Ludvik Bogataj

Stochastic considerations of Grubbström—Molinder model of MRP, input-output and multi-echelon inventory systems☆

Abstract In a recent paper (Grubbstrom and Molinder, 1994) Grubbstrom and Molinder studied theoretically the dynamic relationship between MRP, input-output analysis and multi-echelon inventory systems. In this paper we upgrade this approach including in the model the uncertainty of external demand and decision theory elements. In the model we suppose that we know only the upper and lower bounds of external demand rate and, because of lack of information, that the external demand is uniformly distributed. The complete decision model for our case is considered and the advantages of this approach are demonstrated.

Optimal Lotsizing Within MRP 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 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 Cost approach, or the Net Present Value principle is applied In this paper, we extend this analysis to a general multi-item system of the assembly type. Among other issues, it is shown how the number of internal demand events depends on product structure and number of external demand events. Also the inner-corner condition is proven still to be valid in this somewhat more complex situation, when demand for lower-level items no longer is given from the outset, but instead depends on decisions concerning production of items on higher levels in the product structure. A simple dynamic program procedure is provided offering a solution to maximising the NPV for any general assembly system. To solve for the optimal production plan, the Triple Algorithm may be applied to each stage.

Supply chain coordination in spatial games

Abstract The notion that economies consist of the systems of cities originates from a theory by Christaller and Beckmann, based on a Lochean framework of retailers having an endogenously determined market area. In 1968 it was completed by Tinbergen, who has suggested production approach. This original theory explains the location pattern of the market and manufacturing centers selling goods to the local populations within given market areas. The theory was based on the idea of the urban area being a trading center. Our paper, influenced by the ideas, presents the quantitative method of building up the model of spatial hierarchy as the result of spatial games. The paper by Girlich: “On the Metric Transportation Problems and Their Solution” (Inventory Modelling, Lecture Notes of the International Postgraduate Summer School, Vol. 2, ISIR, Budapest-Portorož, 1995, pp. 13–24), gives us proper foundations for our present research. In this paper customer traveling problem is described, which helps us to define a market area. A mathematical description of market areas, defines the border between two areas under the assumption of shortages of goods in at least one market area. The study is influenced by Grubbstroms work on MRP optimization (for details see R.W. Gubbstrom, Management Science 13(7) (1967) 558–567; The Mathematical Scientist 16 (1991) 118–129; International Journal of Production Economics, in press; International Encyclopaedia of Business and Management, Routledge, London, 1996; R.W. Grubbstrom, A. Bogataj (Eds.), Storlien, 1997, FPP, Portorož, 1998; R.W. Grubbstrom, A. Molinder, International Journal of Production Economics 35 (1994) 289–311.). In our paper the problem is formulated as a non-constant sum game, which is the case of the total market area being constant and the conditions for lower central places, described for different positions of this central place in an urban hierarchy, in the case of two-level production. Differential games are suggested when the total market area changes over the time.

A market game with the characteristic function according to the MRP and input-output analysis model

Abstract In recent papers (Grubbstrom and Molinder, IJPE 1994; Bogataj and Horvat, IJPE 1996) the material requirements planning and input−output analysis have been studied using z-transform for discrete time models, Laplace transform for continuous time models and elements of decision theory for stochastic demand in the model. In the above models a single decision maker, i.e. a company, chooses an optimal decision, i.e. production level, without reference to the effect that the decision has on other decision makers and vice versa. In many business situations (also with an MRP model), however, two or more decision makers, i.e. companies, choose an action, i.e. production level, simultaneously, and the actions chosen by each player affect the rewards, i.e. profits, earned by other companies. We have determined the optimal policy for each company in a three-person game derived by the theory of the Shapley value, where the objective function is estimated by the expected value of the annuity stream using MRP theory (for a detailed coordination case for a two-level model).

Perturbations in living stock and similar biological inventory systems

A differential-delay equation (1) is used to model controlled population dynamics of a closed biosystem consisting of n different co-habitating species. From the macroeconomical and ecological point of view the numbers of members of different biological species represent components of the vector of stock sizes in a very special kind of inventory system, and the corresponding problems will be treated similarly to the standard multipart inventory level optimization problems with quadratic cost functional. Through different control techniques the actual number of specimens of every species considered should be kept as close to the desired (equilibrium) levels as possible, at the simultaneous minimization of the control cost. The theoretical sensitivity results of the first author are generalized to the case of estimating the magnitude of changes in the total cost due to - possibly simultaneous - finite magnitude perturbations in system matrices and both types of delay, i.e. delays in the state and delays in the controls, since in many real-world systems such case is frequently encountered.

The maximin criterion as an alternative to the expected value in the replanning issues

Abstract The maximin criterion is suggested as an alternative to the expected value; it is shown that when it is used in rescheduling issues, the rescheduling is needed only when circumstances change.

Inventory systems optimization for dynamic stochastic and periodical demand

Abstract In this paper a description is given of the optimization model of a production-inventory system where there is a long term increasing demand with regular periodical variations, stochastic by nature. The joint optimization of input flow and warehouse size is described. The costs of input flow depend on the step within the period and on the intensity of production. In this model the costs of risk are included in the criterion function. A dynamic parametric programming model is presented where risk rate is a parameter. The application of this model on a water supply system is described.

Optimisation of short term commercial bank loans to corporates in terms of financing operating activities in Slovenia

This paper presents the mathematical model for determination of the optimal amount of short term commercial bank loans to corporate sector in Slovenia which is based on probabilistic inventory models. The goal of this paper is complete optimisation of cash inventories of corporate sector in the national economy. The results of optimisation are important for the corporate sector and for commercial banks. The optimal order quantity is an amount of short term commercial bank loan to corporates and defines a lending potential of commercial banks in a national economy. As such is important for the central bank when conducting monetary policy. Special emphasis has been given to determinants of optimal order quantity, which reflect market conditions in national economy.

Combinatorial optimization of the location—inventory problem

Abstract In this paper we have studied the joint optimization of the input flows which have integer values, and the sizes of the storehouses (which are continuous variables) situated in the central places of different hierarchical level with different sizes of market areas. Our aim is to minimize the distribution costs (input flows and the sizes of storehouses) and the costs of risk, under some activity allocation conditions. It is supposed to have periodical movement of demand, which is a continuous stochastic variable with known probability density functions on all steps of a period. By a dynamical program on higher level new origins, new input flows could be included in the system. The combination of a parametrical programming method and Gomorys Cutting Plane algorithm is used on the first (lower) level in this optimization system.