Project Presentation   For ECIV 705 - Deterministic Civil and Environmental Systems Engineering  Implementation and Performance Evaluation of a Proposed Integer Model for Yard Crane Scheduling Problem Omor Sharif Spring 2010 University of South Carolina, Columbia

Topics 1. Introduction- Problem Description and Motivation 2. Formulation – Integer Program 3. Solution and Performance Evaluation 4. Results and Conclusion

Problem Description and Motivation Courtesy: (Ng- 2005)

Problem Description and Motivation -To determine the job handling (truck serving) sequence of cranes -Want to minimize waiting time for trucks -Trucks with different arrival time -Jobs are distributed in space -Idling trucks are source of potential emission impacts on the ambient environment -Productivity of yards cranes is crucial for terminals operational efficiency

Project Objectives -To Implement an integer programming model developed by W. C. Ng (2005) -To find optimal and effective allocation of the yard cranes to handle jobs with different ready times -Performance Evaluation of the proposed integer model Courtesy: (Ng- 2005)

Integer Program Definition of Parameters Number of jobs to be handled in a planning period, n Number of cranes available, m Number of slots, Time required to handle a job, h = 8 time units Time required to travel one slot by a crane = 1 time unit Ready time (truck arrival time) of job i is ri Location of Job i in terms of slot number is βi Set of slots the yard crane can possibly be in at period t-1 is p(l) Set of slots the yard crane can possibly be in at period t+1 is s(l) Location of yard crane k at period 0 (initial location in terms of slot number) is αk

Integer Program The objective function used is to minimize the sum of total job completion time. Formally,

Constraints

Notes on Constraints Constraints (1) give the relationship between a job's completion time, ready time and handling time. Constraints (2) ensure that there is only one non-zero completion time for each job given by W. Constraints (3) ensure that during a yard crane job handling operation, the yard crane stays at the job location throughout the operation. Constraints (4) and (5) state the relationship between the locations visited by a yard crane, as implied by Y, in successive periods. Constraints (6) ensure that the movement defined by Y is free of inter-crane inter­ference. Constraints (7) state that a yard crane can only be in one of the slots in the yard zone in each period. Constraints (8) give the relationship between the completion time of a job and that of its successors. Constraints (9) give the relationship between X and W for jobs handled by the same yard crane. Con­straints (10) are simple binary constraints.

the || marks indicates cranes position at t = 0 An Example the || marks indicates cranes position at t = 0 || Row 1 1(1) 3 (6) Row 2 2 (4) 7 (16) Row 3 4 (9) 6 (14) Row 4 5 (11) Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 Optimal Schedule Value of Objective Function = 149 time units Crane 1 handles job {1,4,5,6} Crane 2 handles job {2,3,7} Courtesy: (Ng- 2005)

Value of Objective Function (Time units) Some Results Number of Slots, Number of Jobs, n Value of Objective Function (Time units) Solution time (hr:min:sec)* 5 68 (with h = 4) 00:00:40 10 180 (with h = 4) 00:18:00 15 240 (with h = 2) 01:36:06 75 (with h = 4) 00:00:52 189 (with h = 4) 00:08:36 260 (with h = 2) 08:01:06 90 (with h = 4) 00:01:00 232 (with h = 4) 02:11:32 20 94 (with h = 4) 00:01:11 241 (with h = 4) 06:22:19 * Solution time applies to test cases instance only

Conclusion 1. The integer program finds the optimal schedule for small scale problem sizes 2. Solution time increases much rapidly even if the size of the problem is increased gradually. 3. For large and realistic sized problems the time required to solve the model will most often exceed the time bound within which a solution is desirable.

References 1. Ng, W.C., Crane scheduling in container yards with inter-crane interference. European Journal of Operational Research, 2005. 164(1): p. 64-78. 2. Ng, W.C. and K.L. Mak, Yard crane scheduling in port container terminals. Applied Mathematical Modelling, 2005. 29(3): p. 263-276. 3. Zhang, C., et al., Dynamic crane deployment in container storage yards. Transportation Research Part B: Methodological, 2002. 36(6): p. 537-555.

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