METHODS AND MATHEMATICAL MODELS FOR THE OPTIMIZATION IN LOGISTICS AND PRODUCTION

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(1)Università di Pisa Scuola di Dottorato “Leonardo da Vinci”. Ph.D. Programme in Mechanical Engineering. Ph.D. Dissertation. METHODS AND MATHEMATICAL MODELS FOR THE OPTIMIZATION IN LOGISTICS AND PRODUCTION. Davide Castellano. SSD ING-IND/17 2015.

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(3) Università di Pisa Scuola di Dottorato “Leonardo da Vinci”. Ph.D. Programme in Mechanical Engineering. Ph.D. Dissertation. METHODS AND MATHEMATICAL MODELS FOR THE OPTIMIZATION IN LOGISTICS AND PRODUCTION. Tutors:. Candidate:. Prof. Ing. Marcello Braglia. Davide Castellano. Dott. Ing. Roberto Gabbrielli. (ciclo XXVII). SSD ING-IND/17 2015.

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(5) i. Abstract This dissertation extensively describes the activities carried out during the Ph.D.. With particular regard to the manufacturing industry, it is widely known the key role covered by the Operations Management. In brief, it assures that business operations are efficient in terms of using as few resources as needed, and effective in terms of meeting customer demand. With the aim of providing practitioners with innovative operating tools, the research activity carried out focuses on the development of methods and mathematical models approaching several questions within logistics, production management, and maintenance. When the problem can be faced analytically, i.e., mathematically, the approach is able to give an optimal, or quasi-optimal, solution. The objective of this research is thus to provide a significant contribution to extending the toolbox available to all practitioners in the manufacturing industry, useful to find optimal or heuristic solutions, and to investigate several issues within the Operations Management area. The work done concretized in: (i) five models related to maintenance, (ii) five models concerning production management, and (iii) twelve models dealing with logistic issues. The models and the approaches were developed exploiting several advanced techniques: Artificial Intelligence, Markov Processes, Multivariate Statistics, and Simulation. Since the issues related to Operations Management are still growing, future work may be thus devoted to further extending the research here presented..

(6) ii. Sommario Questa dissertazione descrive estensivamente le attività svolte durante il dottorato. Con particolare riferimento al settore manifatturiero, è ampiamente riconosciuto il ruolo chiave ricoperto dall’Operations Management. In breve, essa assicura che le attività di business siano efficienti, in termini di sfruttamento delle risorse strettamente necessarie, ed efficaci, in termini di soddisfacimento delle richieste del cliente. Con l’obiettivo di dare ai professionisti strumenti operativi innovativi, l’attività di ricerca svolta si focalizza sullo sviluppo di metodi e modelli matematici che approcciano diverse questioni nell’ambito della logistica, della gestione della produzione e della manutenzione. Quando il problema può essere affrontato analiticamente, i.e., matematicamente, l’approccio è in grado di fornire una soluzione ottima o quasi-ottimale. L’obiettivo di questa ricerca è, quindi, fornire un contributo significativo all’estensione dell’insieme degli strumenti a disposizione di tutti i professionisti nel settore manifatturiero, utile per ricercare soluzioni ottimali o euristiche e per investigare diverse problematiche nell’area dell’Operations Management. Il lavoro svolto si è concretizzato in: (i) cinque modelli relativi alla manutenzione, (ii) cinque modelli concernenti la gestione della produzione e (iii) dodici modelli che trattano problematiche di logistica. I modelli e gli approcci sono stati sviluppati fruttando tecniche avanzate: Intelligenza Artificiale, Processi di Markov, Statistica Multivariata e Simulazione. Siccome le problematiche relative all’Operations Management sono in continua crescita, il lavoro futuro può quindi essere dedicato all’ulteriore estensione della ricerca qua presentata..

(7) iii. Preface Generally speaking, firms are characterized by sustaining a twofold relationship within their operating environment. In brief, they operate against competitors to gain and maintain market share. Hence, they are related to the other firms within the industry and to customers through the market. It is further well-known that the operating environment is under a continuous evolution. In fact, the industry tends to become even denser (as the maturity of the sector grows), causing a persistent reduction of profit margins and a high inner turn over. In addition, customer needs become even more diverse, tightening, and changing. Also, the need of firms to cope with rapid technological changes should not be neglected. In such a dynamic context, a firm’s survival depends upon its ability to react to changes, and on the necessity of maintaining the competitive edge. These dictates are strictly linked to the concepts of efficiency, effectiveness, and continuous improvement. In this framework, a strategic role is hold by Operations, which deals with the capacity of a firm to deliver goods and/or services efficiently (in terms of using as few resources as needed), effectively (in terms of meeting customer requirements), and with the objective of continuously improving each activity and process involved. In a broad sense, with Operations we refer to the application of resources (capitals, materials, technology, and human skills and knowledge) to the production of goods and services. The term “operations” also denotes a specific function in an organization, distinct from other functions such as product design, accounting, marketing, finance, human resources, and information systems. People involved in the operations function typically deal with plant scheduling, inventory control, quality assurance, workforce scheduling, materials management, equipment maintenance, capacity planning, and whatever else is needed to deliver the product. Today, the importance of Operations to the health of a firm is greater than ever due to the global competition in the following three essential dimensions: (i) cost, (ii) quality, and (iii) speed (that is related to the so-called Time-Based Competition). Operations are traditionally studied within the Operations Management field. It is an area of management concerned with overseeing, designing, and controlling the process of production and redesigning business operations in the production of goods and services. Concisely, Operations Management is the activity of managing the resources (i.e., the inputs) to produce and deliver goods and services (i.e., the outputs)..

(8) iv. Operations Management exploits the principles of Scientific Management as it uses analytical thinking to deal with any problem related to the execution of business operations. In this regard, Operations Management uses the quantitative methods that the disciplines Operations Research and Management Science (whose names are often treated as respective synonyms) provide. The research activity presented in this dissertation is inserted in this context, and is in particular directed to firms operating within the manufacturing industry. With the aim of providing practitioners with innovative operating tools, the research activity carried out focuses on the development of methods and mathematical models approaching several questions within logistics, production management, and maintenance. When the problem can be faced analytically, i.e., mathematically, the approach is able to give an optimal, or quasi-optimal, solution. The objective of this research is thus to provide a significant contribution to extending the toolbox available to all practitioners in the manufacturing industry, useful to find optimal or heuristic solutions, and to investigate several issues within the Operations Management area. The work done concretized in: (i) five models related to maintenance, (ii) five models concerning production management, and (iii) twelve models dealing with logistic issues. The models and the approaches were developed exploiting several advanced techniques (Artificial Intelligence, Markov Processes, Multivariate Statistics, and Simulation) and within the frameworks Probability Theory and Operational Research. Each model will be presented in a separate chapter, and can be read independently from the others without jeopardize its understanding. That is, the reader is not forced to follow the chapters order to read this dissertation. An overview of each problem studied is now given, taking into account the subdivision in the three areas Maintenance, Production Management, and Logistics.. Maintenance An integer linear programming approach to maintenance strategies selection (Chapter 1) The optimal mix of maintenance policies is the most appropriate allocation between equipments and maintenance activities, where a maintenance activity is one among corrective maintenance, preventive maintenance, condition-based maintenance, proactive maintenance, and outsourcing. The problem studied here is to find the optimal mix of maintenance policies within the Reliability Centered Maintenance (RCM) framework. The question is addressed developing an Integer Linear Programming model, with the objective of maximizing the reduction in the Risk Priority Number, taking into account the maximum available budget and the compatibility constraint between.

(9) v. maintenance policies and equipments. The steps characterizing the approach developed are presented. The model is finally applied to a couple of subsystems in a paper mill. Keywords: Reliability Centered Maintenance; Linear Programming; FMECA. Harmony search algorithm for single-machine scheduling problem with planned maintenance (Chapter 2) This chapter focuses on the single machine scheduling problem, with sequence dependent setup times. Both processing and setup times are deterministic and the objective is to minimize total earliness and tardiness penalties. The novelty of the model can be traced in the fact that the single machine is subjected to breakdowns and that, in order to increase its availability, planned maintenance tasks are also performed. Hence, jobs and maintenance tasks are jointly considered to find the optimal schedule. These features make the problem NP-hard and so, a quasi-optimal solution is searched using a recent meta-heuristic, which integrates harmony search and genetic algorithms. In order to validate the proposed meta-heuristic, a comprehensive set of scheduling problems was fully investigated. Obtained results, compared with those of exhaustive (for small problems) and standard meta-heuristics, confirm both the robustness and the speed of the proposed approach. Keywords:. Earliness-Tardiness. penalties;. Harmony. Search;. Meta-heuristics;. Planned. maintenance; Scheduling; Single machine. Diffusion theory applied to tool-life stochastic modelling under a progressive wear process (Chapter 3) In this chapter, a novel approach to the derivation of tool-life distribution, when the tool useful life ends after a progressive wear process, is presented. It is based on the diffusion theory and exploits the Fokker-Planck equation. The Fokker-Planck coefficients are derived on the basis of the injury theory assumptions. That is, tool-wear occurs by detachment of small particles from the tool working surface, which are assumed to be identical and time-independent. In addition, they are supposed to be small enough to consider the detachment process as continuous. The tool useful life ends when a specified total volume of material is thus removed. Tool-life distributions are derived in two situations: (i) both Fokker-Planck coefficients are time-dependent only, and (ii) the diffusion coefficient is neglected and the drift is wear-dependent. Theoretical results are finally compared to experimental data concerning flank wear land in continuous turning of a C40 carbon steel bar adopting a P10 type sintered carbide insert. The adherence to the experimental data of the tool-life distributions derived exploiting the Fokker-Planck equation is satisfactory. Moreover, the.

(10) vi. tool-life distribution obtained when the diffusion coefficient is neglected and the drift is weardependent is able to well-represent the wear behaviour at intermediate and later times. Keywords: Tool-Wear; Tool-Life; Injury theory; Stochastic Models; Tool-Management; Diffusion. Improving tool-life stochastic control by means of a diffusion-based model (Chapter 4) This work shows how a diffusion-based model to derive the time-to-failure distribution can be exploited to improve tool-life stochastic control. In particular, this is made possible by the peculiarities of a stochastic diffusion process, which make the tool-life density updatable in both mean and variance given the information on both the wear level and the corresponding time instant. In addition, it is shown that the improved capability on tool-life stochastic control provided by such model allows a better exploitation of the tool useful life. In fact, when the tool is replaced prematurely (in terms of existence of further cutting capability) this leads to a loss of useful life. However, the features of a diffusion-based tool-life model allows achieving a better exploitation of the cutting capability. This aspect is then demonstrated with an experimental application. Keywords: tool wear; tool life; injury theory; tool management; diffusion process; stochastic model. Theoretical developments on the injury theory (Chapter 5) This chapter aims at presenting a revisited and extended version of the injury theory. This theory allows deriving tool-life distributions when either a single catastrophic event or a gradual and cumulative wear process terminates the tool useful life. Since these models cover great importance in tool-life modelling, this work would contribute in this direction with further observations and results, giving special attention to the mathematical formulation. In particular, besides deriving known results following a different approach, the theory is extended to the case of multiple machines with both single- and multi-tool configurations. Different expressions of tool-life distribution are derived taking into account particular formulations of the injury rate. Keywords: Wear; Tool life; Stochastic models; Injury theory; Tool management.. Production Management The management of activities in large-scale projects, with an application to the yachting industry (Chapter 6).

(11) vii. This work provides the schema for an innovative and modular computer-based approach to the planning of activities in large-scale projects. Such projects are characterized by tens of thousands of tasks, which are consequently burdensome and difficult to plan manually. This is true to the point that in many shipyards only a low level of detail is used and poor planning is generally performed. The proposed approach is called Computer-Aided Activity Planning (CAAP), and an application in the yachting industry is shown to demonstrate its effectiveness. In particular, the socalled outfitting planning problem is faced. The CAAP system, taking into account the available shipyard resources and the knowledge on the building rules is able to automatically define, sequence, and schedule the activities of the whole outfitting process acting as a "planning configurator". Moreover, it allows the industry-specific knowledge to be stored, maintained, and shared within the (extended) organization. Finally, to verify the actual capabilities of the approach, the CAAP was implemented within a prototypical software. Keywords: Shipbuilding; Project planning; Computer-Aided Activity Planning; Outfitting. Optimizing the number of cards in POLCA-controlled production systems (Chapter 7) The present work is aimed at showing the effective capability of a simulation-driven Genetic Algorithm to optimize an unbalanced POLCA-controlled production system. The objective is that of individuating the correct number of cards and to reduce the overall TTT and the average WIP. In real world applications, the systems to be controlled are designed to process units with very different routings, each with significantly different probability to occur. In all these situations they result particularly unbalanced and difficult to optimize. In particular, one of the major drawbacks to be faced is represented by the setting of the number of cards that are required within each different controlling loop. Little’s law is usually adopted to infer this number from historical data but the obtained number is usually far from the optimum. The proposed approach may provide a valid support tool to overcome these limitations, making the most of POLCA capabilities in many manufacturing configurations. Keywords: POLCA, m-CONWIP, Genetic Algorithms, Simulation. A study on the importance of selection rules within unbalanced MTO POLCA-controlled production systems (Chapter 8) By means of a simulative approach, the behaviour of the POLCA method when the production system is highly unbalanced, in terms of both routings and times, is here investigated. In particular, the study is addressed to verify the impact of how orders are processed at each workstation, to show that POLCA performance can be further improved in such circumstances by adopting an.

(12) viii. appropriate selection rule. In literature, it was proved that POLCA is very effective in reducing the Total Throughput Time (TTT) with respect to the corresponding unconstrained, balanced production systems. Owing to this, it appears evident that POLCA represents a valuable and effective Make-to-Order (MTO) production control method. However, some observations must be carefully addressed when the production system is unbalanced. In fact, often, in real world applications, the systems to be controlled are designed to process units with very different routings, each with significantly different probability to occur. Keywords: POLCA; unbalanced production systems; Make-to-Order systems; selection rules; workload control; simulation. Making the performances of single-loop CONWIP and m-CONWIP comparable (Chapter 9) The present work is aimed at showing, through a simulative approach, that the adoption of a suitable dispatching rule allows to improve the single-loop CONWIP control mechanism, reaching a performance level that is comparable to that of the corresponding m-CONWIP system. The benefits that derive from the adoption of a single-loop CONWIP for the design/management of production systems are obvious, being true that (i) m-CONWIP systems are complex to design and optimize, (ii) the single-loop CONWIP systems can be designed with simpler approaches and that (iii) single-loop CONWIP systems remain easier to manage than the m-CONWIP systems. To optimize the number of cards within the m-CONWIP model, that may be difficult as soon as the number of routings grows up, a Genetic Algorithm has been opportunely configured. In the singleloop CONWIP the Work In Next Queue (WINQ) rule has been tuned and used to guarantee that items are favoured within the routings for which higher capacity is available in the succeeding work centres. Keywords: CONWIP, Material Flow Control Systems, Simulation, Genetic Algorithms. Evaluating the OEE of a manufacturing line (OEEML) and its variability by means of Fuzzy Triangular Numbers (Chapter 10) Lean Manufacturing requires defining appropriate targets and constraints and measuring progress. With respect to the last point, the Overall Equipment Effectiveness (OEE) is a valuable choice to evaluate efficiency but it presents some weaknesses in that it deals with a single machine and gives a static measure of effectiveness. This work proposes a Fuzzy approach that tries to overcome both these limitations, providing practitioners an intuitive and ready to use tool that measures OEE for a whole manufacturing line and its variability as well. The proposed method starts from a specific structure of losses, managing variability by means of the well known Fuzzy Triangular Number..

(13) ix. A case study is given, where a simulation model has been used to illustrate the results of the approach. The proposed method shows to be able to evaluate accurately the variations of the OEE of the line, and therefore both the modal value and the range obtained from its application can be considered respectively as good estimators of the OEEML and of its variability. Keywords: Lean Manufacturing; OEE; Manufacturing Line; Variability; Fuzzy Triangular Numbers.. Logistics Application of the QFD methodology to plant layout evaluation (Chapter 11) A QFD-based methodology allowing the assessment of different plant layout proposals is here presented. Facilities layout is a key factor affecting a firm’s profitability and productivity. The related decisions are thus strategic. In literature, numerous works are focused on the design task. However, along the plant layout life cycle, other not less important activities have to be carried out. Among those activities, the evaluation phase emerges. With respect to other assessing techniques, that are often too vague, the one here presented attempts to face the problem exploiting a more rigorous approach. It is based on the QFD framework, and takes into account several aspects that should not be neglected in the evaluation of a plant layout. In accordance to the QFD methodology, customer needs are first recognized, technical requirements are then identified, and finally a comparison method among several alternatives is presented. Keywords: Plant layout, facilities layout, evaluation, QFD. Just in Time parts feeding policies for paced assembly lines: Possible solutions for highly constrained layouts (Chapter 12) In this chapter, the problem of dimensioning a Just-In-Time (JIT) parts feeding system, for a paced assembly line, is addressed. Specifically, it is considered the quite frequent case of an old fashioned, push-oriented warehouses’ arrangement, characterized by materials and sub-assemblies that are stocked in different and peripherals areas of the manufacturing plant. Typically, in these circumstances, layout changes are almost impossible and warehouses cannot be substituted by supermarkets. Nevertheless, aiming to rationalize material flow and to reduce work in process, a JIT parts feeding system can still be implemented. To this aim, two alternative JIT systems, differentiating in terms of the order with which warehouses and workstations are visited, are presented. To dimension both systems and to estimate their expected performances, a mathematical model is also given. In its basic formulation, the model is very operating and easy.

(14) x. to implement. A more complex formulation (based on Bayesian probability) is also introduced, not much for practical purposes, but rather to demonstrate the potentialities of the proposed JIT systems in very constrained operating settings. Keywords: Applied Probability; Facility Layout; JIT Feeding Systems; Paced Assembly Lines; Vehicles Routing. A study concerning physical space occupation costs in VMI with consignment agreement models (Chapter 13) A new analytic formulation of the Vendor Managed Inventory with consignment agreement is proposed, according to observations that are more realistic. That is, the evaluation of the stockholding related costs is herewith substantially revisited to consider the requirements of a standard VMI relationship with consignment agreement. In particular, the physical space occupation cost held by the buyer is introduced as a component of the overall holding cost and is referred to the maximum stock level, rather than to the average value, as proposed in the past literature. Finally, an implicit analytical solution is given and compared to the original one present in literature. Keywords: consignment stock; JELS; vendor managed inventory; inventory holding cost. A novel approach to safety stock management in a supply chain under VMI with Consignment Stock agreement (Chapter 14) In this chapter, a new analytical approach to safety stock management, within the single-buyer single-vendor framework under VMI with Consignment Agreement (VMI-CS), is presented. The cost of safety stock is evaluated adopting a logistic approximation of the standard normal cumulative distribution. The service level is put in relation to the dimension of the single shipment, to the average demand on the buyer and to the number of admissible stockouts. Once the approximated mean joint total cost function is defined, a quasi-optimal solution is derived. A numerical study is then carried out. First, a sensitivity analysis is shown. Finally, a comparative analysis between the VMI-CS and the standard (i.e., non-VMI-CS) models is presented. Keywords: Consignment stock; Vendor Managed Inventory; JELS; Joint Economic Lot Size; Safety stock cost; Logistic function. Revisiting the multiple-vendor single-buyer integrated inventory model with a variable number of vendors (Chapter 15).

(15) xi. In this work, the original multiple-vendor single-buyer integrated inventory model with a variable number of vendors is revisited. In particular, an alternative solution procedure is presented. Taking into account the same assumptions, the total cost function is reformulated, i.e., the step-type discontinuity is replaced by a logistic approximation and the total costs of the system are computed on the whole set of preselected suppliers. Subsequently, it is shown that the total cost function possesses some properties in terms of convexity and continuity that allow the exploitation of standard constrained nonlinear minimization algorithms. Finally, tests conducted on the set of problems originally considered in literature illustrate the good performances of the solution procedure developed. Keywords: Inventory management; Suppliers selection; Supply Chain Management; Logistic approximation; Joint Economic Lot Size. The (r, q) policy with complete backordering: Approximated optimization and further developments (Chapter 16) This work considers the continuous-review (r, q) policy with complete backordering, when the lead time is deterministic and constant, and the lead-time demand is Gaussian. It is known that such model cannot be optimized in closed form, but only by means of algorithmic approaches. However, closed-form optimal solutions are useful, for example, to be aware of the functional dependence with the parameters of the model. Through the adoption of a logistic approximation of the standard normal loss function, an approximated closed-form minimum-cost solution to the model under analysis is first presented. Then, an alternative formulation of the total expected cost function is provided. In particular, the safety factor is put in functional dependence with the order quantity q. Numerical tests are finally carried out to prove the good performance of the approximations adopted, and to study the sensitivity of the new model presented. Keywords: Inventory; (r, q) policy; Backorders; Approximation; Logistic function. Approximated closed-form minimum-cost solutions to the base-stock policy with complete backordering (Chapter 17) In this chapter, the continuous review (S-1, S) policy with complete backordering, deterministic and constant lead-time, and with a Poissonian lead-time demand is considered. Two different expressions of the total expected cost depending on the customer-service criterion adopted are taken into account. First, the Poissonian lead-time demand is approximated with a well-suited Gaussian random variable. Although this system can be easily solved with a simple algorithmic approach, it is interesting to be aware of the direct analytical functional relationship between the.

(16) xii. optimal value of the decision variable and the parameters of the model. Hence, the second step consists in providing some approximated closed-form minimum-cost solutions for both cost models considered under a Gaussian lead-time demand. An extensive numerical study is finally given to characterize the precision achieved by the approximations developed. Keywords: Inventory; (S-1, S) policy; Base-stock policy; Nonlinear programming; Logistic function; Approximation; Gaussian lead-time demand. The coordinated supply chain with controllable lead time under inflation and time value of money: Approximated solutions and further developments (Chapter 18) This work deals with the integrated single-vendor single-buyer inventory model with controllable lead time, considering both inflation and time value of money. In inventory management, it is known that the present value approach does not allow the derivation of optimal solutions expressed in closed form, even for the simplest model. However, closed-form solutions are useful, for example, to evaluate the functional dependence of the optimal solution with the parameters involved. When feasible, approximated formulations can be derived to respond to such need. An approximated expression of the present value of the joint expected total cost function for the problem under analysis is first presented. Then, a new cost model is shown, where the safety factor is explicitly written as a function of the order quantity. Moreover, two different expressions of the lead time crashing cost are considered: a piece-wise linear-decreasing and a power-law lead time crashing cost function. For different cases, the closed-form expression of the optimal value of all decision variables is derived. In the other circumstances, closed-form formulation for two out of three of them are provided. Finally, numerical tests are conducted to evaluate both the error achieved by the approximation and the sensitivity of the new model developed. Keywords: Present value; JELS model; Integrated inventory; Controllable lead time; Approximation. A novel approach to safety stock and expected shortage management in a coordinated supply chain with controllable lead time (Chapter 19) This work considers the management of safety stock and expected shortage in a coordinated supply chain consisting of a single vendor and a single buyer under continuous review and Gaussian demand. The system is further subject to a backorders-lost sales mixture and a controllable lead time. A novel approach to optimizing the safety stock and the expected shortage in such systems is presented. Traditionally, this may be achieved by means of the satisfaction of the First Order Conditions with respect to both the order (or shipment) quantity and the safety factor, which are.

(17) xiii. thus treated as independent variables. A different perspective is taken by putting the order quantity and the safety factor in functional dependence through the introduction of a new parameter. More precisely, the service level is expressed as a function of the number of acceptable stockouts per time unit and the order quantity. This allows optimizing the safety stock and the expected shortage taking into account the constraint on the number of admissible stockouts per time unit, which is important in practice but rarely considered in the literature. Both exact and approximated minimization algorithms are developed. The approximation algorithm gives a quasi-optimal solution, which is more readily applicable in practice. Numerical examples are finally provided to illustrate the effectiveness of the approximation algorithm, and to investigate the sensitivity of the model to variations in some fundamentals parameters. Keywords: JELS; Safety stock; Backorders; Controllable lead time; Inventory; Stochastic demand. Efficient approximation approaches to inventory models with a backorders-lost sales mixture and a controllable lead time (Chapter 20) This study presents approximation procedures to enhance the practical application of inventory models that include a backorders-lost sales mixture, and a lead time made of both a linear lot sizedependent component and lot size-independent components that can be shortened at a crashing cost. The approximation method adopted is based on a second-order Taylor series expansion of part of the cost function. As application, single- and double-echelon inventory systems, under both periodic and continuous review with Gaussian demand, are considered. The approximation method used allows satisfying in closed form those First Order Conditions that in literature are commonly satisfied by means of an iterative procedure. Numerical tests are finally carried out to verify the error achieved and the computation effort required. Keywords: Inventory; Logistics; Lead time; Backorders; Lost sales; Crashing cost. A periodic-review joint-replenishment model under stochastic demands with backorders-lost sales mixtures and controllable lead times (Chapter 21) In this chapter, the periodic-review Joint Replenishment Problem (JRP) under stochastic, normally distributed demands, with mixtures of backorders and lost sales and controllable lead times, is studied. The purpose is to determine a strict cyclic replenishment policy and the length of lead times that optimize the system performance. However, in presence of stochastic demand, it is known that determining the optimal solution analytically is hard even for the simpler single-item periodic-review system. In the JRP, the problem is much more complicated because of the need to coordinate cycles of different products. The aim is to provide an efficient and more practically.

(18) xiv. applicable approximated solution procedure to the problem under consideration. In this regard, one element of the cost function is replaced with its second-order Taylor series expansion. This allows obtaining a cost function whose expression resembles the deterministic costs structure plus a constant and a quadratic term. Ultimately, the approximation procedure can be performed through the execution of a standard algorithm suitable for the JRP under deterministic demand. Numerical tests show the effectiveness of the approach for a wide range of parameters values. Keywords: Inventory; Joint Replenishment Problem; Stochastic demand; Periodic review; Controllable lead time. Stochastic joint-replenishment problem with distribution-free demands, backorders-lost sales mixtures, and controllable major ordering cost and lead times (Chapter 22) This work studies the periodic-review Joint-Replenishment Problem (JRP) under stochastic demands, with backorders-lost sales mixtures, and controllable major ordering cost and lead times. No assumption is made concerning the distribution of demands. That is, the minimax distributionfree procedure is exploited. The purpose is to determine a strict cyclic replenishment policy, the length of lead times, and the major ordering cost that minimize the total system costs. In presence of stochastic demand, it is known that determining the optimal solution analytically is hard, even for the simpler single-item periodic-review system. In the JRP, the problem is much more complicated because of the need to coordinate cycles of different products. An efficient and more practically applicable approximated solution procedure to the problem under consideration is thus presented. In particular, one element of the cost function related to each single item is replaced with its second-order Taylor series expansion. It is thus obtained a cost function whose expression resembles the deterministic cost structure plus a constant and a quadratic term. Ultimately, the approximation procedure can be performed through the execution of a standard algorithm suitable for the deterministic JRP. Numerical tests show the effectiveness of the approach for a wide range of parameters values. Keywords: Inventory; Joint-Replenishment Problem; Stochastic demand; Periodic review; Controllable lead time; Controllable ordering cost..

(19) xv. Contents. Abstract. i. Preface. iii. Chapter 1 An Integer Linear Programming Approach to Maintenance Strategies Selection. 1. 1. Introduction. 1. 2. A literature review on maintenance strategies selection. 3. 3. The RCM framework. 5. 4. A new RCM-embedded approach to maintenance strategies selection problem. 7. 4.1. The mathematical representation 4.2. Costs estimation. 8 10. 4.2.1. Corrective maintenance. 12. 4.2.2. Preventive maintenance. 12. 4.2.3. Condition-based maintenance. 13. 4.2.4. Proactive maintenance. 14. 4.2.5. Outsourcing strategy. 15. 5. Case study. 16. 6. Conclusions. 23. References. 25. Chapter 2 Harmony Search Algorithm for Single-Machine Scheduling Problem with Planned Maintenance. 28. 1. Introduction. 28. 2. Problem formulation. 30. 2.1. Basic single-machine scheduling problem. 30. 2.2. Extending the model through failure and planned maintenance tasks. 31. 2.3. Optimal number of planned maintenance tasks. 33. 3. Harmony search. 34. 3.1. Basic features. 35. 3.2. Specific features. 39. 4. Numerical assessment. 40. 4.1. Problems generation. 41. 4.2. Parameters optimization and validation. 41. 4.3. Two benchmark heuristics. 42. 4.4. Comparative analysis and results. 44. 5. Additional tests. 46. 5.1. A “smart strategy” to generate certainly optimal solutions. 46. 5.2. A recent Simulated Annealing. 47.

(20) xvi. 5.3. Results. 49. 6. Conclusions. 53. References. 55. Appendix A. 57. Appendix B. 60. Chapter 3 Diffusion Theory Applied to Tool-Life Stochastic Modelling under a Progressive Wear Process. 63. 1. Introduction. 63. 2. Nomenclature. 64. 3. First-order theory. 65. 4. Second-order theory. 66. 5. Probability density function of time-to-failure. 67. 6. Extensions to Fokker-Planck equations with u-dependent coefficients. 70. 7. Comparison between theoretical and experimental results. 72. 7.1. Introduction. 72. 7.2. Fokker-Planck equation with coefficients depending on time only. 76. 7.3. Fokker-Planck equation with coefficients depending on u. 79. 8. Conclusions. 84. References. 86. Appendix A. 88. Appendix B. 91. Chapter 4 Improving Tool-Life Stochastic Control by means of a Diffusion-Based Model. 94. 1. Introduction. 94. 2. Nomenclature. 94. 3. Improving tool-life stochastic control with a diffusion-based model. 95. 3.1. Diffusion theory applied to tool-life stochastic modelling. 95. 3.2. Diffusion theory to improve tool-life stochastic control. 97. 3.3. Diffusion theory to better exploit the tool useful life. 98. 4. Diffusion theory to better exploit the tool useful life: An experimental study 4.1. Experimental evaluation of the drift and diffusion coefficients 4.2. Experimental application. 99 99 101. 5. Discussion and conclusions. 108. References. 109. Chapter 5 Theoretical Developments on the Injury Theory. 110. 1. Introduction. 110. 2. Injury theory. 110. 3. Single-injury model (NI = 1). 113.

(21) xvii. 3.1. Single-tool machine (NT = 1). 114. 3.2. Multi-tool machine (NT = N0 + 1 > 1). 114. 4. Multi-injury model (NI > 1). 116. 4.1. Single-tool machine (NT = 1). 116. 4.2. Multi-tool machine (NT = N0 + 1 > 1). 118. 5. Conclusions. 120. References. 121. Appendix. 122. Chapter 6 The Management of Activities in Large-Scale Projects, with an Application to the Yachting Industry. 124. 1. Introduction. 124. 2. The Computer-Aided Activity Planning (CAAP). 128. 3. A case study: NautiCAAP. 133. 4. Conclusions. 144. References. 146. Chapter 7 Optimizing the Number of Cards in POLCA-Controlled Production Systems. 149. 1. Introduction. 149. 2. The proposed unbalanced POLCA system. 151. 3. The Genetic Algorithm. 153. 4. The case study. 158. 5. Conclusions. 162. References. 164. Chapter 8 A Study on the Importance of Selection Rules within Unbalanced MTO POLCAControlled Production Systems. 165. 1. Introduction. 165. 2. The original balanced model. 166. 3. The unbalanced simulation models. 168. 4. Conclusions. 171. References. 173. Chapter 9 Making the Performances of Single-Loop CONWIP and m-CONWIP Comparable. 174. 1. Introduction. 174. 2. The model. 176. 3. The case study. 180. 4. Conclusions. 187. References. 189.

(22) xviii. Chapter 10 Evaluating the OEE of a Manufacturing Line (OEEML) and its Variability by means of Fuzzy Triangular Numbers. 190. 1. Introduction. 190. 2. The OEE for the manufacturing line. 192. 3. The Fuzzy approach: Fuzzy Triangular Numbers. 194. 4. Case study. 199. 5. Conclusions. 207. References. 209. Chapter 11 Application of the QFD Methodology to Plant Layout Evaluation. 211. 1. Introduction. 211. 2. Quality Function Deployment (QFD). 211. 3. Facilities layout evaluation with QFD. 214. 3.1. Steps of the evaluation procedure. 214. 3.1.1. Step 1: Apply the QFD technique. 214. 3.1.2. Step 2: Generate valid layout alternatives. 220. 3.1.3. Step 3: Alternatives layout evaluation. 220. 3.1.4. Step 4: Rank the alternatives and take the one with the highest score. 221. 4. The case study. 221. 4.1. Introduction. 221. 4.2. Numerical analysis. 222. 5. Conclusions. 225. References. 226. Chapter 12 Just in Time Parts Feeding Policies for Paced Assembly Lines: Possible Solutions for Highly Constrained Layouts. 227. 1. Introduction. 227. 2. Two alternative JIT feeding policies. 229. 2.1. Lines’ feeding policies. 230. 2.2. Preliminary issues and model’s notation. 231. 2.3. Standard Kanban (St-K) policy. 233. 2.4. Clock Kanban (Cl-K) policy. 238. 2.5. Comparison of the alternative policies. 240. 3. Constrained capacity of the picking truck. 243. 3.1. Main hypothesis and simplification of the model. 243. 3.2. St-K policy with limited loading capacity. 244. 3.2.1. Minimum mission time. 244. 3.2.2. Minimal loading capacity. 245. 3.2.3. Further insights concerning the minimal loading capacity. 247.

(23) xix. 3.2.4. A simplification of the analysis. 247. 3.2.5. Stockout probability. 248. 3.3. Cl-K policy with limited capacity. 251. 3.3.1. Minimum mission time. 251. 3.3.2. Minimal loading capacity. 252. 3.3.3. Further insights concerning the minimal loading capacity. 253. 3.3.4. Stockout probability. 254. 3.4. Comparison of the alternative policies with constrained capacity. 256. 3.5. Additional numerical experiments. 258. 4. Conclusions. 260. References. 262. Appendix A. 263. Appendix B. 265. Appendix C. 266. Appendix D. 268. Chapter 13 A Study Concerning Physical Space Occupation Costs in VMI with Consignment Agreement Models. 271. 1. Introduction. 271. 2. Literature review. 272. 3. Analytical study. 274. 3.1. Notation and assumptions. 274. 3.2. Problem definition. 274. 3.3. Model development. 276. 4. Numerical study. 278. 5. Conclusions. 280. References. 282. Chapter 14 A Novel Approach to Safety Stock Management in a Supply Chain under VMI with Consignment Stock Agreement. 284. 1. Introduction. 284. 2. Analytical model. 285. 2.1. Notation and assumptions. 285. 2.2. Model definition and approximated minimum cost solution. 286. 3. Numerical study. 291. 3.1. Sensitivity analysis. 291. 3.2. Models comparison. 299. 3.2.1. Premise. 299. 3.2.2. Results. 301. 4. Conclusions. 312.

(24) xx. References. 314. Appendix A. 315. Appendix B. 317. Appendix C. 322. Chapter 15 Revisiting the Multiple-Vendor Single-Buyer Integrated Inventory Model with a Variable Number of Vendors. 324. 1. Introduction. 324. 2. Model revisited. 325. 2.1. Introduction. 325. 2.2. New model formulation. 326. 3. Solution procedure. 327. 4. Conclusions and further remarks. 329. References. 331. Appendix A. 332. Appendix B. 334. Chapter 16 The (r, q) Policy with Complete Backordering: Approximated Optimization and Further Developments. 336. 1. Introduction. 336. 2. Mathematical model. 336. 2.1. Premise. 336. 2.2. Analysis and quasi-optimal solution. 338. 3. Further developments. 340. 4. Numerical study. 343. 4.1. Error analysis. 343. 4.2. Sensitivity analysis. 350. 5. Conclusions. 356. References. 357. Appendix A. 358. Appendix B. 359. Chapter 17 Approximated Closed-Form Minimum-Cost Solutions to the Base-Stock Policy with Complete Backordering. 368. 1. Introduction. 368. 2. Preliminaries. 369. 3. Approximated closed-form minimum-cost solutions. 371. 3.1. Minimization of (5). 371. 3.2. Minimization of (6). 372. 4. Error analysis. 374.

(25) xxi. 5. Discussion and conclusions. 386. References. 388. Appendix A. 390. Appendix B. 392. Chapter 18 The Coordinated Supply Chain with Controllable Lead Time under Inflation and Time Value of Money: Approximated Solutions and Further Developments. 398. 1. Introduction. 398. 2. Model formulation. 400. 2.1. Notation and assumptions. 400. 2.2. Cost models under analysis. 402. 3. Optimization. 404. 3.1. Premise. 404. 3.2. Approximated standard cost function. 404. 3.2.1. Piece-wise linear-decreasing lead time crashing cost. 404. 3.2.2. Power-law lead time crashing cost. 405. 3.3. Approximated cost function with substitution. 408. 3.3.1. Piece-wise linear-decreasing lead time crashing cost. 408. 3.3.2. Power-law lead time crashing cost. 411. 4. Numerical study. 414. 4.1. Premise. 414. 4.2. Error analysis. 416. 4.2.1. Standard cost model with piece-wise linear-decreasing lead time crashing cost 4.2.2. Standard cost model with power-law lead time crashing cost. 416 419. 4.2.3. Cost model with substitution and piece-wise linear-decreasing lead time crashing cost 4.2.4. Cost model with substitution and power-law lead time crashing cost 4.3. Cost models with substitution: Sensitivity analysis. 422 425 428. 4.3.1. Premise. 428. 4.3.2. Cost model with piece-wise linear-decreasing lead time crashing cost. 429. 4.3.3. Cost model with power-law lead time crashing cost. 431. 5. Conclusions. 433. References. 435. Chapter 19 A Novel Approach to Safety Stock and Expected Shortage Management in a Coordinated Supply Chain with Controllable Lead Time. 437. 1. Introduction. 437. 2. Preliminaries: Notation and main assumptions. 439. 3. The base model. 441.

(26) xxii. 4. A novel approach to optimize inventory replenishment and lead time under the constraint on stockout occurrences. 444. 4.1. The new cost model. 444. 4.2. The optimization procedure. 445. 4.3. An efficient approximated minimization procedure. 447. 5. Numerical study. 450. 5.1. Performance achieved by the approximations introduced. 451. 5.2. Sensitivity analysis. 455. 6. Conclusions. 458. References. 459. Chapter 20 Efficient Approximation Approaches to Inventory Models with a Backorders-Lost Sales Mixture and a Controllable Lead Time. 461. 1. Introduction. 461. 2. The approximated solution procedures. 462. 2.1. Premise. 462. 2.2. Single-echelon inventory system. 465. 2.2.1. The periodic review case. 465. 2.2.2. The continuous review case. 469. 2.3. Double-echelon inventory system. 472. 2.3.1. The periodic review case. 472. 2.3.2. The continuous review case. 475. 3. Numerical study. 478. 4. Conclusions and further remarks. 485. References. 487. Chapter 21 A Periodic-Review Joint-Replenishment Model under Stochastic Demands with Backorders-Lost Sales Mixtures and Controllable Lead Times. 488. 1. Introduction. 488. 2. Single-item model. 490. 2.1. Premise. 490. 2.2. An effective approximation to optimize Eq. (2). 492. 3. Error analysis. 496. 4. Multi-items model: The Stochastic Joint Replenishment Problem (SJRP). 501. 4.1. Premise. 501. 4.2. An approximated solution procedure. 504. 5. Numerical examples. 508. 6. Conclusions. 511. References. 512. Appendix. 514.

(27) xxiii. Chapter 22 Stochastic Joint-Replenishment Problem with Distribution-Free Demands, BackordersLost Sales Mixtures, and Controllable Major Ordering Cost and Lead Times. 515. 1. Introduction. 515. 2. Single-item model. 516. 2.1. Premise. 516. 2.2. An effective approximated solution procedure to problem (6). 519. 3. Numerical study for the single-item model. 523. 4. Multi-item model: the Stochastic Joint-Replenishment Problem (SJRP). 526. 4.1. Premise. 526. 4.2. An approximated solution procedure. 530. 5. Numerical study for the JRP considered. 535. 6. Conclusions. 540. References. 541. Chapter 23 Conclusions. 542.

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(29) Dedicated to O. C..

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(31) 1. Chapter 1. An Integer Linear Programming Approach to Maintenance Strategies Selection 1. Introduction Since poorly maintained machines or equipments can lead to random breakdowns causing a lower utilization and, thus, a lower productivity, an optimum maintenance policy is a key factor to realize competitiveness and profitability of a firm. Maintenance can be defined as the combination of all technical and administrative actions, including supervisory actions, intended to retain an item in, or restore it to, a state in which it can perform a required function (ISO 14224, 2006). An oldfashioned and uneconomic point of view considers maintenance simply a cost and a necessary evil, rather than a profit contributor and provider of business value. Nowadays organizations are oriented through a wiser view of maintenance, i.e. a maintenance management integrated with the corporate strategy to ensure safety requirements, equipment availability, quality products, on-time deliveries, and competitive pricing (Luxhøj et al., 1997). Moreover, 1) maintenance costs can reach 15-70% of production costs, varying according to the industry type (Bevilacqua and Braglia, 2000), and 2) one third of all maintenance costs is wasted due to unnecessary or improper maintenance activities (Mobley, 2002). Thus, the choice of the right maintenance strategies, i.e. the best mix of corrective, preventive, condition-based, and proactive maintenance, for the equipment is fundamental. Throughout the years several approaches to enhance the maintenance management have been developed and applied to a wide variety of organizations and systems. At the industrial level, the maintenance practice was originally based on the concept that interventions are not carried out until a failure has occurred (Sheu and Krajewski, 1994, Quintana et al., 2009, and Viles et al., 2007). This strategy is called corrective maintenance (CM). The first attempt to reduce the equipment failure frequency was done with preventive maintenance (PM). On the basis of the component reliability characteristics, PM reduces the failure probability determining the frequence of checks, replacements, and revisions by means of the failure rate. Some noteworthy examples regarding PM are shown in Fitouhi and Nourelfath (2012), Wu et al. (2011), and Moghaddam and Usher (2011). The second generation of maintenance management initiatives regards the so-called condition-based maintenance (CBM) or predictive maintenance (PDM). It is based on the use of diagnostics devices to survey the working conditions of a machine in order to point out any.

(32) 2. abnormal situation and, if necessary, to stop it before a failure occurs. In Wang et al. (2012) and Tian et al. (2011) two examples of CBM application are shown, and in Jardine et al. (2006) a review of the most recent researches and the developments in diagnostics and prognostics of mechanical systems implementing CBM is exposed. The most known and diffused third generation maintenance management systems are the Reliability-Centered Maintenance (RCM) and the Total Productive Maintenance (TPM). These techniques require the maintenance and production departments to operate with synergy to discern and avoid potential problems. The maintenance objectives are focused on production needs, i.e. to reduce manufacturing lead-times and increase flexibility (Ashayeri, 2007). TPM emphasizes integration between production and maintenance practices and the systematic and continuous improvement, in order to maximize the Overall Equipment Effectiveness performance index (OEE) (Hansson and Backlund, 2003). RCM focuses on the system functions (Rausand, 1998) and directs maintenance efforts at those parts and units where reliability is critical (Garg and Deshmukh, 2006). This approach minimizes maintenance costs balancing the higher costs of corrective maintenance against the cost of preventive maintenance, considering the loss of potential life (Crocker and Kumar, 2000). In addition, the RCM and TPM methodologies make use of the so-called proactive maintenance (PaM). The results of PaM activities are re-engineered and improved machines, systems, and procedures. In this way, the faults are prevented, the maintenance activities are limited, and costs are reduced (Mostafa, 2004). Both RCM and TPM have been widely studied and applied (see, for instance, Pujadas and Chen, 1996, Mokashi et al., 2002, Richet et al., 1995, Chand and Shirvani, 2000, Chan et al., 2005, Eti et al., 2004). At present, the research in the field of maintenance management systems is going ahead in order to improve the existing approaches, or to develop innovative ones. In literature, some interesting works refer to the following techniques: business centered maintenance (Kelly, 2006); age-related maintenance (Crocker and Kumar, 2000); effectiveness-centered maintenance (Pun et al., 2002); intelligent reliability-centered maintenance analysis (Cheng et al., 2008); risk-based maintenance (Arunraj and Maiti, 2007 and Arunraj and Maiti, 2010); reliability and risk centered maintenance (Selvik and Aven, 2011); and strategic maintenance management (Murthy et al., 2002). In this work, a new integer linear programming based approach to the maintenance policies selection problem, within the RCM framework and applicable during the budget monetary resources allocation task, is presented. The first step of the methodology consists in applying the FMECA to the equipments that have to be maintained to calculate the Risk Priority Number (. RPNi ) for each i-th failure. The second step lies in determining the mix of maintenance strategies that maximizes the improvement (i.e. the reduction) of the RPNi , taking into account the.

(33) 3. applicability of the available strategies to each failure, the cost of each strategy, and the maximum available budget. This model has been then applied to a couple of strategic equipments within an Italian paper-mill plant. The chapter is organized as follows. Section 2 shows a literature review on the available approaches to the maintenance policies selection problem. Section 3 presents the main aspects of the RCM. Section 4 describes the developed model while Section 5 punctually illustrates the case study.. 2. A literature review on maintenance strategies selection The selection of the maintenance policies mix is a very important task for a firm. This job is often carried out on the basis of qualitative and subjective choices, thus relying only on the human factor. This fact may lead to the adoption of the wrong maintenance strategies, reducing productivity as well as profitability. Pariazar et al. (2008), combined Factorial Analysis and the Analytic Hierarchic Process (AHP) (improved by means of rough set theory) and firstly recognized nineteen effective criteria in maintenance strategy selection. After using a dimension decreasing technique the authors built a valuable hierarchic structure for the strategies evaluation. Bertolini and Bevilacqua (2006) proposed a combined AHP-Goal Programming approach for the maintenance strategy selection for centrifugal pumps of an oil refinery. The analysis has been conducted following two steps. In the first step the priority levels for the different maintenance policies, with respect to the classical FMECA criteria, have been provided. In the second one a lexicographic Goal Programming model has been formulated and the best set of the maintenance type for the considered failure modes has been identified. Mousavi et al. (2009) used factorial analysis for clustering many decision making criteria into groups. Then they applied a fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method to help decision makers to select the most appropriate maintenance strategy. Khalil et al. (2005) developed a classification grid tool where machine parts are categorized. The classification of the parts is done according to two criteria: 1) machine part failure frequency, and 2) machine part failure cost. This grid allows to decide which maintenance policy would best fit every group of machine parts. Zhaoyang et al. (2011) used a Risk Based Inspection (RBI) methodology to evaluate the maintenance strategy to adopt in the ISOMAX unit of Fujian Oil Refinery. Each equipment has been categorized into five risk zone using a risk matrix. Then, using an AHP model, proper maintenance policy is assigned to each risk level. Pophaley and Vyas (2010) developed a procedure for a mass production system allowing to choose the most economical maintenance strategy among those considered, i.e. corrective maintenance, regular preventive maintenance, and predictive maintenance. The model.

(34) 4. has been developed with the assumption of constant rate of failure. Shyjith et al. (2008) proposed the combination of AHP and TOPSIS to select the suitable maintenance policy for a textile spinning mill ring frame unit. They considered criteria like environmental conditions, component failure, training required, and flexibility. Bevilacqua and Braglia (2000) used an AHP based approach for selecting the best maintenance strategy for an integrated gasification and combined cycle (IGCC) plant in an oil refinery, considering as relevant factors costs, added-value, applicability, and damages. Arunraj and Maiti (2010) presented an approach to maintenance policy selection based on the combined use of AHP and Goal Programming, then applied it to a benzene extraction unit of a chemical plant. The criteria considered in the work are the risk of equipment failure and the cost of maintenance. Peng and Wang (2011) exploited triangular fuzzy number and Relative Membership Grade (RMG) to face with the problem of selecting the best maintenance policy for machines and equipments. A set of benefit and cost attributes is used. This methodology has been then applied to the pumps of a thermal power plant. Bashiri et al. (2011) developed a modified Linear Assignment Method (LAM), namely an interactive fuzzy linear assignment method (IFLAM). Both qualitative and quantitative criteria are evaluated in an interactive manner in order to choose the optimum maintenance strategy. Al-Najjar and Alsyouf (2003) used a fuzzy multiple criteria decision making methodology to select the best maintenance strategy, by adopting the failure causes as attributes. Wang et al. (2007) applied a fuzzy modification of the AHP technique, using a new fuzzy prioritization method. The criteria considered for the evaluation are divided into four main areas: safety, cost, added-value, and feasibility. This system has been applied to the maintenance activities of a Chinese thermal power plant. Dong et al. (2008) decided the optimum maintenance strategy for the equipments of a fossil-fired power plant using a method combining evidential reasoning, with the aim of evaluating criticality on the basis of different qualitative and quantitative factors, and FMEA. Jafari et al. (2008) proposed a complex step-based methodology. First of all, the alternative maintenance strategies have to be chosen and, then, the maintenance goals, that could be either tangible or intangible, have to be defined. Secondly, the Fuzzy Delphi method is used to determine the importance of each goal and the capability of the strategies to satisfy each of them. Thirdly, Yager ranking method is adopted to transform the previously obtained fuzzy numbers to crisp values. Finally, Simple Additive Weighting (SAW) method is used to select the best maintenance strategy for each equipment. In this research a new integer linear programming based methodology for selecting the optimum maintenance strategies mix, within the RCM framework and applicable during the budget monetary resources allocation task, is shown. Given a set of equipments, after having applied the FMECA process, the model allows to determine the mix of maintenance policies such that the.

(35) 5. reduction of the RPNi is maximized, taking into account the applicable strategies to each i-th failure, the cost of each strategy, and the available budget constraint.. 3. The RCM framework Moubray (1997) defines RCM as “a process used to determine what must be done to ensure that any physical asset continues to do what its users want it to do in its present operating context”, characterizing RCM as a process to establish the safe minimum levels of maintenance. In other words, the objective of maintenance should be to keep the equipment doing whatever its users want it to do, rather than to prevent failures for the sake of preventing failures. Hence, determining the operating context and what the user wants the equipment to do is the starting point for the RCM process, which is applied by asking and answering the following seven questions (SAE JA1011, 1999): 1). What is the item supposed to do and what are its associated performance standards?. 2). In what ways can it fail to provide the required functions?. 3). What are the events that cause each failure?. 4). What happens when each failure occurs?. 5). In what way does each failure matter?. 6). What systematic task can be performed proactively to prevent, or to diminish to a satisfactory degree, the consequences of the failure?. 7). What must be done if a suitable preventive task cannot be found?. RCM is thus focused on maintaining the system functions, and it is completely described in the following four characteristics (Mokashi et al., 2002): . Preserve functions;. . Identify failure modes that can defeat the functions;. . Prioritize function need (via the failure modes);. . Select only applicable and effective tasks. Therefore, RCM highlights maintenance needs and considers only those tasks that promote. system reliability. The aim of RCM is to optimize, in a systematic way, the mix of the main maintenance practices, i.e. corrective, preventive, condition-based, and proactive, that must be adopted by the maintenance staff of a firm. These main maintenance policies, rather than being applied independently, are integrated to take advantage of their respective strengths in order to maximize facility and equipment reliability while minimizing life-cycle costs. The RCM logic is implemented as a sequence of activities:.

(36) 6. 1). Identify the systems and sub-systems;. 2). Define the functions of each system;. 3). Characterize all possible functional failures;. 4). Delineate all possible failure modes for each functional failure;. 5). Specify the effects of each failure mode;. 6). Determine the criticality of each failure;. 7). Establish the maintenance action for each failure;. 8). Implement the procedure;. 9). In-service data collection and updating the procedure.. Thus, RCM is a continuous process, i.e. feedbacks are collected, analysed, and the information gained are then used for implementing continuous improvement. In order to determine system functions, functional requirements, functional failures, and cause and consequences of functional failures, the FMECA technique can be adopted. Therefore, following the previous steps and taking into reference the RCM logic tree (Figure 1), it is possible to determine the best maintenance practice for each equipment: . Perform condition-based actions (CBM);. . Perform interval-based actions (PM);. . Determine that redesigning will solve the problem and accept the failure risk, or determine that, since no maintenance action will reduce the probability of failure, installing redundancy is necessary (PaM);. . Perform no action and choose to repair following failure (CM). The classic RCM is often a too qualitative maintenance management approach, i.e. the. maintenance actions to perform are generally determined only on the basis of the RCM decision tree with the help of the FMECA technique. Moreover, the maintenance strategies costs are neglected from a quantitative point of view. Instead, the innovative approach herewith developed allows to face up the maintenance strategies selection problem in a more quantitative and rigorous manner. Therefore, it is particularly suited during the budget monetary resources allocation task. In particular, the strategies mix choice is done considering the cost of each policy, the compatibility constraint between failures and maintenance actions, and the improvement, in terms of reduction of the RPNi index, that each applicable strategy could provide with respect to every failure i..

(37) 7. The failure have a direct and adverse effect on environment, health, security and safety?. NO. YES. The failure have a direct and adverse effect on Mission (quality or quantity)?. NO. The failure result in other economic loss (high cost damage to machines or system)?. YES. Is there an effective Condition-based maintenance technology or approach?. NO. YES. Is there an effective Interval-based task?. Develop & schedule Condition-based task to monitor condition. Develop & schedule Interval-based task to planning maintenance. Perform Condition-based task. Perform Interval-based task. NO. Redesign system or install redundancy. Run-to-fail. Figure 1. Reliability Centered Maintenance (RCM) logic tree.. 4. A new RCM-embedded approach to maintenance strategies selection problem In agreement with RCM principles, the aim of the developed model is to assure the system reliability. However, the classic RCM approach is often too qualitative and, therefore, it is necessary to choose the maintenance strategies mix with more effectiveness. The proposed approach faces the question in a more quantitative and rigorous way through the use of an integer linear programming model. In brief, this model is able to assign to each failure cause i the best maintenance policy, taking into account the cost of each strategy, the compatibility constraint between failures and policies, and the available monetary resources, in order to maximize the whole reduction of the corresponding RPNi. It is possible to consider not only the main four maintenance strategies treated within the classic RCM approach, but the model is also able to take into account the outsourcing option, obviously only after rigorous evaluations. Thus, given a set of equipments that have to be maintained, the starting point to apply the model is the FMECA. In this way it is possible to define all the failure modes, effects and causes, and, finally, to calculate the risk priority number for each of them. Secondly, it is necessary to define the compatibility matrix and to estimate the maintenance strategies costs. The cost of a policy is.

(38) 8. evaluated considering fixed and variable costs, with respect to failures. Lastly, considering the budget constraints, it is possible to resolve the integer linear programming model and to obtain the maintenance policies mix that maximizes the whole reduction of the indexes RPNi. As the whole model is built around the concept of maintenance policies cost, proper and coherent functions for their estimate, valid under certain hypotheses, are presented and punctually discussed in the following paragraphs. It is possible to see the model developed as a process, with a set of inputs, i.e. the failure causes, maintenance strategies, policies cost, etc., and a set of outputs, i.e. the optimum maintenance actions mix and the total maintenance cost (Figure 2). Budget RULES & CONSTRAINTS. OPTIMISATION PROCESS (Integer Linear Programming Model). DATA INPUT. Failure causes (FMECA) Maintenance Strategies:  ΔRPN  Compatibility Matrix  Maintenance Costs. OUTPUT. Maintenance Mix Total Maintenance Costs. Figure 2. Model application process.. Naturally, for an effective application of this approach, the model inputs have to be carefully evaluated by the maintenance managers, particularly for what concerns the cost functions. This can be done with the support of experts in the maintenance sector. 4.1. The mathematical representation The integer linear programming model can be formulated as follows: maximize. subject to.  RPN i. .    b j. k. k  y k. y k. ij.  xij. j. jk. jk.   y jk   vij  xij   G i .   xij  0,  j i. jk.  1,  j. (1).

(39) 9. x. ij.  1,  i. j. zij  xij  0,  i, j xij  0,1,  i, j. y jk  0,1,  j, k ,. where: . i is referred to failures;. . j is referred to maintenance strategies;. . b jk is the fixed cost of strategy j when it is applied on k different failures. The vector B j. would highlight the effect of the economies of scale, that is the cost advantages when strategy j is adopted on several failures (if applicable); . vij is the variable cost of strategy j when it is applied on failure i;. . z ij is equal to 1 if failure i and policy j are compatible;. . xij is equal to 1 if maintenance policy j is applied to failure i, otherwise it is equal to 0;. . y jk is equal to 1 if strategy j is used on k different failures, otherwise it is equal to 0;. . G is the maximum available budget;. . RPN ij  RPN i  RPN ij , i.e. it is the difference between the value of the risk priority. number for failure i, obtained from the FMECA process and before the application of the maintenance strategy j ( RPNi ), and the value of the same index after having applied strategy j ( RPN ij ). The objective function of the model represents the global improvement, i.e. the whole reduction, of the criticality indexes RPNi calculated during the FMECA process. Hence, the quantity RPNij is the criticality reduction of failure i obtained applying strategy j. As reminder, the. indexes RPNi are obtained as product of three factors, that is: RPN i  S f ,i  S r ,i  S i ,. (2). where: . S f ,i rates the likelihood that the failure will occur (parameter probability);. . S r ,i rates the likelihood that the failure mode will be detected by means of checks and. inspections (parameter detectability);.

(40) 10. . Si rates the severity of the potential effects of the failure (parameter severity). The constraints have the following meaning:. . The first one forces the total maintenance strategies mix cost to be smaller or equal to the available budget G;. . The second one determines on how many failures strategy j is applied;. . The third one assures that at most one component of vector Y j  y j1 , y j 2 ,, y jk ,, y jn , for each j, is equal to 1, where n is the number of failures compatible with strategy j;. . The fourth one establishes that each failure has to be treated with at most one policy;. . The fifth one guarantees that, on each failure, only compatible strategies can be applied;. . Finally, the last two constrains characterizes the variables xij and y jk as Boolean type. Thus, to recapitulate, the steps to use the model are the following:. 1). Apply the FMECA process to the systems that have to be maintained and calculate the indexes RPNi , for each failure i;. 2). Define which maintenance strategies are available;. 3). Build the compatibility matrix;. 4). Estimate the maintenance actions cost considering both the fixed and variable components. If a strategy has not certain cost components, then these ones are fixed equal to 0. Moreover, for incompatible pairs i, j  , the corresponding variable cost is fixed equal to 0;. 5). For each compatible pair i, j  , calculate RPN ij , i.e. the criticality reduction of failure i if strategy j is used. Incompatible pairs i, j  have RPN ij  0 ;. 6). For each compatible pair i, j  , calculate RPN ij  RPN i  RPN ij . Greater the effectiveness of j greater the value of RPNij . Incompatible pairs i, j  have RPN ij  0 ;. 7). Insert the data in model (1) and then resolve the problem to obtain the optimum maintenance strategies mix.. 4.2. Costs estimation The objective of this section is to give an acceptable formulation of the cost for applying a certain maintenance policy j to a given failure i, naturally under the hypothesis that j is an appropriate strategy with respect to i. The estimation will be done according to some simplifying hypotheses. It has to be highlighted that the cost components evaluated in this work should not be considered in an absolute sense, as they strictly depend on the contest, i.e. the characteristics of both the.

(41) 11. company and the problem under analysis. However they can be used by the maintenance experts as useful models to define the proper cost structures in real world applications. A maintenance policy cost can be computed as (El-Haram and Homer, 2002):. CM  CD  CI ,. (3). where C M is the cost of a maintenance task, C D are direct costs, and C I are indirect costs. In particular, the direct costs comprise spare parts and materials, labor, equipment and tools. Instead, the indirect costs involve administration and management, overheads, and penalties or loss of revenue. Obviously, between direct and indirect costs the first ones can be more easily estimated than the second ones. Furthermore, the indirect costs assessment, referred to the different maintenance policies, can be a hard and highly time-consuming operation, and, generally, companies that do not apply structured maintenance management programs, like RCM or TPM, are not able to perform it correctly. For this reason, information about the single policies contribution to the total indirect costs often lack. Moreover, the cost components of a maintenance strategy can be grouped also in the following fashion:. C M  C F  CV ,. (4). where C F is the fixed cost component and CV is the total variable cost component. C F is the cost that the firm has to face depending only on the fact of using that policy (for instance, the cost of acquiring a new device for the condition-based maintenance). Instead, CV depends on the failures to which that strategy is applied (in fact, the cost of applying corrective maintenance to two different failures may be diverse). Naturally, the fixed component obeys to economies of scale. The costs formulation is herewith presented under the following hypotheses: 1). since this is a RCM embedded model, historical data on equipment reliability are present;. 2). the possibility to estimate costs due to equipment unavailability (production loss, product quality non-conformity, etc.), when a failure occurs, exists;. 3). other indirect cost components are uniformly allocated on the maintenance policies;. 4). spare parts, materials and tools are present in inventory;. 5). the corrective, preventive, condition-based, and proactive strategies cost is evaluated considering that maintenance activities are not outsourced.. In accordance with these hypotheses, in the following possible formulations for the cost estimation, when a strategy is applied to a certain failure, are given. Naturally, better formulations could be provided, especially if detailed costs information are present. Also, modifications could.

(42) 12. be strictly required if the given hypotheses are not fully respected or in presence of other important constraints. 4.2.1. Corrective maintenance Under the considered conditions, when the maintenance practice agrees to the logic of “run to failure”, its cost is related to: . manpower;. . spare parts;. . additional equipments, tools and materials;. . losses due to equipment unavailability. Thus, considering these aspects, a possible formulation for the cost of the corrective maintenance. practice, when applied to a plausible failure i, i.e. a failure that may be treated using this policy, is:. CiC  M iC  Ti C  C L  EiC  LCi   Ck  pk ,i , k. (5). where: . CiC is the cost to correct failure i;. . M iC is the amount of maintenance personnel occupied in correcting failure i;. . Ti C is the amount of time needed to correct failure i;. . C L is the hourly labor cost per person;. . EiC is the cost of extra equipments, tools, materials used to correct failure i;. . LCi is the cost referred to losses due to equipment unavailability, when failure i occurs,. supposing that corrective maintenance is applied; . C k is the cost per unit of the spare part type k that have to be substituted;. . pk ,i is the number of components type k used to correct failure i.. It is clear that the fixed cost component for the corrective maintenance policy is not present. Therefore, the total cost of this strategy, which coincides with the total variable cost, is:. C V ,C   CiC . i. 4.2.2. Preventive maintenance. (6).