The space of solution alternatives in the optimal lotsizing problem for general assembly systems applying MRP theory

Abstract

MRP Theory combines the use of Input-Output Analysis and Laplace transforms, enabling the development of a theoretical background for multi-level, multi-stage production-inventory systems together with their economic evaluation, in particular applying the Net Present Value principle (NPV). less thanbrgreater than less thanbrgreater thanIn a recent paper (Grubbstrom et al., 2010), a general method for solving the dynamic lotsizing problem for a general assembly system was presented. It was shown there that the optimal production (completion) times had to be chosen from the set of times generated by the Lot-For-Lot (L4L) solution. Thereby, the problem could be stated in binary form by which the values of the binary decision variables represented either to make a production batch, or not, at each such time. Based on these potential times for production, the problem of maximising the Net Present Value or minimising the average cost could be solved, applying a single-item optimal dynamic lotsizing method, such as the Wagner-Whitin algorithm or the Triple Algorithm, combined with dynamic programming. less thanbrgreater than less thanbrgreater thanThis current paper follows up the former paper by investigating the complexity defined as the number of possible feasible solutions (production plans) to compare. We therefore investigate how properties of external demand timing and properties of requirements (Bill-of-Materials) have consequences on the size of this solution space. Explicit expressions are developed for how the total number of feasible production plans depends on numbers of external demand events on different levels for, in particular, the two extreme cases of a serial system and a full system (the latter, in which items have requirements of all existing types of subordinate items). A formula is also suggested for general systems falling in between these two extremes. For the most complex full system, it is shown that the number of feasible plans will be the product of elements taken from Sylvesters sequence (an instance of doubly exponential sequences) raised to powers depending on numbers of external demand events

Designing MRP Model in the Condition of Capacity Constraints and Variability of Delivery for Critical Items in ABC Inventory Model “Case Study in National Company of South’s Oil-rich”

“…(70,309,075$) is minimum damages that has arisen in this project. Which shows the great importance of delay in receiving the goods [20][21][22][23]. Since delay value of goods may be overlapped in analyzing damages, means that simultaneous delay of two or more goods to the same extent causes damages, so in this analysis, the largest time delay is intended as a determinant factor of minimal damage and based on…”

The time-averaged L4L solution – a condition for long-run stability applying MRP theory

“…It is shown that in order for available inventory to be kept at finite levels at any time, the L4L solution must be valid for the time averages of production and demand, irrespective of the policy followed. Recently, [Grubbström et al, 2009, Grubbström andTang, 2012] it has been shown that the L4L solution determines all possible times when internal demand events can occur, so we may regard the currently considered property of the L4L solution as its third rôle.…”

Dynamic lotsizing with a finite production rate

“…This binary view led to the "inner-corner" condition for a production plan to qualify as a candidate for optimality, and it holds even for complex multi-item production structures (Grubbström, Bogataj andBogataj 2009, Grubbström andTang 2012). The inner-corner condition is a geometrical statement, in the discrete time case equivalent with "Inventory is not to be carried into a period where production takes place" (Aryanezhad 1992, pp.…”

“…Curve B (dotted) illustrates the most general case of feasible production (any non-decreasing curve above or possibly touching cumulative demand), Curve C (dotted and dashed) the general case of feasible production taking place in batches, making cumulative production a staircase function, and Curve D (dashed) a batchproduction case which candidates for optimality, since it meets the inner-corner condition. (Grubbström and Tang 2012).…”