By Joanna Jozefowska
As provide chain administration has matured, conserving the suitable move of products to regulate schedules (and reduce inventories) on a just-in-time foundation nonetheless provides significant demanding situations. This has encouraged an array of versions and algorithms to assist make sure the detailed move of parts and ultimate items into inventories to satisfy just-in-time necessities. this is often the 1st survey of the theoretical paintings on computers types and algorithms used in just-in-time scheduling.
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Additional resources for Just-in-Time Scheduling: Models and Algorithms for Computer and Manufacturing Systems
Example text
3 In an optimal schedule one task completes precisely at the due date d. 1 that the sequence of tasks and the start time of the first task are sufficient to determine the schedule. In fact, it is sufficient to assign tasks to two sets: set E of tasks completed before or at the due date and set T of tasks completed after the due date. 2. 3 we can calculate the start time of the first task as d − i∈E pi . 2. We denote by pE i (pi ) the processing time of the i-th task in the ordered set E (set T ). Finally, let nE be the number of tasks in set E and nT be the number of tasks in set T .
The first algorithm solving the MAD problem was proposed by Kanet [140]. We present this algorithm below. Let J denote the set of tasks to be scheduled. 1 (Kanet [140]). 1. Set E = ∅ and T = ∅. 2. Remove task k such that pk = max{pi } from set J. 3. Insert task k into the last position in sequence E. 4. If J = ∅ then remove task k such that pk = max{pi } from set J. 5. Insert task k into the first position in sequence T . 6. If J = ∅ then go to step 2. 7. Concatenate sequences E and T . The final schedule is defined by concatenation (E, T ) and contains no idle time.
Assign the remaining seats to states with the biggest differences calculated in the previous step. A fair method of apportionment should also meet the requirement that if the population of a state increases, then its representation in the parliament should not decrease. This property is called population monotonicity. Formally, a method is population monotone if for any two vectors of populations π, π > 0 and vectors of apportionments a ∈ M (π, h), a ∈ M (π , h ) the following implication holds: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (ai ≥ ai ∨ aj ≤ aj ) or πi πi πi πi ≥ ⇒ = and ⎪ πj πj ⎪ ⎪ π π j ⎪ j ⎪ ⎩ ai , aj can be substituted for ai , aj in a.
