A Mixed Integer Programming Model Formulation for Solving the Lot-Sizing Problem

This paper addresses a mixed integer programming (MIP) formulation for the multi-item uncapacitated lot-sizing problem that is inspired from the trailer manufacturer. The proposed MIP model has been utilized to find out the optimum order quantity, optimum order time, and the minimum total cost of purchasing, ordering, and holding over the predefined planning horizon. This problem is known as NP-hard problem. The model was presented in an optimal software form using LINGO 13.0.Comment: 9 pages, 1 figure, 13 tables; International Journal of Computer Science Issues, Vol. 9, Issue 2, No 1, March 201

Integrated Production-Inventory Models in Steel Mills Operating in a Fuzzy Environment

Despite the paramount importance of the steel rolling industry and its vital contributions to a nation’s economic growth and pace of development, production planning in this industry has not received as much attention as opposed to other industries. The work presented in this thesis tackles the master production scheduling (MPS) problem encountered frequently in steel rolling mills producing reinforced steel bars of different grades and dimensions. At first, the production planning problem is dealt with under static demand conditions and is formulated as a mixed integer bilinear program (MIBLP) where the objective of this deterministic model is to provide insights into the combined effect of several interrelated factors such as batch production, scrap rate, complex setup time structure, overtime, backlogging and product substitution, on the planning decisions. Typically, MIBLPs are not readily solvable using off-the-shelf optimization packages necessitating the development of specifically tailored solution algorithms that can efficiently handle this class of models. The classical linearization approaches are first discussed and employed to the model at hand, and then a hybrid linearization-Benders decomposition technique is developed in order to separate the complicating variables from the non-complicating ones. As a third alternative, a modified Branch-and-Bound (B&B) algorithm is proposed where the branching, bounding and fathoming criteria differ from those of classical B&B algorithms previously established in the literature. Numerical experiments have shown that the proposed B&B algorithm outperforms the other two approaches for larger problem instances with savings in computational time amounting to 48%. The second part of this thesis extends the previous analysis to allow for the incorporation of internal as well as external sources of uncertainty associated with end customers’ demand and production capacity in the planning decisions. In such situations, the implementation of the model on a rolling horizon basis is a common business practice but it requires the repetitive solution of the model at the beginning of each time period. As such, viable approximations that result in a tractable number of binary and/or integer variables and generate only exact schedules are developed. Computational experiments suggest that a fair compromise between the quality of the solutions and substantial computational time savings is achieved via the employment of these approximate models. The dynamic nature of the operating environment can also be captured using the concept of fuzzy set theory (FST). The use of FST allows for the incorporation of the decision maker’s subjective judgment in the context of mathematical models through flexible mathematical programming (FMP) approach and possibilistic programming (PP) approach. In this work, both of these approaches are combined where the volatility in demand is reflected by a flexible constraint expressed by a fuzzy set having a triangular membership function, and the production capacity is expressed as a triangular fuzzy number. Numerical analysis illustrates the economical benefits obtained from using the fuzzy approach as compared to its deterministic counterpart

Primal-Dual Algorithms for Deterministic Inventory Problems

Primal-Dual Algorithms for Deterministic Inventory Problem