ارايه‌ي يك روش آزادسازي لاگرانژي براي يك مدل جديد توليد ـ توزيع در زنجيره‌ي تامين دوسطحي چند‌محصولي

A LAGRANGIAN RELAXATION METHOD FOR A MULTI-PRODUCT, MULTI-FACILITY PRODUCTION-DISTRIBUTION MODEL IN A TWO-ECHELON SUPPLY CHAIN WITH PROBABILISTIC DEMANDS

در اين نوشتار يك زنجيره‌ي تامين دو سطحي ــ شامل چندين مركز توزيع، چند كارخانه با ظرفيت محدود و چند تامين-كننده ــ به‌صورت يكپارچه مدل‌سازي شده است به‌گونه‌يي كه در آن چند محصول، شامل تعدادي قطعه، در جريان است. تقاضاي مراكز توزيع از توزيع نرمال برخوردار است و مدل ارايه‌شده به‌صورت سلسله‌مراتبي، مسيله را به دو سطح استراتژيك و عملياتي تقسيم مي‌كند. در سطح اول، با استفاده از رويكرد لاگرانژ ، مسيله‌ي‌ آزاد ‌سازي و به چهار زير مسيله تقسيم مي شود. با بررسي شرايط بهينگي زيرمسيله‌ها، روش‌هاي حل بهينه و الگوريتم ژنتيك براي آن ها ارايه مي ‌شود. جواب‌هاي حاصل از حل مسيله‌ي سطح اول به‌عنوان ورودي سطح دوم در نظر گرفته مي‌شود؛ مدل سطح دوم نيز كه يك مدل برنامه ريزي رياضي خطي است توسط نرم‌افزارهاي تجاري قابل حل است.

In this paper, we consider a two-echelon supply chain problem with multi-facility, multi-period, multi-product and nondeterministic demands, in which, we assume that demands follow a normal distribution probability function and that each product consists of several pre-determined parts. For solving the introduced model, we propose a hierarchical approach, based on the Lagrangian relaxation method. First, the problem is decomposed into two strategic and operational levels. At a strategic level, we respond to the following questions: Which facilities should be selected, how many demands are assigned to each selected facility, and which suppliers provide the necessary items for each facility. The strategic level problem using the Lagrangian relaxation method leads to four subproblems. The dominance properties of these subproblems are examined, and optimal methods and a genetic algorithm are proposed to solve them. Then, these relaxed subproblems are transformed into a general strategic problem and the Lagrangian coefficients are updated. This procedure will be terminated when stop criteria are satisfied. These criteria are defined based on duality gap percentage, the number of iterations that have not been improved in the upper bound solution, and the total number of iterations. The output of strategic level decisions will be considered as input to operational level decisions. At the operational level, we want to know how many products must be produced during regular work time, how many products must be produced during overtime, and what the inventory level of each item is at the end of period times. The operational level problem is solved using commercial linear programming software. To evaluate the proposed solution algorithms, some random instances of the problem are generated and solved by the algorithms. We generate 18 classes of problem with different sizes, and consider a 120 months planning horizon for all problems. For each class of problem, 10 random instances are generated. All algorithms are run on a PC Pentium 4 with 2.8 GHz processor. The commercial software, Lingo 8.0, was able to solve only small size instances within reasonable computational time. The results of the proposed algorithms are compared with the solutions obtained by Lingo after 180 minutes. The results show the convergence of the proposed solution method based on Lagrangian relaxation to optimal solutions in the early iterations of the method. Also, the duality gaps do not show any trends to mean that the efficiency of the method does not reduce by increasing the problem size.