On the combinatorics of the number of even parts in all partitions with distinct parts

The Ramanujan Journal - Recently, Andrews and Merca obtained two identities concerning the number of even parts in all partitions of n into distinct parts. In this paper, we provide bijective...     

Efficient approaches for furnace loading of cylindrical parts

This paper addresses the heat treatment operation in a manufacturing plant that produces different types of cylindrical parts. The immediate prior process to heat treatment is furnace-loading, where parts are loaded into baskets. The furnace-loading process is complex and involves issues relating to geometry, and heterogeneity in the parts and in their processing requirements. Currently, furnace-loading is accomplished by operator ingenuity; consequently, the parts loaded in heat treatment often do not use furnace capacity adequately. Efficiency in furnace operation can be achieved by improving basket utilization, which is determined by the furnace-loading process. This paper describes the development of integer and mixed integer LP models for 3D loading of cylindrical parts into furnace baskets. The models consider the exact location of parts to be loaded on the basket and incorporate three models with different objectives; the first addresses the nesting of parts within one another, the second addresses the number of basket layers used, and the third addresses the number of baskets used.     

On 5-regular bipartitions with even parts distinct

The Ramanujan Journal - In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts...     

Initial supply and re-order level of new service parts

When a new product is introduced to the market, it may be necessary to estimate in advance the demand for service parts. Historical demand data are by definition lacking, so that standard forecasting techniques cannot be applied. In this paper we present a case study in which the failure rate of the parts and the number of end products in operation were used to estimate the demand and the re-order level during the first few months. Both the applied theoretical model and a few practical results have been included, whereby it is assumed that the number of products in operation grows linearly.     

Compositions with distinct parts

The number of compositionsC(n) of a positive integern into distinct parts can be considered as a natural analogue of the numberq(n) of distinct partitions ofn. We obtain an asymptotic estimate forC(n) and in addition show that the sequence {C(n, k)} of distinct compositions ofn withk distinct parts is unimodal. Our analysis is more complicated than is usual for composition problems. The results imply however that the behaviour of these functions is of comparable complexity to partition problems.     

Partitions and overpartitions with attached parts

We show how to interpret a certain q-series as a generating function for overpartitions with attached parts. A number of families of partition theorems follow as corollaries. Received: 12 April 2006     

A stochastic model for joint spare parts inventory and planned maintenance optimisation

Spare parts demands are usually generated by the need of maintenance either preventively or at failures. These demands are difficult to predict based on historical data of past spare parts usages, and therefore, the optimal inventory control policy may be also difficult to obtain. However, it is well known that maintenance costs are related to the availability of spare parts and the penalty cost of unavailable spare parts consists of usually the cost of, for example, extended downtime for waiting the spare parts and the emergency expedition cost for acquiring the spare parts. On the other hand, proper planned maintenance intervention can reduce the number of failures and associated costs but its performance also depends on the availability of spare parts. This paper presents the joint optimisation for both the inventory control of the spare parts and the Preventive Maintenance (PM) inspection interval. The decision variables are the order interval, PM interval and order quantity. Because of the random nature of plant failures, stochastic cost models for spare parts inventory and maintenance are derived and an enumeration algorithm with stochastic dynamic programming is employed for finding the joint optimal solutions over a finite time horizon. The delay-time concept developed for inspection modelling is used to construct the probabilities of the number of failures and the number of the defective items identified at a PM epoch, which has not been used in this type of problems before. The inventory model follows a periodic review policy but with the demand governed by the need for spare parts due to maintenance. We demonstrate the developed model using a numerical example.     

A direct bijection between descending plane partitions with no special parts and permutation matrices

We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.     

Crucial combinations of parts for handwritten alphanumeric characters

A combining approach has been studied previously to integrate different parts of handwritten characters for their analysis and recognition. Perfect combinations, by which the characters can be identified with certainty, are important to pattern analysis and character recognition. However, a large number of possible combinations (e.g., 63 combinations for a character partitioned into six parts), also produce a lot of perfect combinations. Hence, it is necessary to determine which of them are most important. In this paper, we propose a methodology of finding the basic crucial combinations, and algorithms to compute them. Compared with perfect combinations, such basic crucial combinations are most significant to the character distinctiveness. Similarly, the largest confusion regions are also identified.Experimental studies have also been conducted using the 89 most frequently used patterns of 36 alphanumeric handprints, to obtain their basic crucial combinations and largest confusion combinations. The results indicate that the ratio of the number of basic crucial combinations to perfect combinations is only 12.6%, and the ratio of the number of the largest confusion regions to the total confusion combinations is 15.6%.     

Integer parts of powers of Pisot and Salem numbers

Let a \alpha be a Salem number or a Pisot number. We prove that integer parts of its powers [aⁿ] [\alpha^n] are composite for infinitely many integers n.     

Recognizing polygonal parts from width measurements

Automatic recognition of parts is an important problem in many industrial applications. One model of the problem is: given a finite set of polygonal parts, use a set of “width” measurements taken by a parallel-jaw gripper to determine which part is present. We study the problem of computing efficient strategies (“grasp plans”), with the goal to minimize the number of measurements necessary in the worst case. We show that finding a minimum length grasp plan is -hard, and give a polynomial time approximation algorithm that is simple and produces a solution that is within a log factor from optimal.     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the number of partitions with distinct even parts

Let ped(n) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for ped(n)mod2 and mod4 by the theory of Hecke eigenforms.     

A new partial pooling structure for spare parts networks

Motivated by real-life spare parts networks, we introduce a new spare parts inventory model with lateral transshipment. We consider a multi-item, multi-location, single-echelon system with base stock control and aggregate mean waiting time constraints. The local warehouses are divided into two types: main and regular local warehouses. Lateral transshipment is allowed from main local warehouses only. A practical advantage of this structure is that only a limited number of local warehouses has to be equipped to provide lateral transshipment. This structure represents a new form of partial pooling, with no pooling (zero main locals) and full pooling (zero regular locals) as special cases. We develop an accurate and fast approximate evaluation method, and exploit this method in a heuristic procedure for the base stock level determination. We show that only a small number of main locals is sufficient to obtain most of the full pooling benefits. We also apply our methods to case data of ASML, an original equipment manufacturer in the semiconductor supplier industry. As a result of our work ASML was able to improve spare parts planning.     

Optimal provisioning strategies for slow moving spare parts with small lead times

When an expensive piece of equipment is bought, spare parts can often be bought at a reduced price. A decision must be made about the initial provisioning of spare parts. Furthermore, if at a certain time the stock drops to zero, because a number of failures have occurred, a decision must be made about the number of parts to be ordered. We focus on one specific expensive slow-moving part that is essential for the functioning of the equipment. The lead time of that part is small compared to its lifetime. We seek an ordering strategy for this part that covers the entire lifetime. Such a strategy should also be applicable to cases where there is uncertainty about the lifetime of the machine, the reliability of the components and the cost of failure. The main result is the development of a simple strategy that performs well in almost all cases. Furthermore, this strategy can easily deal with uncertainty and changes in the parameters.     

Computing fence designs for orienting parts

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of a polygonal part by a sequence of fences. We show that any polygonal part can be oriented by a sequence of fences placed along a conveyor belt, thereby settling a conjecture by Wiegley et al. (1997), and present the first polynomial-time algorithm to compute the shortest such sequence. The algorithm is easy to implement and runs in time O(n³ logn), where n is the number of vertices of the part.     

The number of partitions of a natural number n into parts each of which is not less than m

Mathematical Notes - We present recurrence formulas for the number of partitions of a natural number n whose parts must be not less than m. A simple proof of Euler’s formula for the number of...     

A two-step method for forecasting spare parts demand using information on component repairs

Forecasting spare parts demand is notoriously difficult, as demand is typically intermittent and lumpy. Specialized methods such as that by Croston are available, but these are not based on the repair operations that cause the intermittency and lumpiness of demand. In this paper, we do propose a method that, in addition to the demand for spare parts, considers the type of component repaired. This two-step forecasting method separately updates the average number of parts needed per repair and the number of repairs for each type of component. The method is tested in an empirical, comparative study for a service provider in the aviation industry. Our results show that the two-step method is one of the most accurate methods, and that it performs considerably better than Croston’s method. Moreover, contrary to other methods, the two-step method can use information on planned maintenance and repair operations to reduce forecasts errors by up to 20%. We derive further analytical and simulation results that help explain the empirical findings.     

Divisions of symmetric designs into two parts

We answer a question of Peter Cameron by showing that no non-degenerate symmetric design has a division into two parts each of which is a symmetric design. We also exhibit a class of symmetric designs each member of which has a division into two parts such that the parts are symmetric tactical configurations with the same parameters. These parts arise as tactical configurations consisting of (i) absolute elements, and (ii) non-absolute elements, of certain highly uniform polarities.     

On the number of partitions of n into k different parts

We study the number of partitions of n into k different parts by constructing a generating function. As an application, we will prove mysterious identities involving convolution of divisor functions and a sum over partitions. By using a congruence property of the overpartition function, we investigate values of a certain convolution sum of two divisor functions modulo 8.