Knut Richter

The EOQ repair and waste disposal model with variable setup numbers

In a previous paper the author discussed an EOQ model in which the stationary demand can be satisfied by newly made products and by repaired used products. In the modelled situation some share of the used products is collected and later repaired; the other products are disposed outside according to some waste disposal rate. In the present paper this model is extended to the case of variable setup numbers n and m for production and repair within some collection time interval. First, for a fixed waste disposal rate the cost optimal setup numbers and the minimum cost are determined. Secondly, the minimum cost is analysed as a function of this rate and it is shown to be convex for small and medium waste disposal rates and to be concave for large rates. Thirdly, the existence and generation of cost optimal waste disposal rates are discussed.

The extended EOQ repair and waste disposal model

Abstract In this paper an EOQ model is studied in which the stationary demand can be satisfied by newly made products and by repaired used products. This model assumes that the used products are collected and later repaired at some rate and the other products might be disposed outside according to some waste disposal rate. This model extends previous studies to the case of variable setup numbers n and m for production and repair within some variable collection time interval. The behavior of the setup numbers, the length of the collection interval and the minimum cost regarded as function of the waste disposal rate is discussed.

Remanufacturing planning for the reverse Wagner/Whitin models

Abstract In this paper the reverse Wagner/Whitin’s dynamic production planning and inventory control model and some of its extensions are studied. In such reverse (product recovery) models, used products arrive to be stored and to be remanufactured at minimum cost. For the reverse model with given demand the zero-inventory-property of optimal solutions is proved, the corresponding Wagner/Whitin algorithm is presented and the stability of optimal solutions is discussed for the case of a large quantity of low cost used products. Furthermore, the model of the alternate application of remanufacturing and manufacturing processes is analysed. Again, for the case of a large quantity of low cost used products the zero-inventory-property is proved, the Wagner/Whitin algorithm is applied to determine the periods in which used products are remanufactured or new products are produced, and the stability of Silver/Meal solutions for this model is sketched.

The reverse Wagner/Whitin model with variable manufacturing and remanufacturing cost

Abstract In this paper, the reverse Wagner/Whitin model is extended to the case with additional variable manufacturing and remanufacturing cost. This model can be regarded as a combination of the classical Wagner/Whitin model and a pure reverse Wagner/Whitin model with given returns of used products. The combined model is more appropriate to practice, if the alternate application of remanufacturing and manufacturing processes is analysed. For the case of time-constant cost and demand data we prove the optimality of a policy starting with remanufacturing before switching to manufacturing and give an estimation for the optimal switching point. Finally, it is investigated how the disposal of excess inventory would change the solution.

Pure and mixed strategies for the EOQ repair and waste disposal problem

In this paper the analysis of the EOQ repair and waste disposal model with variable setup numbers for production and repair within some collection time interval is continued. The cost analysis is now extended to the extreme waste disposal rates and it is shown that the pure (bang-bang) policy of either no waste disposal (total repair) or no repair (total waste disposal) dominates the strategy of mixing waste disposal and repair. Moreover, the different behavior of the minimum cost, of the optimal setup numbers, lot sizes and collection intervals for small and large waste disposal rates is discussed.ZusammenfassungIn diesem Beitrag wird die Analyse des EOQ-Reparatur- und Entsorgungsproblems mit variablen Loszahlen für die Produktion und die Reparatur innerhalb eines Sammelzeitintervalls fortgesetzt. Die Kostenanalyse wird hier auf extreme Werte der Entsorgungsrate ausgedehnt und es wird gezeigt, daß die reine (bang-bang) Politik des vollständigen Verzichts auf Entsorgung (vollständige Reparatur) oder auf Reparatur (vollständige Entsorgung) die Mischstrategie von Entsorgung und Reparatur dominiert. Weiterhin wird das unterschiedliche Verhalten der minimalen Kosten, der optimalen Loszahlen, Losgrößen und Sammelzeitintervalle für geringe und große Entsorgungsraten diskutiert.

Analysis of the EOQ repair and waste disposal problem with integer setup numbers

In this paper EOQ repair and waste disposal model with variable setup number n and m for production and repair, respectively, within some collection time interval is considered as an integer nonlinear program. The cost analysis is extended to the extreme waste disposal rates and it is shown that the pure strategy for either total repair or total waste disposal is dominant. Furthermore, the minimal cost function and the optimal solution for that integer program is characterised and optimal and approximate solutions are presented for several subclasses of the problem.

An O(T3) algorithm for the capacitated lot sizing problem with minimum order quantities

This paper explores a single-item capacitated lot sizing problem with minimum order quantity, which plays the role of minor set-up cost. We work out the necessary and sufficient solvability conditions and apply the general dynamic programming technique to develop an O(T3) exact algorithm that is based on the concept of minimal sub-problems. An investigation of the properties of the optimal solution structure allows us to construct explicit solutions to the obtained sub-problems and prove their optimality. In this way, we reduce the complexity of the algorithm considerably and confirm its efficiency in an extensive computational study.

Stability of the constant cost dynamic lot size model

Abstract A mathematical analysis of the dynamic lot size model with constant cost parameters is provided. First, stability regions for so called generalized and optimal solutions are found, which show how the cost input may vary, leaving the solution valid. Based on these results a BAsic dialog program has been designed to display the optimal solution and the stability regions to the decision maker. Secondly, an estimation of the initial optimal solution is given for the case, when the cost inputs leave the stability region.

Solving a multiobjective traveling salesman problem by dynamic programming

In this paper Klotzlers method of multiobjective dynamic programming is applied to the solution of a two-dimensional traveling salesman problem. In this way Bellmans and Held/Karps dynamic programming approach to one-dimensional traveling salesman problems is extended to the multiobjective case.

The market-oriented dynamic product recovery model in the just-in-time framework

Abstract A market-oriented deterministic dynamic product recovery problem and its multi-echelon model formulation are discussed in this paper. For this general non-linear alternate deterministic dynamic product recovery model a dynamic programming solution procedure is developed. Furthermore, the subcases of the Just-in-time situation with regard to the suppliers as well as customers are considered, for which the two-stage problem is transformed to easily solvable one-stage models. It will be seen that the regarded recovery problem in this formulation is just a special multi-echelon problem with two alternative production options.