Sample Average Approximation Method for Task Assignment with Uncertainty

불확실성을 갖는 작업 할당 문제를 위한 표본 평균 근사법

  • 김광 (조선대학교 경영학부)
  • Received : 2022.11.22
  • Accepted : 2022.12.29
  • Published : 2023.02.28


The optimal assignment problem between agents and tasks is known as one of the representative problems of combinatorial optimization and an NP-hard problem. This paper covers multi agent-multi task assignment problems with uncertain completion probability. The completion probabilities are generally uncertain due to endogenous (agent or task) or exogenous factors in the system. Assignment decisions without considering uncertainty can be ineffective in a real situation that has volatility. To consider uncertain completion probability mathematically, a mathematical formulation with stochastic programming is illustrated. We also present an algorithm by using the sample average approximation method to solve the problem efficiently. The algorithm can obtain an assignment decision and the upper and lower bounds of the assignment problem. Through numerical experiments, we present the optimality gap and the variance of the gap to confirm the performances of the results. This shows the excellence and robustness of the assignment decisions obtained by the algorithm in the problem with uncertainty.

최상의 에이전트-작업 할당을 결정하는 문제는 조합 최적화(combinatorial optimization)의 대표적인 문제이자 NP-난해(NP-hard)임이 알려져 있다. 본 연구에서는 에이전트와 작업의 할당 시 결정되는 작업 수행 확률(completion probability)이 불확실한 상황에서의 문제를 다룬다. 에이전트나 작업 내부의 요인 혹은 시스템 외적인 요소로 인한 작업 수행 확률은 고정적이기보다 불확실성을 갖는 것이 일반적이다. 불확실성을 고려하지 않은 할당 결정은 변동성이 있는 현실 상황에서 효과적이지 않은 결정이 될 수 있다. 작업 수행 확률의 불확실성을 수학적으로 반영하기 위해 본 연 구에서는 추계적 계획법(stochastic programming)을 활용한 수리 모형을 제시한다. 본 연구에서는 효율적으로 문제를 풀기 위해 표본 평균 근사법(sample average approximation)을 활용한 알고리즘을 제안한다. 본 문제 해결 방법론을 이용해 효과적인 할당 결정과 상한값과 하한값을 구할 수 있고, 결과의 성능을 확인하기 위해 최적 격차(optimality gap)와 분산을 실험을 통해 제시한다. 이를 통해 알고리즘으로 구한 할당 결정의 우수성 및 강건성을 보인다.



이 논문은 2022학년도 조선대학교 학술연구비의 지원을 받아 연구되었음.


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