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Fairness in Repetitive Scheduling

Danny Hermelin, Hendrik Molter, Rolf Niedermeier, Michael Pinedo, Dvir Shabtay

TL;DR

The paper addresses fairness in repetitive scheduling by proposing a generic $\alpha|\beta,\mathrm{rep}|\max_j {Z_j}$ framework with per-day decisions for $n$ clients over $q$ days and QoS measures $Z_{i,j}$, aggregated as $Z_j=\sum_i Z_{i,j}$ and constrained by a fairness threshold $K$.It analyzes three canonical objectives corresponding to $Z_{i,j}\in\{C_{i,j},W_{i,j},L_{i,j}\}$, classifies the computational complexity across parameter regimes, and develops a fixed-parameter-tractable approach via $n$-fold IP for certain cases.Key results include a polynomial-time solution for $q=2$ in the $1|\mathrm{rep}|\max_j \sum_i C_{i,j}$ case, a spectrum of NP-hardness (weak/strong) as $q$ grows, and a formal quantify of the price of fairness that bounds the global efficiency loss due to fairness constraints.The framework enables broad applicability to diverse domains and opens avenues for further research on additional machine models, alternative fairness notions, and approximation algorithms.

Abstract

Recent research found that fairness plays a key role in customer satisfaction. Therefore, many manufacturing and services industries have become aware of the need to treat customers fairly. Still, there is a huge lack of models that enable industries to make operational decisions fairly, such as a fair scheduling of the customers' jobs. Our main aim in this research is to provide a unified framework to enable schedulers making fair decisions in repetitive scheduling environments. For doing so, we consider a set of repetitive scheduling problems involving a set of $n$ clients. In each out of $q$ consecutive operational periods (e.g. days), each one of the customers submits a job for processing by an operational system. The scheduler's aim is to provide a schedule for each of the $q$ periods such that the quality of service (QoS) received by each of the clients will meet a certain predefined threshold. The QoS of a client may take several different forms, e.g., the number of days that the customer receives its job later than a given due-date, the number of times the customer receive his preferred time slot for service, or the sum of waiting times for service. We analyze the single machine variant of the problem for several different definitions of QoS, and classify the complexity of the corresponding problems using the theories of classical and parameterized complexity. We also study the price of fairness, i.e., the loss in the system's efficiency that results from the need to provide fair solutions.

Fairness in Repetitive Scheduling

TL;DR

The paper addresses fairness in repetitive scheduling by proposing a generic $\alpha|\beta,\mathrm{rep}|\max_j {Z_j}$ framework with per-day decisions for $n$ clients over $q$ days and QoS measures $Z_{i,j}$, aggregated as $Z_j=\sum_i Z_{i,j}$ and constrained by a fairness threshold $K$.It analyzes three canonical objectives corresponding to $Z_{i,j}\in\{C_{i,j},W_{i,j},L_{i,j}\}$, classifies the computational complexity across parameter regimes, and develops a fixed-parameter-tractable approach via $n$-fold IP for certain cases.Key results include a polynomial-time solution for $q=2$ in the $1|\mathrm{rep}|\max_j \sum_i C_{i,j}$ case, a spectrum of NP-hardness (weak/strong) as $q$ grows, and a formal quantify of the price of fairness that bounds the global efficiency loss due to fairness constraints.The framework enables broad applicability to diverse domains and opens avenues for further research on additional machine models, alternative fairness notions, and approximation algorithms.

Abstract

Recent research found that fairness plays a key role in customer satisfaction. Therefore, many manufacturing and services industries have become aware of the need to treat customers fairly. Still, there is a huge lack of models that enable industries to make operational decisions fairly, such as a fair scheduling of the customers' jobs. Our main aim in this research is to provide a unified framework to enable schedulers making fair decisions in repetitive scheduling environments. For doing so, we consider a set of repetitive scheduling problems involving a set of clients. In each out of consecutive operational periods (e.g. days), each one of the customers submits a job for processing by an operational system. The scheduler's aim is to provide a schedule for each of the periods such that the quality of service (QoS) received by each of the clients will meet a certain predefined threshold. The QoS of a client may take several different forms, e.g., the number of days that the customer receives its job later than a given due-date, the number of times the customer receive his preferred time slot for service, or the sum of waiting times for service. We analyze the single machine variant of the problem for several different definitions of QoS, and classify the complexity of the corresponding problems using the theories of classical and parameterized complexity. We also study the price of fairness, i.e., the loss in the system's efficiency that results from the need to provide fair solutions.
Paper Structure (22 sections, 19 theorems, 28 equations, 1 table)

This paper contains 22 sections, 19 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contribution
  4. Techniques
  5. Paper organization
  6. A Model for Fairness in Repetitive Scheduling
  7. Formal model definition
  8. Three specific problems
  9. An illustrative example
  10. Further example applications
  11. Possible extensions
  12. The $1\mid\mathop{\mathrm{rep}}\nolimits\mid\max_j \sum_i C_{i,j}$ Problem
  13. A polynomial-time solvable special case
  14. Two weakly NP-hardness results
  15. Strong NP-hardness
  16. ...and 7 more sections

Key Result

Lemma 3

For any $K$, there exists a $K$-fair schedule for a $1\mid\mathop{\mathrm{rep}}\nolimits\mid\max_j \sum_i C_{i,j}$ instance with $q=2$ if and only if there exists a $K$-fair schedule where the schedule for day 1 is in the reverse order of the schedule of day 2.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Definition 5
  • Theorem 6
  • proof
  • ...and 33 more