流水線環境中考慮內生性良率的生產與預防性保養之整合規劃問題

Abstract

本研究探討流水線環境中單產品的生產與預防性保養之整合規劃問題。當工廠機台未進行保養時，機況會隨時間下降，導致生產成本上升；另一方面，儘管保養有助於改善機況，但是過於頻繁的保養將增加機台停機時間，進而提升存貨成本或缺貨成本。因此，如何決定適當的保養時機與對應的生產計畫，以最小化總成本，便成為工廠能否在成本與效益間取得平衡的關鍵議題。本研究透過一個非線性整數規劃模型來精確描述此問題。由於在問題規模龐大時，求解該數學模型需要耗費大量時間，因此本研究提出數種演算法，以在多項式時間內求得最佳解或近似解。當生產環境僅包含單一階段時，本研究首先證明保養週期與存貨週期呈現巢狀結構，並利用此特性，進一步根據保養時機將規劃時程分割為若干個子問題，最後透過線性規劃求解子問題，以及將整體問題轉化為最短路徑問題，來求得最佳解。當生產環境包含兩個階段時，本研究證明兩個階段的保養時機遵循同步或相鄰的模式，並延伸單階段演算法，設計一套具雙層最短路徑結構的精確演算法來求解。當生產環境超過兩個階段時，本研究設計一套啟發式演算法，先將問題依階段分解，再透過單階段演算法逐步求解，以獲得近似解，同時設計數值實驗來驗證其品質與效能。最後，本研究展示如何將此啟發式演算法擴展至良率遞減率為隨機的情境，並同樣透過數值實驗說明其表現。

In this study, we investigate a single-product flow shop joint production-maintenance planning problem in a deteriorating system. The yield rates of machines decline over time but can be increased through preventive maintenance. However, machines must be shut down during maintenance periods, which may lead to higher inventory costs or demand shortage costs. Determining the appropriate timing for maintenance and the associated production plan to minimize the total cost is therefore a crucial issue for the factory to remain cost-effective. When the system consists of only one stage, we show that the maintenance and inventory cycles are nested. This property allows us to decompose the planning horizon into subproblems by maintenance and reformulate our problem as a shortest-path problem, where the edge costs can be solved by linear programming. When the system consists of only two stages, we show that the maintenance timing across the two stages follows either a synchronized or a neighboring pattern. An exact algorithm which exhibits a two-layer shortest-path structure is then developed to find an optimal solution. For problems with more than two stages, we develop a heuristic algorithm that decomposes the problem by stage and utilizes the single-stage algorithm to generate a near-optimal solution. Finally, we illustrate how our heuristic algorithm can be extended to problems with stochastic yield declining rates by using a rolling schedule approach. Through numerical experiments, we demonstrate the effectiveness of our heuristic algorithm under both deterministic and stochastic settings.