Mata Kuliah: Penjadwalan Produksi

Teknik Industri – Universitas Brawijaya

Hybrid Flow Shop

Problems

Scheduling Problems

HFS

A set of 𝑛 jobs are to be processed in a series of 𝑚 stages optimizing a given objective function.

• Parallel machine scheduling (PMS) problem

Key decision: allocation of jobs to machines

• Flow shop scheduling (FSS) problem Key decision: sequence of jobs through the shop

Main decisions: To assign and to schedule the jobs to the machines in each stage.

HFS

• Standard problem: – all jobs and machines are available at time zero, – machines at a given stage are identical, – any machine can process only one operation at a time

and any job can be processed by only one machine at a time,

– setup times are negligible, – preemption is not allowed, – the capacity of buffers between stages is unlimited , – and problem data is deterministic and know in

advance.

HFS

• Characteristics of HFS variants: – The number of processing stages 𝑚 is at least 2,

– Each stage 𝑘 has 𝑀(𝑘) ≥ 1 machines in parallel and in at least one of the stages 𝑀(𝑘) > 1,

– All jobs are processed following the same production flow: stage 1, stage 2, ..., stage 𝑚. A job might skip any number of stages provided, it is processed in at least one of them,

– Each job 𝑗 requires a processing time 𝑝𝑗𝑘 in the 𝑘. We shall refer to processing of job 𝑗 in the stage 𝑘 as operation 𝑜𝑗𝑘.

HFS

• Application:

– Manufacturing:

• electronics, paper, textile, photographic film, others.

– Non-manufacturing:

• civil engineering, internet service architecture, container handling systems.

• Standard problem “template”:

HFS

HFS

• 𝑗: identifies a job (1, 2, 3, … , 𝑛) • 𝑘: identifies a stage (1, 2, 3, … ,𝑚) • 𝑙: 𝑙𝑡ℎmachine of a given stage

• 𝑝𝑗𝑘: processing time required by job 𝑗 in stage 𝑘

• 𝑐𝑗𝑘: completion time of job 𝑗 in stage 𝑘 (𝑐𝑗𝑘 = 0) • 𝑍: objective function (𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡𝑕𝑒

𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑖𝑜𝑛 𝑡𝑖𝑚𝑒 𝑜𝑓 𝑡𝑕𝑒 𝑙𝑎𝑠𝑡 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑡𝑜 𝑓𝑖𝑛𝑖𝑠𝑕 𝑖𝑡𝑠 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑖𝑛𝑔, 𝑖. 𝑒.𝑚𝑎𝑘𝑒𝑠𝑝𝑎𝑛)

• (3): guarantees that all operations are assigned strictly to one machine at each stage

• (4): restrict the starting time of operartion 𝑜𝑗𝑘 to be greater or equal to its release time from previous stage

• (5) & (6): prevent any two operations from overlapping in a common machine

• (7), (8), (9): define the domains of the decision variables

HFS

HFS

• ERMA1

HFS

• ERMA1 – (𝐴): express the fact that the jobs must be processed

at every stage

– (𝐵): imply that there is at most one job in position 𝑙 on each machine

– (𝐶) & (𝐷): enable us to calculate the completion times at each stage (both routing and disjunctive contraints on the machine)

– (𝐸) & (𝐹): define the criteria 𝐶𝑚𝑎𝑥 and 𝐶

– (𝐺): express the bound on the criterion 𝐶𝑚𝑎𝑥 for the solution sought

HFS

• ERMA2

HFS

• ERMA2 – (𝐴): express the fact that the jobs must be

processed at every stage

– (𝐵): imply that there is at most one job in position 𝑙 on each machine

– (𝐶) & (𝐷): enable us to calculate the completion times at each stage (both routing and disjunctive contraints on the machine)

– (𝐸) & (𝐹): define the criteria 𝐶𝑚𝑎𝑥 and 𝐶 to be minimised

Referensi

• Multicriteria Scheduling. Vincent T’kindt, Jean-Charles Billaut. Springer. 2002.

• Review and Classification of Hybrid Flow Shop Scheduling Problems from A Production System and A Solutions Procedure Perspective. Imma Ribas, Rainer Leisten, Jose M. Framinan. Computer & Operations Research 37: 1439 – 1454. 2010.

• The Hybrid Flow Shop Scheduling Problem. Ruben Ruiz, Jose Antonio Vazquez-Rodirguez. European Journal of Operational Research 205: 1 – 18. 2010.