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The purpose of this paper is to present a polynomial time algorithm which determines the lot sizes for purchase component in Material Requirement Planning (MRP) environments with deterministic time-phased demand with zero lead time. In this model, backlog is not permitted, the unit purchasing price is based on the all-units discount system and resale of the excess units is possible at the ordering time. The properties of an optimal order policy are argued and on the basis of them, a branch and bound algorithm is presented to construct an optimal sequence of order policies. In the proposed B&B algorithm, some useful fathoming rules have been proven to make the algorithm very efficient. By defining a rooted tree graph, it has been shown that the worst-case time complexity function of the presented algorithm is polynomial. Finally, some test problems which are randomly generated in various environments are solved to show the efficiency of the algorithm.

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The overall cost of companies dealing with the distribution tasks is considerably affected by the way that distributing vehicles are procured. In this paper, a more practical version of capacitated vehicle routing problem (CVRP) in which the decision of purchase or hire of vehicles is simultaneously considered is investigated. In CVRP model capacitated vehicles start from a single depot simultaneously and deliver the demanded items of several costumers with known demands where each costumer must be met once. Since the optimal vehicle procurement cost is a function of total distance it traverses during the planning horizon, the model is modified in a way that the decision of purchasing or hiring of each vehicle is made simultaneously. The problem is formulated as a mixed integer programming (MIP) model in which the sum of net present value (NPV) of procurement and traveling costs is minimized. To solve the problem, a hybrid electromagnetism and parallel simulated annealing (PSA-EM) algorithm and a Shuffled Frog Leaping Algorithm (SFLA) are presented. Finally, the presented methods are compared experimentally. Although in some cases the SFLA algorithm yields better solutions, experimental results show the competitiveness of PSA-EM algorithm from the computational time and performance points of view.

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In this study, a two-echelon supplier-manufacturer system with finite production rate and lead time is proposed. It is assumed that shortage is not permitted and the lot size of manufacturer (second echelon) is m-factors of the lot size of supplier (first echelon) and supplier can supply the manufacturer’s lot size in several shipments in each cycle. So, the production rate of supplier is greater than manufacturer’s. The proposed model aims to determine the optimal lot-size of each echelon such that the total cost of system is minimized. First, the problem is studied regardless of lead time and the optimal value of the lot sizes and the number of shipments is determined through analytical relations. Then, an exact solution algorithm for the problem is presented for the case with non-zero lead time. Finally, the performance of the proposed algorithm is reviewed by solving some numerical instances of the problem.

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Dynamic lot sizing problem is one of the significant problem in industrial units and it has been considered by  many researchers. Considering the quantity discount in  purchasing cost is one of the important and practical assumptions in the field of inventory control models and it has been less focused in terms of stochastic version of dynamic lot sizing problem. In 
this paper, stochastic dynamic lot sizing problem with considering the quantity discount is defined  and  formulated.  Since  the  considered  model  is  mixed  integer  non-linear programming,  a  piecewise  linear  approximation  is  also  presented.  In  order  to  solve  the mixed integer non-linear programming, a branch and bound algorithm are presented. Each node in the branch and bound algorithm is also MINLP which is solved based on dynamic programming framework. In each stage in this dynamic programming algorithm, there  is a sub-problem which can be solved with lagrangian relaxation method. The numeric results found in this  study indicate that the proposed algorithm solve the problem faster than the mathematical  solution  using  the  commercial  software  GAMS.  Moreover,  the  proposed algorithm for  the  two  discount  levels  are  also  compared  with  the  approximate  solution  in mentioned software. The results indicate that our algorithm up to 12 periods not only can reach to the exact solution, it consumes less time in contrast to the approximate model.

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The travel times among demand points are strongly influenced by traffic in a supply chain. Due to this fact, the service times for customers are variable. For this reason, service time is often changes over a time interval in a real environment. In this paper, a time-dependent periodic green vehicle routing problem (VRP) considering the time windows for serving the customers and multiple trip is developed with this assumption that urban traffic would disrupt timely services. The objective function of proposed problem is to minimize the total amount of carbon dioxide emissions produced by the vehicle, earliness and lateness penalties costs and costs of used vehicles. At first, a novel linear integer mathematical model is formulated and then the model is validated via solving some test problems by CPLEX solver. Finally, the sensitivity analysis is carried out to study the role of two critical parameters in the optimal solution.