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In this paper we attempt to derive properties of the distribution of the net present value NPV, when the un-derlying cash flow is affected by stochastic disturbances in the form of a random walk process, the disturbances thus jumping continuously between each of two values. A leverage is introduced, affecting the size of each jump in relation to time.A given time period is split into a number of sub-periods of equal length. During each sub-period, a fraction of altogether one monetary unit is transferred. This cash flow is subject to the random walk, and is discounted by a given constant discount rate. We study consequences on the distribution of the net present value as the number of sub-periods increases beyond all bounds.Two different processes are studied, which turn out to have rather different properties. In the first, Case A, the two levels between each jump, are chosen as each others negative, and in the second, Case B, except for a timing factor, as each others inverse. Among other consequences, it is shown that the probability distribution of the NPV in Case A is close to, but different from, a lognormal distribution, and that in Case B, the NPV indeed has a lognormal distribution, but lies stable at a constant value. Also is shown that in Case A, the leverage must be at least 1/2 for a limiting distribution to exist, and in Case B at least unity. For leverage values above these values, respectively, the random walk will not affect NPV, and the net present value becomes deterministic.

MRP theory concerns production-inventory systems, in which produced items are made up of sets of sub-items, either produced or purchased (imported into the system) and required to be available a lead-time before the product is completed. The hierarchical dependence of relations between items is captured by the use of input matrices from Input-Output Analysis, the necessary advanced timing by applying the methodology of Laplace transforms, and the production-economic consequences by the net present value (NPV). This theory has been developed for about 40 years, but little attention has been given to transient aspects, i.e. when there happens to be an initial available inventory of items usable in future production. In this paper, we attempt to highlight the theoretical consequences from having a positive initial available inventory.

In this paper we compare the two basic risk preference measures suggested independently by John W. Pratt (1964) and Kenneth J. Arrow (1965) as to their impact on simple production decisions with uncertain economic outcomes. We develop a further related concept by introducing a risk aversion leverage measuring the dependence of absolute risk aversion on wealth. We also discover a remarkable approximate relation between risk aversion measures of different leverages making the corresponding risk preference functions similar and leading to a method for estimating different Certainty Monetary Equivalents using zero-leverage expressions. We concentrate on three simple cases of production decisions of the Newsboy type.

In this paper we base our study on the application of the Laplace transform to risk preference theory. With a constant measure of absolute risk aversion (Pratt, 1964; Arrow, 1965), the Certainty Monetary Equivalent (CME) of a risky project previously has been developed into an expression involving the logarithm of the bilateral Laplace transform of the probability density of its stochastic economic outcome. The internal risk aversion (IRA) is the break-even level of the absolute risk aversion, between making the project favourable or unfavourable. Below, we apply this methodology to determining an expression for the incremental risk rate of interest in the case of standard investments with payments having probability distributions. A simple approximate formula is derived explaining how the incremental risk rate of interest depends on the absolute risk aversion and the discount rate, i.e. on measures of risk preference and of time preference.

In this paper, we apply the Laplace transform to risk preference theory (decision theory, decision analysis). We show that with a constant measure of absolute risk aversion, the certainty monetary equivalent (CME) can be developed into an expression involving the logarithm of the bilateral Laplace transform of the probability density of the outcome of risky projects. We also introduce a measure named internal risk aversion (IRA). This is in analogy with the earlier application of the transform to net present value problems. Properties of the CME function are examined. Basic examples of application are introduced followed by applications to portfolio theory and option pricing.

In this paper we attempt to provide a partial answer to the question of why energy is a scarce resource. Scarcity is a fundamental concept in the science of economics. If resources, goods or services were not in scarce supply, we need not economise when utilising them. Indeed, free commodities we need not pay for, their prices are zero, we attach no economic value to them, and their supply is in abundance - at least beyond the point at which our needs and wants are satisfied. However, energy is regarded as a scarce resource, although energy - as such - is not scarce. To describe energy as a useful and therefore a valuable quantity, to which a price may be attached, energy will thus have to be characterised in further dimensions than energy content alone. Apart from quantity, there is a need for a uniform qualitative measure of energy. The obvious field to revert to for such considerations is thermodynamics, which offers a method for defining a uniform measure for the qualitative content of energy, namely exergy. Although exergy is defined from purely physical properties, it is shown to have an important role to play when comparing the economic value of energy in different forms. In particular, this paper will focus on the economic value of heat, especially heat delivered through a district heating system. The concept of exergy is defined from maximising a work output reversibly taking an infinite time. However, for processes to run within finite horizons, entropy must be generated. This leads us to add finite time considerations from examining consequences from the assumed availability of so-called endo-reversible processes. In a small case example we show that heat appears to be overpriced compared to electricity from an exergetic point of view and that this is even more pronounced adopting finite time considerations. (C) 2015 Elsevier B.V. All rights reserved.

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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 Harriss 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.

In this paper we extend and apply MRP theory towards reverse logistics including the considerations of transportation consequences. Our aim is to demonstrate the versatility obtained from using MRP theory when combining Input-Output Analysis and Laplace transforms. This enables an analysis of a supply chain including four sub-systems, namely manufacturing, distribution, consumption and reverse logistics, where the geographical distance between the activities play an important role. The main focus in this paper is on reverse logistics (recycling, remanufacturing). Especially we wish to model the evaluation of disposal and reverse activities far away from agglomerations, which often means an improved environment for nearby inhabitants. This is also illustrated in a numerical example. We use the Net Present Value as a measure of the economic performance. Our ambition is to show that supply chain sub-systems may accurately be described using input and output matrices collected together in corresponding matrices for the system as a whole. Activity levels in each sub-system govern the speed of the respective processes, and these activity levels, in general, will be considered as decision variables. We now analyse reverse logistics activities in which the flows of materials and goods are typically divergent (arborescent processes), similar to properties of the distribution sub-system, and recent results on the extensions of basic MRP theory introducing the concepts of output delays and the generalised output matrix are also introduced here, when modelling the reverse logistics sub-system.

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 prod

Exergy is a concept that is gaining an increasingly wider recognition as a proper measure for the actual energy resources consumed, when energy is used. Energy as such is indestructible, but exergy is not. As entropy is generated while energy is used, exergy is consumed. Exergy can be interpreted as the qualitative content of energy, or as energy in its highest quality. Therefore, there is an interest in investigating this concept from as many theoretical aspects as possible. In earlier papers the author has developed formulae for the exergy potential of a system of finitely extended objects, not necessarily having any environment. It was there shown that the classical formula for exergy obtains as one of the objects grows beyond all bounds thereby taking on the rôle as an environment. In this current paper formulae are derived for the exergy content of an isolated body in thermodynamic disequilibrium, viewed as a system of infinitely many objects each with infinitesimal extension and in microscopic equilibrium.

MRP Theory has been developed during the last 25 years for capturing processes concerning multi-level, multi-stage production-inventory systems in a compact way. Input-output analysis has been used to describe structures, and Laplace transforms to describe the timing relations. This theory has mainly dealt with assembly systems, in which each item has only one successor. The lead times for the assembly of an item have usually been constants and equal for all items entering a given successor. For such systems, the equations describing the flows of components may be written to include the generalised input matrix as the product of an input matrix containing needed amounts, and a diagonal lead time matrix with lead time operators along its main diagonal. less thanbrgreater than less thanbrgreater thanOn occasion, there has been a need to deviate from this representation enabling lead times to vary depending on which input item that is considered. This paper deals with how to represent lead times and similar output delays (in diverging, arborescent systems), when the assumption of equal times is relaxed, in order to retain the basic structure of the fundamental balance equations involved. The intention of this paper is to create a general taxonomy for the representation of timing in algebraic form for a variety of systems covering both assembly systems and arborescent systems (such as extraction, distribution and remanufacturing), as well as systems with mixed properties. For instance, this method may be used directly for the evaluation of investments in capacity or in the location of activities in a production network, or even in a global supply chain.

MRP theory provides a theoretical background for multi-level, multi-stage production-inventory systems (material requirements planning in a general sense) together with their economic evaluation, in particular applying the net present value principle. The theory combines the use of input-output analysis and Laplace transforms, the former for capturing product structures, and the latter for incorporating timing, including time lags, lead times, and output delays. In this paper, we consider any production policy, when given any external demand as a vector-valued function of time. It is shown that in order for available inventory to be kept at finite levels at any time, the lot-for-lot (L4L) solution must be valid for the time averages of production and deliveries, irrespective of the policy followed. This analysis is carried out using properties the Laurent expansions of the transforms involved.

Inefficient locations for production, distribution and reverse logistics plants will result in excess costs no matter how well material requirements planning (MRP), inventory control, distribution and information sharing decisions are optimized. In this paper we study ways in which aspect of activity cell location decisions can be analyzed within an extended MRP model. This model has previously been extended by including distribution and reverse logistics components in a compact form, presented in Grubbstrom et al. (2007). Our aim is to demonstrate the basic differences between an approach to location problems with MRP "under the same roof" as the global supply chain, in which transportation time delays and direct transportation costs have substantial influence. We discuss possibilities of how to present location aspects in the supply chain model obtained from combining input-output analysis and Laplace transforms in four sub-systems, namely manufacturing, distribution, consumption and reverse logistics, and show how the transportation costs and lead time influenced by the location of all these activities affect the resulting net present value (NPV). Our aim is to build a model supporting decisions concerning the structure of a supply chain as an alternative to a mixed integer programming formulation. The model developed is based on the use of continuous functions describing spatial distributions of cost and customer demand. Continuous functions are embedded in the MRP extension previously introduced in Grubbstrom et al. (2007). Location decisions influence (i) production costs, because timing influences the cost of activities involved in creating a product, cf. (Grubbstrom and Bogataj, submitted for publication), and (ii) logistics costs, which refer to the procurement and physical transmission of materials through the supply chain. In this current paper we wish to combine both of these aspects into a comprehensive model, where we show the interaction between the "

The Newsboy (Newsvendor) problem is probably the simplest of all stochastic inventory problems, involving a one-time purchase decision and a stochastic sales outcome. As an investment, it can be interpreted as the simplest stochastic version of the point-in, point-out investment problem of Jevons [Jevons, W.S., Theory of Political Economy, Macmillan, London 1871]. This paper provides a compound variation of the Newsboy problem. Instead of demand simply being known as to its distribution, here demand is generated by customers arriving at different points in time requiring amounts of varying size. Arrivals follow a renewal process, and amounts required are each taken from a second independent distribution. It is shown how the optimal purchase quantity in explicit form depends on properties of the two distributions, maximising the expected net present value (NPV) of the payments involved. The solution to the compound problem will be the solution to the classical problem, if designing a special distribution for the demand process. The developments make use of the relation between the NPV and the Laplace transform, simultaneously using the Laplace transform as a moment-generating function.

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) In this paper we concentrate our attention on the question of optimal lotsizing decisions within the MRP Theory framework. MRP Theory has mainly dealt with assembly structures by which items produced downstream (on a higher level in the product structure) contain one or more sub-items on lower levels, but at each stage, the assembly activity produces only one type of output. This enables the input matrix, after enumerating all items suitably, to be organised as a triangular matrix, with non-zero elements only appearing below its main diagonal. The introduction of a diagonal lead time matrix capturing the advanced timing when required Inputs are needed, enables compact expressions to be obtained, explaining the development of key variables such as available inventory and backlogs in the frequency domain. Central in this theory is the generalised input matrix showing when and how much the internal (dependent) demand amounts to for any production plan. Previously has been demonstrated that in a one-product case, inner-corner conditions for an optimum production plan in continuous time reduce the number of possible replenishment times to a finite set of given points at which either a replenishment is made, or not. The dynamic lotsizing problem is thus turned into choosing from a set of zero-one decisions with 2(m-1) alternatives, of which one (or possibly several equivalent) solution(s) must be optimal, where m is the number of requirement (demand) events. Given these points in time and the corresponding staircase function describing cumulative demand, the optimal plan may be obtained, for instance by employing the Triple Algorithm of dynamic lotsizing. This applies either an Average C

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Material Requirements Theory (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 (NPV) principle. The time scale may be continuous ("bucketless" in MRP terminology) or discrete. Since the average cost measure may be viewed as an approximation of NPV, the theory may be applied to cases, when average costs are preferred as the objective function.

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