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Sharing delay costs in stochastic scheduling problems with delays

J. C. Gonçalves-Dosantos, I. García-Jurado, J. Costa

TL;DR

The paper tackles fair sharing of delay costs in stochastic scheduling where activity durations follow known distributions. It defines the Shapley-based rule $SSh(SP)=\varPhi(v^{SP})$ with $v^{SP}(S)=E(C(x_S,X^0_{N\setminus S}))$, and proves a balancedness-based axiomatic characterization that yields a unique allocation. Through examples, it demonstrates how $SSh$ captures the impact of stochastic delays and can assign rewards for not being late, not just penalties. It also provides a practical computational framework using simulation to approximate the Shapley allocations, with an $O(n^4)$ procedure and empirical timings up to large problem sizes, enabling application in real-world project management under uncertainty.

Abstract

An important problem in project management is determining ways to distribute amongst activities the costs that are incurred when a project is delayed because some activities end later than expected. In this study, we address this problem in stochastic projects, where the durations of activities are unknown but their corresponding probability distributions are known. We propose and characterise an allocation rule based on the Shapley value, illustrate its behaviour by using examples, and analyse features of its calculation for large problems.

Sharing delay costs in stochastic scheduling problems with delays

TL;DR

The paper tackles fair sharing of delay costs in stochastic scheduling where activity durations follow known distributions. It defines the Shapley-based rule with , and proves a balancedness-based axiomatic characterization that yields a unique allocation. Through examples, it demonstrates how captures the impact of stochastic delays and can assign rewards for not being late, not just penalties. It also provides a practical computational framework using simulation to approximate the Shapley allocations, with an procedure and empirical timings up to large problem sizes, enabling application in real-world project management under uncertainty.

Abstract

An important problem in project management is determining ways to distribute amongst activities the costs that are incurred when a project is delayed because some activities end later than expected. In this study, we address this problem in stochastic projects, where the durations of activities are unknown but their corresponding probability distributions are known. We propose and characterise an allocation rule based on the Shapley value, illustrate its behaviour by using examples, and analyse features of its calculation for large problems.
Paper Structure (4 sections, 1 theorem, 22 equations, 2 figures, 3 tables)

This paper contains 4 sections, 1 theorem, 22 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The problem
  3. Shapley rule for stochastic scheduling problems with delays
  4. Computational Analysis

Key Result

Theorem 3.1

The Shapley rule is the unique rule for stochastic scheduling problems with delays that satisfies the balancedness property.

Figures (2)

  • Figure 1: PERT graph of the project in Example \ref{['example2']}
  • Figure 2: Density estimations of the variables $Z^1_i$ (solid line) and $Z^2_i$ (dotted line)

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Example 3.1