[Paper Review] Minimum Dissatisfaction Personnel Scheduling

Mugurel Ionuţ Andreica, Romulus Andreica+1 more|arXiv (Cornell University)|Aug 8, 2008

Scheduling and Timetabling Solutions9 references3 citations

TL;DR

This paper proposes an optimal scheduling framework for personnel allocation that minimizes overall dissatisfaction, modeled as a time-dependent linear function per employee. It introduces two novel scheduling problems—single-shift and multi-shift—with efficient algorithms based on greedy assignment and dynamic programming, achieving optimal solutions in polynomial time.

ABSTRACT

In this paper we consider two problems regarding the scheduling of available personnel in order to perform a given quantity of work, which can be arbitrarily decomposed into a sequence of activities. We are interested in schedules which minimize the overall dissatisfaction, where each employee's dissatisfaction is modeled as a time-dependent linear function. For the two situations considered we provide a detailed mathematical analysis, as well as efficient algorithms for determining optimal schedules.

Motivation & Objective

• Address the challenge of scheduling personnel to minimize collective dissatisfaction in work allocation.
• Model individual dissatisfaction as a linear function of time, reflecting employee preferences for shift timing.
• Formulate two distinct scheduling problems: single-shift and multi-shift personnel assignment.
• Develop efficient algorithms that guarantee optimal solutions under the dissatisfaction minimization objective.
• Provide a mathematical foundation and algorithmic framework applicable to real-world workforce scheduling with preference constraints.

Proposed method

• Model each employee's dissatisfaction as a linear function of the time they are assigned to work, with steeper slopes indicating stronger time preferences.
• Formulate the scheduling problem as an assignment optimization task, minimizing the sum of individual dissatisfaction functions.
• For the single-shift case, apply a greedy algorithm that sorts employees by dissatisfaction slope and assigns shifts in order of increasing preference.
• For the multi-shift case, use dynamic programming to explore feasible shift assignments while maintaining optimality under time-dependent cost functions.
• Prove correctness and time complexity (polynomial) of both algorithms through mathematical analysis.
• Leverage algorithmic techniques later applied to competitive programming problems, validating practical efficiency.

Experimental results

Research questions

• RQ1 How can personnel scheduling be optimized to minimize the total dissatisfaction of employees when dissatisfaction is time-dependent?
• RQ2 What is the optimal assignment strategy when employees have varying preferences for shift timing, modeled as linear dissatisfaction functions?
• RQ3 Can an efficient polynomial-time algorithm be designed for single-shift scheduling under linear dissatisfaction models?
• RQ4 What is the computational complexity and algorithmic approach for multi-shift scheduling with time-dependent dissatisfaction?
• RQ5 How do the proposed algorithms ensure global optimality while respecting individual employee preferences in shift allocation?

Key findings

• The single-shift scheduling problem admits an optimal solution in O(n log n) time using a greedy algorithm based on dissatisfaction slope ordering.
• The multi-shift scheduling problem is solved optimally using dynamic programming with a time complexity that remains polynomial in the number of employees and shifts.
• The proposed algorithms guarantee minimal total dissatisfaction across all employees, outperforming heuristic or greedy approaches that ignore preference gradients.
• Mathematical proofs confirm the correctness and optimality of both algorithms under the defined dissatisfaction function model.
• The algorithmic techniques were later reused in competitive programming, demonstrating practical efficiency and scalability.
• The framework provides a foundation for real-world scheduling systems incorporating employee preferences and time-sensitive cost functions.