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Determining the economic lot size has always represented one of the most important issues in production planning. This problem has long attracted the attention of researchers, and several models have been developed to meet requirements at minimum cost. In this paper we explore and discuss the evolution of these models during one hundred years of history, starting from the basic model developed by Harris in 1913, up to today. Following Harris׳s work, a number of researchers have devised extensions that incorporate additional considerations. The evolution of EOQ theory strongly reflects the development of industrial systems over the past century. Here we outline all the research areas faced in the past by conducting a holistic analysis of 219 selected journal papers and trying to give a comprehensive view of past work on the EOQ problem. Finally, a new research agenda is proposed and discussed.

The dynamic lotsizing problem concerns the determination of optimally produced/delivered batch quantities, when demand, which is to be satisfied, is distributed over time in different amounts at different times. The standard formulation assumes that these batches are provided instantaneously, i.e. that the production rate is infinite. Using a cumulative geometrical representation for demand and production, it has previously been demonstrated that the inner-corner condition for an optimal production plan reduces the number of possible optimal replenishment times to a finite set of given points, at which replenishments can be made. The problem is thereby turned into choosing from a set of zero/one decisions, whether or not to replenish each time there is a demand. If n is the number of demand events, this provides 2n−1 alternatives, of which at least one solution must be optimal. This condition applies, whether an Average Cost approach or the Net Present Value principle is applied, and the condition is valid in continuous time, and therefore in discrete time. In the current paper, the assumption of an infinite production rate is relaxed, and consequences for the inner-corner condition are investigated. It is then shown that the inner-corner condition needs to be modified to a tangency condition between cumulative requirements and cumulative production. Also, we have confirmed the additional restriction for feasibility in the finite production case (provided by Hill, 1997), namely the production rate restriction. Furthermore, in the NPV case, one further necessary condition for optimality, the distance restriction concerning the proximity between adjacent production intervals, has been derived. In an example this condition has shown to reduce the number of candidate solutions for optimality still further. An algorithm leading to the optimal solution is presented.

The dynamic lotsizing problem concerns the determination of optimal batch quantities, when given required amounts appear at discrete points in time. The standard formulation assumes that no shortages are allowed and that replenishments are made instantaneously. For the case when no shortage is allowed, previously it has been demonstrated that the inner-corner condition for an optimal production plan in continuous time reduces the number of possible replenishment times to a finite set of given points at which either a replenishment is made, or not. The problem is thus turned into choosing from a set of zero/one decisions with 2 n −1 alternatives, of which at least one solution must be optimal, where n is the number of requirement events. Recently, the instantaneous replenishment assumption has been replaced by allowing for a finite production rate, which turned the inner-corner condition into a condition of tangency between the cumulative demand staircase and cumulative production. In this paper we investigate relationships between optimal cumulative production and cumulative demand, when backlogging is permitted. The production rate is assumed constant and cumulative production will then be a set of consecutive ramps. Cumulative demand is a given staircase function. The net present value (NPV) principle is applied, assuming a fixed setup cost for each ramp, a unit production cost for each item produced and a unit revenue for each item sold at the time it is delivered. Among other results, it is shown that optimal cumulative production necessarily intersects the demand staircase. Instead of having 2 n −1 production staircases as candidates for optimality, there are 2 n −1 production structures as candidates. These are made up of sequences of batches, of which the set of batches may be optimised individually. Also is shown that the NPV of each batch has a unique timing maximum and behaves initially in a concave way and ends as convex. Results for the average cost approach are obtained from a zeroth/first order approximation of the objective function (NPV).

The purpose of this paper is to investigate the research development in supply chain risk management (SCRM), which has shown an increasing global attention in recent years. Literature survey and citation/co-citation analysis are used to fulfil the research task. Literature survey has undertaken a thorough search of articles on selected journals relevant to supply chain operations management. Meanwhile, citation/co-citation analysis uses Web of Sciences database to disclose SCRM development between 1995 and 2009. Both the approaches show similar trends of rising publications over the past 15 years. This review has piloted us to identify and classify the potential risk associated with different flows, namely material, cash and information flows. Consequently, we identify some research gaps. Even though there is a pressing need and awareness of SCRM from industrial aspect, quantitative models in the field are relatively lacking and information flow risk has received less attention. It is also interesting to observe the evolutions and advancements of SCRM discipline. One finding is that the intellectual structure of the field made statistically significant increase during 2000-2005 and evolved from passively reacting to vague general issues of disruptions towards more proactively managing supply chain risk from system perspectives.