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Purpose This paper aims to deal with integrated planning and scheduling problem in multi-site manufacturing environment and provides a comprehensive review of literature. Classification schemes and various aspects of planning and scheduling problem in multi-site manufacturing are highlighted. Design/methodology/approach A structured review methodology is adopted to classify the relevant literature. Taxonomy for classification of the problem is presented, followed by review of modelling approaches, solution strategies and challenges faced in multi-site integrated planning and scheduling problem. Findings The paper is concluded with interesting research findings and a short view on directions related to modelling approach, solution strategy and technique for further developments in the area of multi-site integrated planning and scheduling. Research limitations/implications The findings of this study would be helpful for future researchers and practitioners to provide a knowledge base and to further work in this area. Originality/value This study attempts to consolidate the diverse literature available and highlight the various aspects of planning and scheduling in multi-site manufacturing.

Small and Medium-sized Enterprises (SMEs) play a pivotal role in economic growth, industrial output, and employment creation. However, SMEs are often ignored by researchers as far as the adoption o...

PurposeThis paper aims to discuss the concept of leanness and provide an effective assessment tool for measuring leanness of small and medium enterprises (SMEs).Design/methodology/approachA hybrid interpretive structural modelling–analytical network process (ISM-ANP) approach is used to develop leanness index for SMEs. The parameters for leanness are extracted from the literature survey, and the inter-relationships among them are identified through ISM approach. Further, the ANP tool is used to derive the weights of the parameters, and the leanness index is developed for SMEs.FindingsA leanness assessment model is developed, which considers the interdependencies among leanness parameters. Continuous improvement, Just in Time and active management participation, respectively, get first, second and third ranks for leanness measurement in SMEs.Research limitations/implicationsThis study is based on the expert’s opinion, and it may tend to be biased. However, future study will be performed as empirical research to catch more explicit concept and more insights of leanness in context of SME sector.Practical implicationsThis paper provides guidelines to the managers of SMEs for measuring the leanness index and planning for future. This leanness index gives information regarding the degree of lean adoption in the organization.Originality/valueThe proposed model has been developed by the expert opinion of academicians and practitioners. The proposed model can provide guidelines and directions for managers for leanness assessment in SME context.

Supplier selection is one of the most important activities of an industry. The goal of present paper is to exhibit key elements of supplier selection and ranking of potential suppliers. A bearing manufacturing company was considering two criteria of suppliers selection, i.e., quality rating and service rating. In the current paper, six criteria have been considered instead of two for improving the supplier’s selection process which are product quality, product cost, location, delivery time, information system and service rating. First of all, the key factors involved in supplier selection have been identified, a survey has been conducted for data collection from purchase department in the company. Fuzzy AHP method and fuzzy TOPSIS method are used to calculate the criteria weights for the suppliers’ selection and to determine the ranking of the suppliers. The contribution of this study is to give improved suppliers’ selection process to the company.

PurposeGlobal competition has intensified pressure on small- and medium-sized enterprises (SMEs) to implement lean. Recently, the debate has converged to the role of lean implementation barriers (LIBs). The purpose of this paper is to contribute to this debate by exploring the LIBs in SMEs through three case studies.Design/methodology/approachA case study approach was employed followed by interpretive structural modelling (ISM) to model the interrelationship among the LIBs.FindingsThis study reveals that lack of management commitment, leadership and resources are the key barriers to lean implementation in SMEs in India. Furthermore, poor communication between different levels of the organisation and inadequate dissemination of the knowledge of lean benefits also creates hindrance in lean implementation. Managerial implications of the identified barriers for lean implementation in SMEs have been discussed.Originality/valueThe research regarding lean implementation in SMEs is scarce. This study is the first attempt of its kind to identify the lean barriers in a small industry setup through mathematical analysis.

This paper addresses an integrated production and distribution planning problem for a two-echelon supply chain network comprising of multiple manufacturers serving multiple selling locations. A novel fuzzy multi-objective mixed integer programming model is formulated considering multi-product, multi-period, and multi-site manufacturing environment. Minimization of total cost, delivery time, and backorder level are the three fuzzy objectives represented by piecewise linear membership function. Three important production and distribution aspects viz. capacity of the heterogeneous transportation, backordering for unfulfilled demand, and set-up cost/time for different products at manufacturing site are incorporated in an integrated manner to represent closeness to the real-life problem. An illustrative example inspired from a real-world case of an automobile industry is taken to demonstrate analytical results of the proposed approach. The outcome of the research indicates the practical applicability of the approach. Sensitivity analysis on objective function values is conducted for analyzing the effect of change in aspiration levels of objective function values on decision maker’s satisfaction level.

Purpose The purpose of this paper is to gain insights about the applicability of Lean principles in software industry along with the Lean implementation issues. Design/methodology/approach Exploratory case studies have been carried out in five software companies. Data were collected by observation and semi-structured interviews with project managers. In this paper, case organizations are categorized as product vs project software development (SD) organization. Findings It is found that although Lean principles are being adopted in SD projects, application of all Lean principles is not visible. This research reveals that value and flow are more relevant in SD organizations, while value stream, pull and perfection are not conspicuous. Originality/value This paper would prove invaluable to lean practitioners and researchers to gain knowledge in lean SD. The paper puts forward the key issues that should be addressed for successful adoption of lean in SD. This study set out to determine the practitioners’ perception of the applicability of lean principles in SD projects. It provides a sound basis for further empirical research on adoption of lean principles in SD organizations.

The aim of every manufacturing organization is to fulfil the demand of their customers at minimum total cost or at maximum profit. To fulfil the demand of geographically dispersed customers, manufacturing organizations need to produce products at multiple manufacturing sites located close to the customer. Although planning function in such a scenario becomes very complex, this multi-site environment has been found to improve efficiency and provide better services to customers. It involves interlinked decisions involving procurement, production and distribution at different plants and distribution sites. Production and distribution planning are very important aspects of planning in multi-site manufacturing. To compete in this globalized market, enterprises should focus on optimizing and integrating production and distribution functions simultaneously. The authors in the research work present an integrated production and distribution planning problem for a two-level supply chain consisting of multiple manufacturing sites serving multiple selling locations. The problem is formulated as a multi-objective mixed-integer programming model considering three important aspects of production and distribution planning, viz. set-up, backorder and transportation capacity. Total cost, delivery time and backlog level are the three conflicting objectives that need to be minimized. The proposed multi-objective mathematical problem is solved using pre-emptive goal programming method. The performance of the proposed model is illustrated through an example problem instance. Further analysis is conducted to visualize the effect of changing priority level on objective function and deviation variable values.

PurposeAlthough lean thinking is deemed to be a gold standard of modern production management, a lot of scepticism still remains regarding its applicability in small- and medium-sized enterprises (SMEs). The purpose of this paper is to understand the perception of lean in SMEs and establish the relationship between lean adoption and operational performance.Design/methodology/approachWith the help of a survey, data were collected from 425 SMEs in India and analyzed using structural equation modeling.FindingsOperational performance of the firms was found to be positively related to lean implementation.Originality/valueThis study also furnishes practitioners with a better understanding of lean thinking in SMEs and its impact on performance.

In the epoch of open economy and with the emergence of availability of individualised products over the Internet, Indian manufacturing industries are facing an enormous pressure to become more flexible and responsive, to accomplish customer’s varied and increasing demands. Mass customisation (MC) is about developing a customised product on demand for a particular customer after reception of a real order and producing it with the similar operational efficiency as one would anticipate from a mass-produced product. MC takes into account the merits of both the earlier systems of production, i.e. mass production and craft production. The craft production satisfies the personalised demands of customers and the mass production produces a limited variety of products at lower cost. Industries in developing countries such as India confront pressure from several perspectives to adopt MC. This study has been presented in the context of Indian manufacturing industries, and particularly for footwear industries ...

In this paper we study the degree of approximation of functions (signals) in a Besov space by trigonometric polynomials using deferred Cesàro mean. We also deduce a few corollaries of our main result and compare them with the existing results.

Rapidly increasing customer demands, competition, continuous changing scenario and accelerating pace of technological developments have put tremendous pressure on the business organization to deliver quality products at lower cost. On the same lines the software development (SD) companies need to deliver quality codes with new features at reduced cost. This can be achieved through Lean to software development projects. As the lean has been considered in different ways and has been implemented to varying extent in different sectors of the economy this paper aims to investigate as to how “lean” is viewed in software development projects and status of implementation in software development projects. First, application of lean in different types of projects viz. construction, healthcare, aerospace, new product development and service is discussed. Secondly, application of lean to SD projects is investigated at three levels: philosophy, principles (value, value stream, flow, pull and perfection) and practices/tools. The effect of lean on performance (inventory, lead time, customer satisfaction, cost, and business value) of SD projects is also analyzed. Further, “Leagile” software development and agile dominance is explored through this paper.

In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z_{p}^{(\omega)}$\end{document}Zp(ω) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p \ge1$\end{document}p≥1) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results.

This paper presents a new heuristic for solving a facility layout problem. The proposed heuristic works on the non-greedy systematic pairwise exchange of two facilities, that is 2-exchange neighbourhood search based on non-greedy strategy. The proposed heuristic is applied on a large number of test problems provided by different authors in the quadratic assignment problem (QAP) library (Burkard, Karisch, Rendl, J Glob Optim 10(1):391–403, 1997) with problem size ranging from 12 to 256. Out of the 135 test problems available in the QAP library, the proposed heuristic reached optimal solutions for 64 test problems and matched the best known available solution for the other 15 test problems. For the remaining 56 test problems, the proposed approach reports highly encouraging solutions for 44 test problems (within the 2 % of deviation from the optimal/best known solutions), and for the remaining 12 problems, the proposed approach provides fair solution in reasonable time. Comparison with other meta-heuristic approaches (Ro-TS, RE-TS, GEN and SA) shows the effectiveness of the proposed heuristic.

The manufacturing environment is now so competitive that the companies must not only continuously improve their performance, but also do it faster than others. Small and Medium Enterprises (SMEs) significantly contribute to the industrial output of an economy, and must be competitive for the national economy to be competitive in this era of international business. With almost all productivity improvement (PI) tools requiring a working knowledge of statistics, the Indian SME manager with his semi-literate workforce with little or no technical and mathematical training needs a simple and heuristic PI tool that addresses his particular needs. This article proposes a simple scheduling heuristic to improve productivity quickly and effectively and demonstrates its application and benefits through a case study in a spring manufacturing SME.

In competitive world, it becomes essential for the firms to consider the risks and uncertainties as the core concerns. To control the risks, various strategies like multiple sourcing, horizontal/ vertical integrations, buffer stocks etc are employed. But, out of many available risk mitigation strategies, SMEs usually prefer to keep excess safety stocks to manage the operational risks as they have feeble control and position in supply chain as well as limited means to employ other strategies, which requires skills and resources. Keeping appropriate safety stocks becomes a very complex and crucial problem for SMEs as excess stocks reduces the efficiency and shortage of it may make their position vulnerable in supply chains. Concerning to this issue, this paper focuses on operational risks with SME perspectives and explores the appropriate safety inventory levels using artificial neural network models and the results are further simulated with various settings of risk scenarios.

In this paper, a multi agent based decentralized decision making approach is presented for allocating resources and scheduling of distributed multiple projects. The resource allocation and scheduling is performed cooperatively and collectively by a group of autonomous project agents, resource agents and a mediator agent. We propose a price mechanism that allows project agents to bid for the needed resources. A new contract net protocol (CNP) concept is used for negotiations and coordination among the agents. This approach is successfully applied on an example problem.

PurposeToday's customer‐focused paradigm of business environment puts tremendous pressures of quality, delivery, dependability, flexibility and cost on the manufacturing organisation. Automatic manufacturing systems offer several advantages and are increasingly being adopted as a strategy to improve the performance of manufacturing organisations. Automatic manufacturing systems are highly sophisticated and expensive, and it is therefore important to maximise their productivity. Yet, one can improve only what one can measure. Performance measurement is the key to improving performance, and is a prerequisite to diagnosing, trouble‐shooting and improving the production system. Accordingly, performance measurement has been attracting increasing attention over the last two decades, and several frameworks have emerged for the design, review, evaluation and improvement of performance measurement systems for businesses and manufacturing organizations. The performance measurement, monitoring and continuous productivity improvement of automatic manufacturing systems has assumed special significance on account of their high investments and operating costs.Design/methodology/approachA review of the current literature is undertaken to determine the current status and the status of performance measurement in automated production systems.FindingsOverall equipment effectiveness (OEE) has emerged as one important and universally accepted metric for measuring the overall performance of single automatic equipments. OEE has been further adapted and extended into several variations for use as a metric for automatic manufacturing systems consisting of several automatic machines.Originality/valueThis paper reviews the recent developments and the current status of performance measurement of automatic manufacturing systems.

Mittal, Rhoades [5] , [6] , [7] , [8] and Mittal et al. [9] , [10] have initiated a study of error estimates E n ( f ) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4] , where he has weakened the conditions on { p n } given by Chandra [2] , to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15] .

Let B n ( x ) denote the n th term of the conjugate series of a Fourier series of function f . Mohanty and Nanda [R. Mohanty, M. Nanda, On the behavior of Fourier coefficients, Proc. Am. Math. Soc. 5 (1954) 79–84] were the first to establish a result for C 1 -summability of the sequence { nB n ( x )}. Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795] improved it for the product summability H 1 · C 1 , which was generalized by various investigators using different summability methods with different set of conditions. In this note, we extend the result of Mittal [M.L. Mittal, On the ‖ T ‖ · C 1 summability of a sequence of Fourier coefficients, Bull. Cal. Math. Soc. 81 (1989) 25–31], which in turn generalizes the results of Prasad [K. Prasad, On the ( N , p n ) · C 1 summability of a sequence of Fourier coefficients, Indian J. Pure Appl. Math. 12 (7) (1981) 874–881] and Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795].

The manufacturing organizations around the globe have been observing a paradigm shifting from mass production to mass customization (MC). Mass customization capability (MCC) is the ability of a firm to produce varieties of customized products quickly, on a large scale and at a cost comparable to mass-production through flexible production processes and responsive supply chain. The goal of this paper to report a existing mass customization capability level (MCCL) appraisement of Indian bicycle manufacturing firm using multi-grade fuzzy approach. Through this study, a MCCL measurement model integrated with multi-grade fuzzy approach was designed. After the computation of MCCL index, the areas for improvement have been identified to adopt and implement mass customization strategy in the case firm. The approach contributed in this paper could be employed as appraisement equipment for estimating an organization’s capability to turn out to be a mass customizer.

In continuation of a recent work by Mittal [M.L. Mittal, On strong Norlund summability of Fourier series, J. Math. Anal. Appl. 314 (2006),75-84], the present authors obtain a sufficient condition for the summability [N, p n (1) , 2] of the conjugate Fourier series. In conjunction with the known Tauberian theorem on the strong Norlund summability, which was also considered earlier by Mittal [M.L. Mittal, A Tauberian theorem on strong Norlund summability, J. Indian Math. Soc. (N.S.) 44 (1980), 369-377], our result gives a sufficient condition for the summability [C, 1,2] of the conjugate Fourier series. Our main theorem generalizes the results given earlier by Prasad [G. Prasad, On Norlund Summability of Fourier Series, Ph.D. thesis, University of Roorkee, Roorkee, 1967] and Singh [U.N. Singh, On the strong summability of a Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India Part A 13 (1947), 319-325].

In this note, a sufficient condition for summability [ N , p n ( 1 ) , 2 ] of Fourier series has been obtained which in conjunction with the author's Tauberian theorem [M.L. Mittal, A Tauberian theorem on strong Norlund summability, J. Indian Math. Soc. 44 (1980) 369–377] on strong Norlund summability gives a sufficient condition for summability [ C , 1 , 2 ] of a Fourier series. This generalizes results due to Prasad [G. Prasad, On strong Norlund summability of Fourier series, Univ. Roorkee Res. J. 9 (1966–1967) 1–10] and Varshney [O.P. Varshney, Note on H 2 summability of Fourier series, Boll. Un. Mat. Ital. 16 (1961) 383–385].

To compete in this globalised market, enterprises should focus on optimising and integrating production and distribution functions simultaneously. This study deals with the problem of integrated production and distribution planning in multi-site manufacturing environment. A multi-objective mixed integer programming model is formulated considering three important aspects of production and distribution planning; capacity of heterogeneous transportation, setup and backordering. Three conflicting objectives that need to be minimised are total cost, delivery time and backorder level. To solve the proposed multi-objective mathematical model, preemptive goal programming approach is implemented. A real world case example of an automobile industry is considered to demonstrate analytical results of the proposed approach. Further analysis is conducted to visualise effect of changing priority level on objective functions and deviation variables. The results demonstrate flexibility of decision maker by adjusting priorities of objective functions according to importance and aspiration level with respect to desired target values.

In today's competitive globalised market, it is imperative for enterprises to improve their efficiency, provide better service to customers, and facilitate better production and distribution activities. To do so, organisations are opting for producing at multiple manufacturing sites situated at geographically distinct locations. In the last two decades, there has been seen an increasing trend in articles working in the field of multi-site production and distribution planning. This paper defines multi-site production and distribution planning problem and provide direction for researchers and practitioners to understand the field and develop mathematical models in this area. The main objective of this paper is to analyse the existing mathematical models for identification of current research trend and future research directions. The literature is classified based on the problem formulation approach, aspects of the model, solution approach and applications. Following the classification of the articles; discussion, research findings and future research directions are indicated.

Increase in the demand of products causes a company to change its manufacturing system from single-plant to multi-plant (or, multi-site) with an objective to gain competitive advantage in the global manufacturing scenario. Multi-site production and distribution (P-D) planning have been studied and practiced by the researchers and practitioners over the past two decades. A good size of literature has been published and trend is continuing. In the light of impact of multi-site manufacturing in gaining commercial advantage in global competition scenario, this review is dedicated to multi-site aggregate production planning (MSAPP) and multi-site aggregate P-D planning (MSAPDP). This review focuses on analysing the major research gaps and unaddressed areas of MSAPP/MSAPDP. It has been felt that switching from discrete to continuous and pure mathematical to simulation and agent based models along with consideration of economical and environmental aspects have become very important to effectively deal with real world MSAPP/MSAPDP problems.

ON STRONG MATRIX SUMMABILITY

Mittal and Rhoades (Int. J. Math. Game Theory Algebra 9(4), 259–267, 1999 [9]; J. Comput. Anal. Appl. 2(1) 1–10, 2000 [10]) and Mittal et al. (J. Math. Anal. Appl. 326(1) 667–676, 2007 [7]; Appl. Math. Comput. 217(9), 4483–4489, 2011 [8]) initiated the studies of error estimates \(E_{n}(f)\) through trigonometric-Fourier approximation (tfa) for situations in which the summability matrix T does not have monotone rows. In this paper, we extend the results of Mittal et al. (Appl. Math. Comput. 217(9), 4483–4489, 2011 [8]) to a more general \(C_{\lambda }\)-method in view of Armitage and Maddox (Analysis 9, 195–204, 1989 [1]), which in turn generalizes the several previous known results due to Mittal and Singh (Int. J. Math. Math. Sci., Art. ID 267383, 1–6, 2014 [11]), De\({\tilde{\mathrm{g}}}\)er et al. (Proc. Jangjeon Math. Soc. 15(2), 203–213, 2012 [4]), Leindler (J. Math. Anal. Appl. 302, 129–136, 2005 [6]), Chandra (J. Math. Anal. Appl. 275, 13–26, 2002 [3]) and Quade (Duke Math. J. 3(3), 529–543, 1937 [15]).

ON MULTIPLIERS FOR THE ABSOLUTE MATRIX SUMMABILTTY

ON A SEQUENCE OF FOURIER COEFFICIENTS

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order α: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result.

We compute the degree of approximation of functions, in the H$ouml;lder metric,using lower triangular nonnegative matrices. Ourwork generalizes that of Singh [5], and other authors, since we do notassume any monotonicity condition on the rows of the matrix.

The study of ( F )-effectiveness of the N p -method was initiated by Hille and Tamarkin. In this direction, Dikshit investigated ( F 1 )-effectiveness of the product method C 1 N p -method. Recently Kumar and Prasad established the necessary condition for ( F 1 )-effectiveness of the product method. In this note, we extend the corresponding sufficient part of Kumar and Prasad to the C 1 T -method.

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimatesEn(f)through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrixTdoes not have monotone rows. In this paper, the first author continues the work in the direction forTto be aNp-matrix. We extend two theorems on summability matrixNpof Deger et al. (2012) where they have extended two theorems of Chandra (2002) usingCλ-method obtained by deleting a set of rows from Cesàro matrixC1. Our theorems also generalize two theorems of Leindler (2005) toNp-matrix which in turn generalize the result of Chandra (2002) and Quade (1937).

Different authors have used different definitions of strong Norlund summability [ N , p n , q ]. In this paper, after discussing these definitions, we have obtained a theorem which gives an equivalence between the definitions of Borwein and Cass and Srivastava under certain conditions on { p n }. We have also proved several other necessary and sufficient conditions for strong Norlund summability.

Mittal, Rhoades (1999–2001), Mittal et al. (2005, 2006, 2011) have initiated a study of error estimates through trigonometric Fourier approximation (tfa) for the situation in which the summability matrix [Formula: see text] does not have monotone rows. Recently Mohanty et al. (2011) have obtained a theorem on the degree of approximation of functions in Besov space [Formula: see text] by choosing [Formula: see text] to be a Nörlund ([Formula: see text])-matrix with non-increasing weights [Formula: see text]. In this paper, we continue the work of Mittal et al. and extend the result of Mohanty et al. (2011) to the general matrix [Formula: see text].

This paper proposes a non-greedy local search heuristic for solving facility layout problem. The proposed heuristic works on non-greedy systematic pair wise exchange of two facilities, that is 2-exchange local search based on non-greedy strategy. Pair wise exchanges are accepted if the objective function value after the exchange is lowered or smaller than the average objective function increment divided by an intensity factor. Proposed heuristic is tested on commonly used Nugxx series problems and computational results show efficiency and effectiveness of proposed heuristic.

In order to be more efficient, firms have adopted strategies such as outsourcing, global partnerships and lean practices. Although such strategies have tremendous abilities to improve the efficiencies but simultaneously they make the firms vulnerable to market uncertainties, dependencies and disruptions. Moreover, natural calamities and manmade crises have also put negative impact on strategic, operational and tactical performance of supply chains. These factors have triggered the interest of academia and industry to consider the risk issues as prime concerns. To capture the more fine-grained elements of diversified risk issues related to the supply chain we employ a multi-layered top town taxonomy to classify and codify the literature and put forward the probable dimensions for future research. We further study the pool of SCRM literature focusing on coordination, decision making and sector-wise SCRM implementation issues and derive relevant propositions.

1.1 Origin The traveling salesman problem (TSP) were studied in the 18th century by a mathematician from Ireland named Sir William Rowam Hamilton and by the British mathematician named Thomas Penyngton Kirkman. Detailed discussion about the work of Hamilton & Kirkman can be seen from the book titled Graph Theory (Biggs et al. 1976). It is believed that the general form of the TSP have been first studied by Kalr Menger in Vienna and Harvard. The problem was later promoted by Hassler, Whitney & Merrill at Princeton. A detailed dscription about the connection between Menger & Whitney, and the development of the TSP can be found in (Schrijver, 1960).

Mohanty and Nanda (1959) were the first to establish a result for the $(C,1)$ i.e. $C_1$-summability of the sequence $\{n B_n(x)\}$. Varshney (1959) improved the result for $(N, \frac{1}{n+1} ) \cdot C_1$ summability which was generalised by several investiga- tors such as Sharma (1970), Singh (1963), Lal (1971), Khare and Singh (1988) etc. In this note, we have·discussed $||T||\cdot C_1$-summability of the sequence $\{n B_x(x)\}$ which includes the result due to Khare and Singh (1988).

Publisher Summary This chapter deals with fluid flow in a Sudden Expansion Geometry for moderately high Reynolds number. Fluid flow in sudden expansion geometries, with or without solid particles, is an important technological process for many industrial and scientific applications, especially for combustion processes. For such geometries, even with moderately high values of the Reynolds number, the separated boundary-layer becomes unstable, and the flow exhibits repeated vortex formation and shedding at the corner point near the sudden expansion. The formation and persistence of large scale coherent vortex structures within turbulent wake-like regions give rise to turbulent behavior. The full time dependent Navier-Stokes equations are solved in this geometry. To resolve all the prevailing excited scales of the fluid flow, a fine staggered grid is used. The fine mesh and small time steps necessitate computation on scalable parallel machines. The chapter discusses the performance of parallel implementation and computations. It concludes that the performance of the parallel computations of the fluid flow for the present application is very encouraging as seen from the speedup curve, although millions of floating point operations per second (Mflops) and per processor performance is well below the optimal.

This paper proposes a new heuristic approach for solving facility layout problem (FLP) which is traditionally formulated as quadratic assignment problem (QAP). In this paper, FLP is formulated as linear assignment problem (LAP) consisting of facility pair and location pair. Being linear in nature LAP can be solved efficiently. Solution of LAP provides lower bound on corresponding QAP formulation of FLP. Heuristic procedure is applied to solve FLP from sets of LAP solution. Proposed heuristic is tested on benchmark instances taken from literature and compared with other heuristics available in literature. Computational results show that proposed heuristic provides a good quality approximate solution. This solution can be taken as initial solution for any improvement heuristic to get optimal/near optimal solution for FLP.