A mathematical programming approach to fuzzy queues with batch arrivals

This paper proposes a mathematical programming approach for constructing the membership functions of the performance measures in batch-arrival queueing systems with constant batch size and the arrival rate and service rate being fuzzy numbers. The basic idea underlying the proposed method is to transform a fuzzy batch-arrival queue to a family of conventional crisp queues with batch arrivals by applying the α-cut approach. Then the family of crisp batch-arrival queues is described by formulating a pair of parametric nonlinear programs, through which the membership functions of the performance measures can be derived. A numerical example is solved successfully to demonstrate the validity of the proposed approach. Since the performance measures are completely expressed by membership functions rather than by crisp values, more information is provided for designing queueing systems. The successful extension of batch-arrival queues to fuzzy environments permits queueing models to have wider applications in the real world.     

A mathematical programming approach to supply chain models with fuzzy parameters

Supply chain (SC) models play an important role in supply chain management (SCM) for reducing costs and finding better ways to create and deliver value to customers. An approach to deriving the membership function of the fuzzy minimum total cost of the multi-product, multi-echelon, and multi-period SC model with fuzzy parameters is proposed in this article. On the basis of α-cut representation and the extension principle, a pair of mathematical programs are formulated to calculate the lower and upper bounds of the fuzzy minimum total cost at possibility level α. The membership function of the fuzzy minimum total cost is constructed by enumerating different values of α. To demonstrate the validity of the proposed procedure, a four-echelon five-period SC model with fuzzy parameters is solved successfully. Since the objective value is expressed by membership functions rather than by crisp values, they completely conserve the fuzziness of input information when some of the SC data are ambiguous. Thus the proposed approach can represent SCs with fuzzy parameters more accurately, and more information is provided for designing SCs in real-world applications.     

An EOQ model when production standards are assumed to be normally distributed

In this paper an EOQ model is developed especially for a production environment where the inventory to be controlled is raw material. The fundamental difference between this model and others is that a stochastic demand is placed upon the inventory in terms of the time for an individual to consume one unit of inventory rather than the number of units consumed in some time interval. A shortage cost for inventory is imposed via the idle machine (man) cost. The optimal re-order quantity is determined as a function of the stockout probability and the optimal stockout probability is obtained via a one-dimensional search, over some prescribed set of feasible stockout probabilities.     

Non-linear programming for the optimization of machine repair problems in fuzzy environments

Machine repair models have wide applications in many systems such as production line systems and maintenance operations. A procedure is developed to derive the fuzzy objective value of the cost-based machine repair optimization problem, in that the cost coefficients and the machine breakdown rate are fuzzy numbers. On the basis of the extension principle, a pair of non-linear programs are formulated to calculate the lower and upper bounds of the fuzzy minimum expected total cost at the possibility level α. The membership function of the minimum expected total cost is constructed by enumerating different values of α. A numerical example is solved successfully to demonstrate the validity of the proposed method. Since the minimum expected total cost is completely expressed by a membership function rather than by a crisp value, the fuzziness of the input data is conserved, and more information is provided for decision-making. Furthermore, since the optimum repair rate obtained is fuzzy, a crisp optimum repair rate based on the Yager ranking indices is recommended for practical use.     

带有积压定单的最优模糊存储模型

本文提出一个新的具有积压定单的关于模糊总需求的模糊存储模型。在模糊函数原理下，给出了模糊总存储成本。为了寻找最优解，使用积分均值法白化模糊总存储成本，利用Lingo8．0求解不等式约束问题，我们发现最优解都是确定的实数。此外，当模糊总需求是确定的实数时，我们提出模型的最优解与经典的具有积压定单存储模型具有相同的结果。     

Multi-item reliability dependent imperfect production inventory optimal control models with dynamic demand under uncertain resource constraint

In this paper, imperfect multi-item production inventory models are considered over a finite time horizon with known dynamic demands. The production rates are functions of time which are taken as control variables. In the production process, reliability plays an important role to improve the quality of products and to decrease the defective rate. The said defective units are partially or fully reworked. The unit production cost is a function of production rate and also dependent on raw material cost, development cost due to reliability and wear-tear cost. There is a constraint on the total production cost termed as budget constraint which is crisp/imprecise/random in nature. The objective of the present investigation is to fix the optimum reliabilities of the production system to have maximum return. Thus, the models are formulated as optimal control problems for the maximisation of profit and solved using Hamiltonian (Pontryagin’s Maximum Principle), fixed-final time and free-final state system, Kuhn–Tucker conditions and Generalised Reduced Gradient Method. Several particular cases are derived from the general model. The models are illustrated numerically and graphically and some managerial decisions are derived.     

A membership function approach for aggregate production planning problems in fuzzy environments

This paper proposes a novel approach for the aggregate production planning (APP) problem with fuzzy parameters. Different from the results of previous studies, in this paper the membership function of the fuzzy minimal total cost is constructed based on Zadeh's extension principle and fuzzy solutions are provided. A pair of mathematical programs parameterised by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy total cost at α. By enumerating different values of α, the membership function of the fuzzy total cost is constructed. To illustrate the validity of the proposed approach, the example studied by Lai and Hwang (1992 Lai, YJ and Hwang, CL. 1992. Fuzzy mathematical programming: methods and applications, Berlin: Springer.  [Google Scholar]) using Chanas's approach is investigated. Since the objective value is expressed by a membership function rather than by a crisp value, the proposed approach can represent APP systems more accurately, thus obtained solutions which contain more information can offer more chance to achieve the feasible disaggregate plan, and it is beneficial to the decision-maker in practical applications. The proposed approach can also be applied to APP problems with other characteristics.     

AN ITERATIVE ANALOG INVENTORY POLTCY COST MODEL

This article describes the use of an analog computer, operating in the iterative mode, to perform simulations of inventory policy situations. The inventory policies studied are the re-order level, re-order cycle and (s, S) policies, and within each of these inventory policies a replenishment order policy of either a fixed or variable type can be operated. All policies are subject to stochastic sized customer demand and lead-time durations and a cost model is developed to compare the operating costs of the various policies under these conditions. This cost model is used further to investigate in some detail the self-adaptability and the optimal character of the (s, S) policy.     

Ranking alternatives via maximizing set and minimizing set based fuzzy MCDM approach

Abstract

This work suggests a maximizing set and minimizing set based fuzzy multiple criteria decision‐making (MCDM) model, where criteria are classified into cost and benefit criteria. The final fuzzy evaluation value of each alternative is developed based on the concept of subtracting the summation of weighted normalized benefit ratings from that of weighted normalized cost ratings. Using interval arithmetic of fuzzy numbers can develop the membership functions for the final fuzzy evaluation values. Chen's maximizing set and minimizing set is then applied to defuzzify all the final fuzzy numbers for ranking alternatives. Formulas for the membership functions and ranking procedure of the final fuzzy numbers are clearly presented. The suggested method provides an extension to the fuzzy MCDM techniques available. A numerical example demonstrates the computational process of the proposed method.     

On the time-window fulfillment rate in a single-item min-max inventory control system

This paper considers an inventory control system with a constraint on the service level. We focus on a popular service target in practice, the time-window ready rate, which is defined as the percentage of periods in which demands are completely fulfilled within a pre-specified time window. We explore the cost impact of different time-window ready rate settings by proposing evaluation and optimization methods for a single-location single-item min-max (or (s, S)) inventory system. New optimization procedures have been developed because the existing methods, which are based on a Lagrangian multiplier, cannot generally handle the current problem. More specifically, in some cases the time-window ready rate is a complicated function of the re-order point s, which renders existing algorithms inapplicable. Our algorithms are based on the monotone properties of cost and service constraint functions, which allow us to search the optimal policy efficiently within an identified region of policy-parameter values. Numerical experiments show the non-linear trade-off between cost and service, and indicate that extending or shrinking the time window can have a significant impact on the inventory cost.     

Optimal age replacement scheduling for a random work system with random lead time

System maintenance and spare parts are two closely related logistics activities since maintenance generates the demand for spare parts. Most studies on integrated models of preventive replacement and inventory of spare parts have focused on age replacement scheduling, while random replacement policy, which is sensible and necessary in practice, is rarely discussed and applied. The purpose of this paper is to present a generalised age replacement policy for a system which works at random time and considers random lead time for replacement delivery. To model an imperfect maintenance action, we consider that the system undergoes minimal repairs at minor failures and corrective replacements at catastrophic failures. Before catastrophic failures, the system is replaced preventively at age T or at the completion of a working time, whichever occurs first. The main objective is to determine an optimal schedule of age replacement that minimises the mean cost rate function of the system in a finite time horizon. The existence and uniqueness of optimal replacement policy are derived analytically and computed numerically. It can be seen that the proposed model is a generalisation of the previous works in maintenance theory.     

Effective investment to reduce lost-sales rate in a periodic review inventory model

In 1973 Montgomery et al. proposed an (R, T) type of a periodic review inventory model in which the lost-sales rate caused by stockout is given. The purpose of this article is to investigate in this heuristic periodic review inventory model with partial lost-sales to effectively increase investment and to reduce the lost-sales rate. We discuss two models: complete (normal distribution) and partial (distribution free) information about the protection interval (i.e., review period plus lead time) demand distribution. For each model, two commonly used investment cost functional forms, logarithmic and power, are employed for lost-sales rate reduction. That is, for these two investment cost functions, we first assume that the protection interval demand follows a normal distribution, and then we assume that only first and second moments of the probability distribution of protected interval demand are known. Two algorithms are developed to find the optimal investment decision, and six numerical examples are given to illustrate the results.     

Using fuzzy coding with qualitative data: example with subjective data in human-computer interaction

This article shows the role that fuzzy sets may play in the prospect of analysing qualitative data. To underline this role, a human-computer interaction (HCI) study is presented. The data coming from 20 experts concerns their judgment regarding 33 questions related to the use of HCI approaches in order to support interactive system development phases. Each response scale features three main modalities, that is Agree, Partially agree and Disagree. The dataset example is analysed using multiple correspondence analysis (MCA) with both crisp and fuzzy coding models where the intermediate modality, Partially agree, is removed and considered with ½ membership values to the two extreme modalities. A comparative analysis is performed and the discussion states the interest of fuzzy coding with several kinds of qualitative factors or measurement variables. With qualitative measurement variables (our example), the main drawback of fuzzy coding could be the information loss, which is counterbalanced by the possibility of having fewer modalities and therefore of simplifying the multivariate analysis.     

Optimal echelon stock policies for multi-stage supply chains in fuzzy environments

This paper investigates decision-making in multi-echelon serial supply chain management in the presence of imprecision or uncertainty arising from human reasoning, emphasising the computational resolution. The proposed analysis method is based on a combination of the extension principle and the alpha-representation in fuzzy theory and optimisation theory. The problem is first formulated as a fuzzy optimisation model with several fuzzy parameters. To conserve the fuzziness of the input information of the supply chain, such as forecast market demands and inventory costs, a pair of two-level mathematical programs is proposed to identify the lower and upper bounds of the fuzzy performance at different possibility levels, so that the complete membership function can be described. Four example scenarios are solved to demonstrate the validity of the proposed analysis method. The proposed methodology is widely applicable with different types of membership functions for fuzzy parameters, positive lead times or other more complicated cases. The managerial implications are also discussed for reference by decision-makers.     

Impact of random defective rate on lot size focusing work-in-process inventory in manufacturing system

Literature has focused inventory models with intensive emphasis on imperfect production processes in recent past. However, the work-in-process-based inventory models have been ignored, relatively, in general and the impact of random defects in the form of reworkable and non-reworkable defect rate on lot size and total cost function in particular. This paper develops mathematical models for work-in-process-based inventory by incorporating the effect of random defects rate on lot size and expected total cost function. Our proposed models assume that defective products produced during the production process follow random distributions. Defective products, either in the form of reworkable or rejected production units, follow four types of distribution density functions: uniform, triangular, double triangular and beta distribution. Mathematical models are derived for optimum lot size based on minimization of expected total cost function through the analytical optimization approach. Numerical examples and detailed sensitivity analysis are carried to illustrate and compare the proposed models at different levels of distribution functions’ parameters.     

The Acquisition of Automation Subject to Diminishing Returns

In this paper, it is assumed that the purchase of automation has the effect of reducing the per unit production and in-process inventory costs. The optimal dynamic mix of automation and manual output is derived to minimize the cost incurred over the planning horizon due to (1) deviating from the future goal levels of output, (2) production, (3) inventory, (4) changing the level of manual output, and (5) acquiring automation. The cost of purchasing automation and the effectiveness of automation on reducing the per unit. production and in-process inventory costs are expressed as functions of time to include the impact of learning or anticipated technological advancement. Furthermore, the effectiveness of automation on reducing the per unit production and in-process inventory costs is assumed to decrease as the level of automation held by the organization increases due to diminishing returns. We show that it is rarely optimal to increase automation without simultaneously modifying the level of manual output. In addition, a numerical solution algorithm is presented to compute the optimal times and levels of changes in automation and manual output.     

Optimal inventory policies for uniform replenishment systems with time-dependent demand

This study develops an analysis of lot size inventory systems where the replenishment rate is uniform and demand follows a power demand pattern. Shortages are not allowed. Holding cost, replenishing cost and purchasing cost are considered in inventory system control. The objective of the study is to find the economic production quantity that minimises total inventory cost per unit of time. We conclude that optimal inventory policies depend on the demand pattern index chosen to represent customer demand. Theoretical results are illustrated with a business case study. A sensitivity analysis is proposed to describe the optimal policy behaviour.     

The use of rush deliveries in periodic review assemble-to-order systems

We calculate optimal safety stock in a periodic review (T,S) assemble-to-order system having multiple components and multiple finished goods (FGs). Customer orders for FGs arrive according to independent Poisson processes, and cannot be neither backlogged nor lost. In case of potential component stock-out, the studied system uses rush deliveries from suppliers. For this setting, approximate expressions of the optimal safety stock that minimise the sum of inventory holding and rush ordering costs are developed. Exact optimal safety stocks are calculated using Discrete Event Simulation, and compared numerically to the approximate expressions. The model is applied to a first-tier automotive supplier and yields to a significant reduction in terms of inventory holding and rush ordering costs. A sensitivity analysis on relevant system parameters such as components demand, assembly coefficients and unit rush ordering cost is conducted.     

Waste treatment: an environmental,economic and social analysis with a new group fuzzy PROMETHEE approach

Most complex decisions involve several stakeholders and therefore need to be solved using a group multi-criteria decision method. However, stakeholders or decision-makers often have divergent views, especially in the environmental sector. In order to integrate this divergence, a new group fuzzy PROMETHEE approach is introduced to combine the traditional environmental criteria of life cycle assessments with social and economic criteria. The modelling of uncertainty within the group of decision-makers using a fuzzy approach makes this method unique. The proposed fuzzy approach differs significantly from the standard one. The decision-makers express their judgments in crisp forms. In order to take into account the intrinsic dispersion of judgments within the group, a posteriori fuzzification procedure is applied. The crisp values are not simply aggregated; they are converted into a triangular fuzzy number based on the given evaluations. As a consequence, the definition of fuzzy membership functions, as required in standard fuzzy logic, is not required, which simplifies the process and makes it more reliable. The new approach is illustrated with a real case study concerning the selection of the best waste treatment solution in a natural park from among a traditional incinerator and an innovative integrated plant.