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A single machine scheduling problem with bin packing constraints

Abstract : We consider a single machine scheduling problem with additional bin packing constraints. The origin of the problem comes from the production of chemotherapy drugs (Mazier et. al. 2010). In this production environment, raw materials are called monoclonal antibodies and can be stored in vials for a long time before use (Billaut 2011). However, once a vial is opened or once the active agent has to be mixed with some water, it must be used before a given time limit, in order to keep intact the properties of anticancer active agents. The maximum delay of use after opening depends on the agent and may vary between several hours to several days. In the mean time, the product has to be stored in a fridge for temperature and darkness reasons. The cost of these drugs is not negligible and the economic impact of saving these products is very important. The preparations that are scheduled and which use the same vial have to be completed before the product perishes. In other words, the total processing time of the jobs assigned to a same vial cannot be greater than the life time of the raw material. Furthermore, the total consumption cannot be greater than the total volume of the vial. Because each preparation has to be delivered to a patient for a given due date, one objective of the problem is to minimize the maximum lateness related to these due dates. Notice that in such a context, the deadline for the use of the raw material becomes a variable of the problem, which is directly related to the production scheduling decisions: once the life duration or the capacity of the vial is exceeded, a new vial is opened. 1 Problem statement and notations We consider a simplied version of the above problem, with only one machine and one type of raw material. We consider a set of n jobs to schedule on a single machine. W.l.o.g., the jobs are supposed to be numbered in EDD order, i.e. d 1 ≤ d 2 ≤ ... ≤ d n. To each job j ∈ {1, ..., n} is associated a processing time p j , a consumption b j and a due date d j. The life duration of the product after opening is equal to T and the volume of one vial is equal to V. We assume always w.l.o.g. that p j < T and b j < V , ∀j, 1 ≤ j ≤ n. The number of vials is not limited but supposed to be bounded by n. We denote by C j the completion time of j, L j the lateness dened by L j = C j − d j. The maximum lateness is dened by L max = max 1≤j≤n L j. We assume that the maximum lateness is bounded by Q. Minimizing the quantity of lost raw materials is equivalent to minimize the number of vials that are opened. Therefore, the problem is a mixed between a scheduling problem and a two-constraint bin packing problem. Without due dates (or with extremely large due dates), the problem is a bin packing problem. For this reason, the problem is clearly NP-hard. With a huge T and a huge V , the problem is the trivial single machine problem with the L max minimization. In the following, we call a bin, the set of jobs performed with the same vial. Example: We consider a set of six jobs with the following data and T = 10 and V = 10.
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Contributor : Jean-Charles Billaut <>
Submitted on : Wednesday, February 7, 2018 - 3:33:12 PM
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Jean-Charles Billaut, Federico Della Croce, Andrea Grosso. A single machine scheduling problem with bin packing constraints. 14th International Conference on Project Management and Scheduling (PMS’2014), Mar 2014, München, Germany. ⟨hal-01245902⟩



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