non-consecutive days off on each day apart from the constraints and the considerations which are considered vital for smooth running of the organization. Such constraints and considerations may be to maintain a minimum number of bus trips into operation on each day for convenience of the public. In other words, a predetermined number (in a range) of crews may be given days off. As far as possible, there should be a maximum of crews having consecutive days off and minimum number of crews must be given nonconsecutive days off having better labour relation and improving morale of the crews. The problems having

tPresented at XIV Annual Convention of ORSI held at NITIE, Bombay from 12 to 14 March 1982. 207

such multiple objectives, which some times even may be conflicting, need an efficient tool for solution. In the last decade, some models have been developed to deal with crews planning of plants/service operations. Monroe[l] presented an arithmetical procedure for assigning staff personnel to shifts and their days off (both consecutive and non-consecutive) for the service operations manned seven days a week. This technique entails as much intuition as analysis because of its tentative nature as expressed by the author himself. Rothstein[2] presented a linear programming model containing 15 constraints and 15 variables to overcome the shortcomings of Monroe’s model. Additional constraints have also been suggested for the consecutive days off. This may result in infeasible solution because of its higher value. Tibrewalap] suggested two solution algorithms to find the schedules with both one and two days off in a week, for minimizing the total number of workers while satisfying the daily requirement. The algorithms do not take care of non-consecutive days off and also the manpower assignment when the manpower available is less than the requirement. Baker [4] developed a two phase algorithm for solving the staffing problem. First phase of the algorithm deals with the problem for which noslack solution exists, i.e. workforce is equal to the one-fifth of the total number of employees required on all days and the number of employees scheduled for days off for each day is equal to the maximum number of employees who could be assigned days off for the respective days. Second phase of the algorithm deals with the general case, i.e. when there may be some slack in any allocation of full time employees. In this phase, the lower bound is imposed on the increase in workforce size for maintaining the feasibility in the solution. This process appears to be complex and requires separate model formulation for slack and nonslack.

208

J.

hAKASH

An additional drawback of all the above mentioned approaches is their inability to entertain multiple objectives which invariably exist in all real world problems. For this reason, goal programming[5-101 surfaces as a logical solution alternative. Also, since goal programming attempts to achieve a satisfactory rather than optimal solution in the face of conflicting goals, it appears to be more realistic approach. The following sections present a goal programming model formulation for preparing a schedule of days off for the crews of bus transportation. Since the number of crews can not be in fractions, the integer option of modified simplex algorithm[5] is used for solving this type of problems.

et

at.

and

xi+x,+,+y,>b,, x7 +

XI +

y,

i=j=1,2

By including positive and negative deviations, the goal constraints would be represented as xi+xi+1 +y,-di+=aj, x7+x1+y7-d7+

variables,

deviation variables and con-

., 7 pairs of days Sunday-Monday, Monday-Tuesday,-Saturday-Sunday, respectively Tues192,. . ., 7 for days Monday, day,-Sunday, respectively xi number of crews having consecutive days off on the days pair i yj number of crews having non-consecutive days off on the day j d, + /d, positive/negative deviation from t th goal aj/bj upper/lower bound on the total number of crews having off on the day j c, lower bound on the number of crews having consecutive day off on the days pair i

1,2,.

i M

m W

set of the numbers of the pairs of the consecutive days to be restricted in second goal total number of elements in the set M total workforce of crews

2.2 Goal constraints 2.2(a). Limits on the total number of crews having days off on each day. The number of bus trips on different days of the week vary resulting in the needs of differing number of bus crews. Further at times, more number of crews remain absent than that planned for a specific day (based on planned number of trips), number of trips in operation is reduced causing inconvenience. To avoid such sudden changes, some extra crews are maintained, though it incurs some more costs. Therefore, both lower and upper limits are prescribed for providing days off to the crews on different days of the week, i.e.

x7 + xl + y7 < a7

i=j=l,2

,...,

6

i=j=1,2

,...,

6.

=a7

and

2.2(b). Limit on the number of crews having consecutive days ofl In a five days working week, most of the crews prefer to take consecutive days off on the particular pairs of consecutive days, therefore, the number of crews having consecutive days off on the desired pairs of consecutive days should not be less than the number of crews who have shown their desire to have consecutive days off for those pairs off days, i.e. X, > ci, iEM. The goal constraint would be written as

by adding negative deviation,

x, + d,-+ 14= c,

,...,

6

i EM

m for lirst, second, . . . . and last where t=l,2,..., element of the set M. 2.2(c). Minimization of the number of crews having non-consecutive days off. Generally, all crews prefer to have consecutive days off for their convenience and better utilization of days off, hence, the total number of crews having non-consecutive days off should be minimum, i.e.

C

Yj

The formulation of goal constraints, system constraints and the objective function is given below.

x,+x,+,+yj
i=j=l,2

x, + x, + y, + d,i = b,.

FORMULATION

The decision variables, deviation variables and constants for model formulation are as follows: 2.1 Decision stants

6.

b,.

2

xi+xi+,+yj+di+,=bj, 2. MODEL

,...,

G 0.

j=1

By including positive deviation, would become

1

Yj -

the goal constraint

dG+1s = 0.

,=I

2.3 System constraints 2.3(a). Each crew must take two days off either consecutively or non-consecutively per week, therefore, the sum of the crews having consecutive and half of non-consecutive days off should be equal to the total workforce of crews, i.e.

2.3(b). For providing two days off per week to each crew, it is essential that two nonconsecutive days off should be assigned to those crews who could

209

Bus transportation crews planning by goal programming not be assigned consecutive days off in such a way that, in no case, the same crew could be assigned both days off on the same day, i.e. 2yj-

i y,
,...,

ints, system constraints 3.1 Goal constraints 3.1(a).

7.

j=l

2.4 Model assumptions The model assumptions are that all decision variables should have non-negative integer values, all deviation variables should be non-negative and product of positive and negative deviation variables of the same goal should be zero, i.e. xi, yj = non-negative d,+,d,-

20

integers,

and d,+ x d,- =0

x,+x2+y,-d,+

=5

xz+x,+y,-dd,+

=7

x,+x,+y,-d,+

=6

x,+x5+y,-d,+

=8

xs+x,+ys-d,+

=4

xg+x,+y,5-ds+

=2

xT+x,+y,-d,+

=5

x,+x,+y,+d,-

=3

x,+x,+y,+$-

=5

x9+x,+y,+dk=4

for all i, j and t. 2.5 The objective fiction The objective function consists of positive deviations for under achievement, negative deviations for over achievement and both negative and positive deviations for exact achievement of the goals. On the basis of this analysis, the positive deviations, d, +, d2+, . . . d, + and negative deviations d8 -, h-, . . . d, of goal one, negative deviations d;, d,, . . . d;, ,., of goal two and positive deviation d:,,, of goal three would be included in the objective function. The objective function also includes the pre-emptive priority factor indicating the ordinal ranking of different goals. The priority ranking of the goal in the present paper is as follows (in order of decreasing priority level, i.e. Pj & Pi + J

p, PZ pj

limits on the total number of crews having days off on each day, limit on the number of crews having consecutive days off for the pairs of days, minimization of the number of crews having non-consecutive days off,

and the objective function:

x,+x,+y,+d,=6 x5+x,+y,+d,z=2 x,+x,+y,+dh=

1

x,+x,+y,+d,=3.

3.1(b). x,+d,=2 x, + d, = 3.

3.1(c). y,+y,+y,+y,+y,+y,+y,-d:,=O. 3.2 System constraints 3.2(a).

3.2(b).

The above priority ranking of the goals would result in the following objective function

i y,,
2yj-

j=l

3.3 xi, yj = non-negative integers d,+,d,-

30

and d,+ xd,-

=0

for all i, j and t. 3.4 The objective function 3. NUMERICAL EXAMPLE

The numerical example considered for demonstration of the model consists of workforce of 15 crews. The goals are that the total number of days off on Monday, Tuesday, Wednesday, Thurdsay, Friday, Saturday and Sunday should be between 3 and 5, 5 and 7,4 and 6,6 and 8,2 and 4, 1 and 2, and 3 and 5 respectively. The consecutive days off on Friday-Saturday and Saturday-Sunday should be 2 and 3 respectively. The non-consecutive days off should be minimum. This problem results in the following goal constra-

+ f’dd, + 42 + PM,)

.

4. RESULTS AND DISCUSSION The

above problem is solved by using a modified simplex algorithm and the following results were obtained:

J.

210

PRAKASH

et al.

Decision variables x, = 1

d,- =2 d,+ =3

x2 = 4

d,+ =l

x, = 2

d&=1

xq = 4

d:,=4

x5 = 2

d&=2.

x, = 2 Deviational variables d,- = 1 d4- =2 d5- =2 d,- =2 d,+ =2

d$?+= 1 d,=2

d&=1 d,=2 d,=

All other decision and deviational variables = 0. The above results indicate that 4, 5, 5, 6, 2, 5 and 3 crews would have days off on Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday respectively. Therefore, goal one is not met exactly. Goal two has been met indicating the number of crews having days off on the pairs of consecutive days Friday-Saturday and Saturday-Sunday as 2 and 3 respectively. Goal three has not been met indicating one crew should be assigned two non-consecutive days off one each on Tuesday and Thursday. The analysis of the above two runs shows that goal two has been met at the costs of goal one and three.

1. 5.

All other decision and deviational variables = 0. The above results show that goal one has been met exactly indicating the number of crews having days off on Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday as 5, 6, 6, 6, 2, 2 and 3 respectively. Goal two has not been met showing that no crew has consecutive days off on the pair of days Friday-Saturday and 2 crews have consecutive days off on the pair of day Saturday-Sunday against the goal of 2 and 3 crews respectively. Goal three has been met exactly indicating that no crew was non-consecutive days off. If the decision maker thinks that goal two should be met, it can be done by assigning higher priority to this goal, at the cost(s) of other goal(s). For the varification of the above statement, we rerun the model by inter-changing the priority levels of the goal one and two and got the following result. Decision variables x2 = 4 x, = 5 X6 = 2 x, = 3 Y2=

1

Y4=

1

Deviational

variables

d,- =l d,- =2 d,- = 1 d4- =2 d5- =2

CONCLUSION

The model presented in this paper can not be used to minimize the number of crews. This may only be utilized to help the transportation management for finding the number of crews having consecutive and non-consecutive days off on each for proper utilization of existing crews depending upon given constraints and objectives. The model also provides information about the extent of achievement of each of the objectives, which may be helpful in modifying the objectives. authors are grateful to Prof. J. P. Agarwal, Department of Mathematics, LIT., Kharagpur for his consistent help at various stages of the work. We are

Acknowledgements-The

also indebted to the referee for his useful suggestions. REFERENCES

1. G. Monroe, Scheduling of manpower for service oper, ations. Industrial Engng l&17 (Aug. 1970). 2. M. Rothstein, Scheduling manpower by mathematical programming. Industrial Engng 29-33 (April 1972). 3. R. Tibrewala, D. Philippe and J. Browne, Optimal scheduling of two idle periods. Management Sci. 19, 71-75 (1972). 4. K. R. Baker, Scheduling a full-time workforce to meet cyclic staffing requirements. Management Sci. 20, 1561-1568 (1974). 5. J. P. Ignizio, Goal Programming and Extensions. Lexington Books, Lexington, Mass. (1976). 6. A. Chames and W. W. Cooper, Management Models and Industrial Applications of Linear Programming. Wiley, New York (1961). 7. Y. Ijiri, Management Goal and Accounting for Control. Rand McNally, Chicago (1965). 8. A. J. Houghes and D. E. Grawiog, Linear Programming: An Emphasis on Decision Making. Addison-Wesley, Reading, Mass. (1972). 9. S. M. Lee, Goal Programming for Decision Analysis. Auerbach, Philadelphia (1972). IO. S. M. Lee, Linear Optimization for Management. Petrocelli, New York (1976).