Dịch và giải thích cho tôi đoạn văn này đi. Multi-bucket Optimization for Integrated Planning and Scheduling in the Perishable Dairy Supply Chain C. Sela1, B. Bilgena , J.M. Bloemhof-Ruwaardb , J.G.A.J. van der Vorstb a Industrial Engineering Department, Dokuz Eylul University, 35397 Buca/Izmir, Turkey b Operations Research and Logistics Group, Wageningen University, Hollandseweg 1, 6706 KN Wageningen, The Netherlands Abstract This paper considers a dairy industry problem on integrated planning and scheduling of set yoghurt production. A mixed integer linear programming formulation is introduced to integrate tactical and operational decisions and a heuristic approach is proposed to decompose time buckets of the decisions. The decomposition heuristic improves computational efficiency by solving big bucket planning and small bucket scheduling problems. Further, mixed integer linear programming and constraint programming methodologies are combined with the algorithm to show their complementary strengths. Numerical studies using illustrative data with high demand granularity (i.e., a large number of small-sized customer orders) demonstrate that the proposed decomposition heuristic has consistent results minimizing the total cost (i.e., on average 8.75% gap with the best lower bound value found by MILP) and, the developed hybrid approach is capable of solving real sized instances within a reasonable amount of time (i.e., on average 92% faster than MILP in CPU time). Keywords: Integrated planning and scheduling, mixed integer linear programming, constraint programming, hybrid approach, perishability, dairy supply chain

1 Corresponding author. Email: [email protected], Tel: 90 232 301 76 34 Fax: 90 232 301 76 08 Page 3 of 60 Accepted Manuscript

1. Introduction The yoghurt production is a semi-continuous process and subject to individual characteristics. Yoghurt is a notably perishable product within the category of dairy industry (Lütke Entrup 2005). The perishability highly restricts its storage duration and delivery conditions. It has a wide variety of retail cup sizes or labels, contents and special ingredients with numerous flavoured and coloured types. When it comes to producing large numbers of products from a few initial product recipes, product dependent cleaning, sterilizing, re-tuning issues of pipes and mixing units arise to avoid contamination (Montagna et al., 1998). Especially, long sequence-dependent setup times and high costs are considerable at the filling and packaging stages of the yogurt production and, they cause a noticeable reduction of available production times and increase the costs. Hence, planning and scheduling of the yoghurt production require specific models to support decision making. Mixed integer linear programming (MILP) models provide mathematical frameworks to represent specific characteristics of problems and to get optimal solutions. The MILP is an extensively accepted tool in the dairy industry for well-defined problems (e.g., Banaszewska et al., 2013; Kopanos et al. 2011a, 2012a). Bilgen and Çelebi (2013) present a MILP model addressing the production scheduling and distribution planning problem in a yoghurt production line of multi-product dairy plants. They consider the yoghurt production with perishability and sequence-dependency issues by focusing on the packaging stage operating with parallel units sharing common resources. Sel and Bilgen (2014a) state that integrated multi-echelon, multi-period planning and scheduling models accounting for multi-stage semi-continuous yoghurt production particularities are found to be of practical use in the field. Accordingly, the contribution of this study is aligned with the gap pointed out by Sel and Bilgen (2014a) presenting a literature review and discussion on quantitative models for supply chain management within dairy industry. Page 4 of 60 Accepted Manuscript In this paper, we consider a production and distribution problem for a two-stage semi-continuous set type yoghurt production which is also comparable to other dairy production processes (e.g., cheese, butter and ice cream). The production side fundamentally corresponds to packaging and fermentation/incubation operations. The distribution side considers the storage of products and the delivery to distribution centres (DCs). The scope of the considered problem is illustrated in Figure 1. For the problem, we introduce a multi-echelon, multi-period integrated MILP model with shelf life consideration. Our integrated MILP model is an extension of the formulation previously proposed by Bilgen and Çelebi (2013). The model is extended by considering timing and capacity constraints with respect to the incubation operation of set type yoghurt. The scheduling constraints corresponding to both packaging and incubation operations are reformulated efficiently inspired by the generic MILP model of parallel machine scheduling with sequence dependent setup times, which is studied by Guinet (1993) as a vehicle routing formulation.