Fuzzy-goal-programming-for-inventory-management.doc

Fuzzy goal programming for inventory management: A bacterial foraging approach 库存管理模糊目标规划：细菌觅食算法a b s t r a c t摘要An efcient inventory planning approach in todays global trading regime is necessary not only for increasing the prot margin, but also to maintain system exibility for achieving higher customer satisfaction. Such an approach should hence be comprised of a prudent inventory policy and ar clesatisfaction of stakeholders goals. Relative signicance given to various objectives in a supply chain network varies with product as well as time. In this paper, a model is proposed to ll this void for a single product inventory control of a supply chain consisting of three echelons. A generic modication proposed to the membership functions of the fuzzy goal-programming approach is used to mathematically map the aspiration levels of the decision maker. The bacterial foraging algorithm has been modied with enhancement of the algorithms capability to map integer solution spaces and utilised to solve resulting fuzzy multi-objective function. An illustrative example comprehensively covers various decision scenarios and highlights the underlying managerial insights.在今天的全球贸易体制下，一个有效的库存规划方法不仅能够提高边际利润，还能为达到较高的客户满意度维持一定得的系统弹性。因此这个方法应该由谨慎的库存策略和利益相关者目标的清晰的满意度组成。要根据产品和时间的不同给予供应链网络中的不同目标以不同的重要性。本文提出了一个包括三个层次的模型去填补供应链上单一产品的无效性。模糊目标规划方法的从属函数的通用修正可以用于映射决策者的数学期望水平。细菌锻造算法已经被放大修正了算法容量，以映射整数解空间和用于求解由此产生的多目标函数。一个说明示例要全面涵盖不同决策方案和强调基础管理的洞察力。1.introductionLeading manufacturing and distribution companies strive to make inventory management as one of their core competencies. This fact ascribes pivotal importance to this eld among the research community and industrial practitioners. Global nature of operations, volatile market conditions and erce competition make it imperative to devise new policies to maintain higher service levels in the lowest possible cost and time. Inventory management has thus become a vital part of any supply chain. Inventory management seeks to integrate the decision procedure for the whole supply chain and nds optimal inventory policies to meet the current aspiration levels from the supply chain. Efcient Inventory management policy not only regulates the costs incurred but also makes the rm more agile.1.引言主营生产和销售的企业都在努力使库存管理成为他们的核心竞争力。这是由于这一问题对从研究团体到工业从业人员来说都至关重要。全球的运营状况，动荡的市场环境和激烈的竞争使得我们有必要去设计一种新的策略去以尽可能低的成本和时间来维持较高的服务水平。在任何一个供应链中库存管理都是非常重要的。库存管理就是要试着整合整个供应链的决策过程，从而去发现最优的库存策略以满足目前供应链的期望水平。有效的库存管理策略不仅要能够控制发生成本，还要是企业运行更加敏捷。A review of supply chain management literature brings out two streams of approaches generally adopted, one considering the individual entities as independent and proposing coordination mechanisms to achieve synergies while other one with centralized control over entire supply chain. Thomas and Grifn (1996) have reviewed the literature addressing coordinated planning between two or more stages of the supply chain. In the other hand, the rst fundamental results on multi-echelon inventory systems were given by Clark and Scarf (1960). Diks et al. (1996) reviewed multi-echelon systems with a concentration on the interactions between the elements that constitute such a multi-echelon system, in order to determine several performance measures. One of the representative analytical models that consider the whole SC network is the model developed by Cohen and Lee (1988). They presented a comprehensive model framework for linking decisions and performance throughout the material-productiondistribution supply chain.Emerson et al. (2009)has considered dynamic reconguration of supply chains over time and developed a knowledge- based framework to consider the effect of inventory constraints and goodwill. It is a deviation from extant literature which assumes static supply chain network conguration. Certain models on inventory storage have brought out useful managerial insights.Chopra and Meindl (2001)emphasized on storage of inventory at local retailers to reduce the delivery times and increase responsiveness, but incurring higher operation costs. Among recent works,Schmitt et al. (2010) present closed-form approximate solution for inventory system of a retailer facing the risk of supply disruption with stochastic demand. This paper considers a single product multi echelon supply chain and proposes appropriate inventory policy to tune supply chain according to aspirations以往的供应链管理方面的文献主要有两个普遍采用的方法，一是单个实体公司做为独立的个体并提出协调机制以实现协同增效效应，二是集中控制整个供应链。Thomas and Grifn (1996)引用文献综述了在两层及以上供应链中解决协作规划的问题。另一方面，Clark and Scarf (1960)给出了多层库存系统的最重要的结果。Diks et al. (1996)对多层系统中各元素的相互作用进行了综述，从而去确定各元素在多层系统中的绩效。Cohen and Lee (1988)提出了把整个供应链网络都考虑在内的一个有代表性的解析模型，他们提出了一个复杂的模型框架，以连接供应链上从原材料-生产-销售的全部过程。Emerson et al. (2009)研究了随时间变化而进行动态重组的动态供应链，提出了考虑到库存约束和商业信誉作用的一个知识框架，这一论述与现存文献假设供应链网络结构是静态的有一些偏差，这些模型对于库存管理能产生有用的管理洞察力。Chopra and Meindl (2001)强调要本地零售商处惊醒存货储备以减少交易时间，提高响应能力，但是这会产生较高的运行成本。在最近的研究中，Schmitt et al. (2010)提出了零售商库存系统的近似闭型解以应对随机需求下供应中断的风险。本文主要研究单一产品在多层供应链中的情况，提出恰当的库存策略以使供应链的运作与预期一致。.There are multiple performance metrics for assessment of any supply chain. Kleijnen and Smits (2003) critically analysed these by using simulation and then used balanced scorecard method to evaluate the overall performance of a rm. This problem can also be modelled as a multi-criteria optimisation problem or a goal programming problem. Tamiz et al. (1998) provide a detailed overview of goal programming(GP) techniques and discuss the connection between GP and other multi-objective programming techniques as well as their utility interpretation. However, determining the target values for various goals can be difficult. To incorporate this uncertainty and imprecision into the formulation, the fuzzy set theory was initially proposed by Zadeh (1965) in the eld of conventional MODM problems. Zimmermann (2001) emphasized on fuzzy theory to handle situations with vague and imprecise parameter values, because it allows the model to easily incorporate decision makers aspirations in developing critical parameter estimates. Narasimhan (1980) initially presented fuzzy goal programming (FGP) by using membership functions. Hannan (1981) demonstrated how fuzzy or imprecise aspirations of the decision maker(DM) can be quantied and presented for the use of fuzzy goal programming and with the maximisation of the membership function. Tiwari et al. (1987) have used different weights for various goals in order to reect relative importance of goals, and considered the weights as coefcients of the objective function. Mohamed (1997) has discussed goal programming (GP) and fuzzy programming (FP) and has highlighted the similarities between GP and FP. Among nonlinear goal programming models, Dhahri and Chabchoub (2007) incorporated decision-makers preferences in forecasting and inventory processes by usage of satisfaction functions. The present paper proposes a novel modication to the fuzzy goal programmings membership function which improves the solution quality. In real-world integrated problems in supply chains, the decision maker must simultaneously handle conicting goals regarding the use of the resources within organizations. One of the recent works in this area is done by Wee et al. (2009) where fuzzy multi-objective joint replenishment inventory model of deteriorating items is developed. Petrovic et al. (1998) carried out fuzzy modeling and simulation of a supply chain (SC) in an uncertain environment, with an objective to determine the stock levels and order quantities for each facility in an SC during a nite time horizon to obtain an acceptable delivery performance at a reasonable total cost for the whole Supply chain. Liang (2007) applied fuzzy goal programming to production/transportation planning decisions in a supply chain with multiple conicting goals in uncertain environments.有多种性能指标可以评估供应链。Kleijnen and Smits (2003)利用仿真和平衡计分卡理论精细的评估公司的全部绩效。这一问题也可以看做是多目标优化问题或目标规划问题。Tamiz et al. (1998)详细阐述了目标规划技术，并论述了目标规划和其他多目标规划以及他们的实用的解释间的联系，尽管如此，对不同目标的目标值的确定仍然是很困难的。为了将这种不确定性和不精确性集成到公式中，Zadeh (1965)最早提出将模糊集理论用于传统的多目标规划问题中。Zimmermann (2001)强调用模糊理论去解决模糊和不精确参数值的问题，因为这比较容易将决策者的期望融入到临界参数的估计中。Narasimhan (1980)最早将模糊目标规划用于从属函数。Hannan (1981)阐述了了决策者模糊和不精确程度量化的结果决定了模糊目标规划和从属函数最大化的利用。Tiwari et al. (1987)为了反映目标的相对重要性给目标赋予了不同的权重，然后将权重作为目标函数的系数。Mohamed（1997）研究了目标规划和模糊规划，认为二者之间存在一定得相似性。在非线性目标规划模型中，Dhahri and Chabchoub (2007)提出利用满意度函数将决策者的偏好融入预测和存货程序。目前的文献对模糊目标规划从属函数提出了一个新颖的修正，提高了求解的质量。在实践中，决策者在处理供应链整合问题时，必须同步解决组织内资源的分配这个此消彼长的问题。Wee et al. (2009)就易变质物品的模糊多目标联合补货模型展开了深入的研究。Petrovic et al. (1998)实现了在不确定环境下对供应链进行模糊建模和仿真工作，他根据供应链中企业的客观情况，在保证有限的时间水平，合理的总成本，可接受的交货能力下，确定存货水平和订单数量。Liang (2007)在不确定环境下，将模糊目标规划法用于存在多重矛盾目标的供应链的生产/运输规划决策。This work tries to connect a supply chains goals directly to operational parameters. The paper also makes a detailed analysis of inherent relationship among various parameters. It proposes a generic modication to the membership functions of the fuzzy goal-programming model, used to map the vagueness in the aspiration levels of the decision maker into a crisp mathematical model. As non-linearity and large number of decision variables are found in the objective function, heuristic solution procedure is argued. Hence, recently proposed bacterial foraging algorithm (Passino, 2002) is utilised after proper modication. Numerical example has been solved and analysed to infer the underlying management principles.The paper is organized as follows. In Section 2, the background of fuzzy goal-programming and the modications have been discussed. Section 3 covers a detailed account of the proposed model. Section 4 is devoted to description of the bacterial foraging algorithm and discussion related to implementation procedures of the algorithm. Illustrative example and analysis is included in Section 5. Managerial insights and concluding remarks are given in Section 6.本文试着将供应链目标与运行参数相联系，并详细分析各内部参数间的关系。本文将对模糊目标规划模型的从属函数做一个通用修正，用于把决策者模糊的期望水平映射到清晰的数学模型中，由于目标函数中有非线性的、大量的决策变量，所以在这里也可以考虑启发式算法，从而最近有人提出利用适当修正后的细菌觅食算法来解决这类问题(Passino, 2002)。已经求解并分析的数值算例可以推断基础管理原则。本文共分为6部分，第二部分主要阐述模糊目标规划及其修正的背景，第三部分详细阐述本文提出的模型，第四部分是关于细菌觅食算法及其补充算法的步骤，第五部分是案例分析，第六部分得出管理方面的见解及结论。2. Implementation of fuzzy goal programming2.1. Fuzzy goal programming: IntroductionFuzzy based approaches become the mode of choice when crisp models undermine the subtlety of decision variables. The fuzzy goal programming approach is aimed to provide a body of concepts and framework to model the vagueness and impreciseness to a crisp mathematical formulation appropriate for various solution procedures. This approach gives an advantage by mapping various goals of different type and scale using membership functions into satisfaction levels that may be incorporated into a multi-objective function by applying appropriate weightages. A typical formulation of multi-objective fuzzy goal programming is given below.2.模糊目标规划的实施2.1 模糊目标规划简介当清晰的模型不能表达决策变量的微妙之处时，模糊就变成了可选择的模式。模糊目标规划方法就是要提供一个基本的概念和框架，将模糊和不精确的问题用清晰的数学模型去解决。这种方法的优势在于通过将从属函数用于融入了权重多目标函数的满意度水平，来映射各种不同类型和规模的目标，最典型的多目标模糊目标规划公式如下：这里的主要目的就是确定多维向量X，以优化模糊目标。The objective is to determine the multi-dimensional vector, x to optimize the fuzzy goals.Subject to Where represent the goal constraints, and the are the system constraints. The operator represents fuzzified such that the decision maker is satisfied even if the objective value overshoots above the goal up to a tolerance of thus operator provides an allowance of on the right extreme of the goal ,the operator provides such a allowance on both sides of its goal, whilecreates room of on the left. The tolerance limits dene the depreciating satisfaction of the goal, with only partial satisfaction within tolerance limits and no satisfaction beyond the limits. The goals can be of minimising or maximising type, where the parameter has to minimised or maximised, respectively, or may have to be equal to some predened value for the solution to be optimum. Now the fuzzy membership functions are elaborated for these goal types.在这里代表目标约束，代表系统约束。运算符代表模糊小于，在这里，即使客观价值超过了目标的允许范围，而这一运算符可以在目标右极限上提供一个可行的范围，这样就可以满足决策者的要求。当在目标左端的极限时，运算符在目标的两端各提供一个许可范围。偏差范围决定了目标的不满意水平，只有部分满意度在公差范围内，没有超过极限的满意度。目标参数分别进行最大化或最小化，或等于最佳方案的预定义值时，目标就可以被化成最大化或最小化类型。模糊从属函数可以被分解成如下目标类型。For fuzzy minimum 模糊最小化For fuzzy maximum模糊最大化For fuzzy equal 模糊相等In this way the membership function have been established for each goal type and the nonlinear fuzzy objection is formulated with weights attached to each goals.对每一个目标类型和赋权非线性模糊目标联系在一起进行建模。Max:Such that 满足条件n=1,2,N,i=1,2,.mNext, we propose the modifications to original membership functions.之后，我们提出对原始从属函数的修正。2.2. Modication in the fuzzy membership functionThe membership functions of fuzzy goal programming problem (Eqs. (3)(5) have been subjected to a slight modication in view of the solution procedure adopted that uses greedy search heuristics algorithm. In the present case, bacterial foraging algorithm (Passino, 2002) is employed to obtain a near optimal solution vector for a given scenario.The greedy search algorithms work best when used against undulating curves, but the degree of randomness dominates over guided search in case of piecewise linear function such as those of original fuzzy goal programming problems membership function. Under the original problem consideration, the middle section of the curve (objective function) does well in providing the needed direction on the other hand the former and the later section undermine the directed search reducing the search just to a multidimensional random search. This randomness may seriously degrade the quality of the obtained solution making it a chance-dependent event. Also the fuzzy function modelled presents various solutions at the same objective value thus provides a set of possible solutions to select from a set of equivalent solutions. The original approach evaluates members of the solution set at an equal scale, but certainly some will be more rewarding than the others. Considering these issues, a modication has been proposed in the solution procedure.2.2.模糊从属函数的修正考虑到解决问题的修改过程中应用了贪婪搜索的启发式算法，模糊目标规划问题的从属函数(Eqs. (3)(5) 要服从这个修改。在目前的情况下，细菌觅食算法可以得到给定情况下的近似最优解向量。当不使用起伏的曲线时，贪婪搜索算法的效果最佳，但是随机的程度要超过引导搜索，比如：分段线性函数那些原始模糊目标规划问题。基于原始问题来考虑，曲线的中间部分（目标函数）可以很好的提供所需方向的搜索，另一方面，曲线的前后两部分则没什么方向，搜索主要是多维随机搜索。这种随机性可能严重降低解决方案的质量，使其成为一个机会依赖事件。同时，模糊函数建模对于同一目标价值可以提出各种解决方法，所以，在这里会有一套可行的解决方案，从中进行方案选择。原始的方法来评估同等规模的解集成员，从中肯定有些解会比另外一些解更好。考虑到这些问题，在解决程序中我们提出对问题的修正。A gentle gradient has been applied to the previously at (horizontal) sections of Fuzzy function, yet maintaining continuity. This will best serve its purpose by not only directing the search but also in assisting the selection of more rewarding solution within the solution set.The modied membership functions are formulated withas maximum value of ,whileas the minimum value and a parameterwith .Also the case for non bonded variable has been considered.The goals as stated previously can be of minimising type, or may have to be equal to some predefined value for the solution to be optimum.一个温柔的梯度应用于以前的模糊函数的平（卧式）部分，然而也能保持连续性。这不仅能够引导搜索，而且能够协助我们从方案集中选择绩效好的方案，更好的服务于目标。从属函数的修正用作为的最大值，作为最小值，参数和。非相互作用的变量被认为是这样的。如前所述的目标，可以最大限度的减少类型，或可能等于最佳解决方案的预定值。For fuzzy maximum模糊最大化For fuzzy minimum模糊最小化For fuzzy equal模糊等式For the goals that do not have a close boundary a hyperbolic function may be used. say for example, if some minimization type goal does not have a right end bounded value so for that region of the may be replaced by:Figure1 Original and modified fuzzy minimum membership function.原始和修正后的模糊最小从属函数Fig1 differentiates between on original and modified fuzzy minimum membership functions. From the Fig.1 it can be inferred as delta approaches zero the modified function approaches original function.Next section gives a detailed account of the model and description of various goals.对于目标来说，没有一个封闭边界的双曲线函数可以应用。举个例子来说，一些最小化类型的目标如果没有右端界值，则对于，的区域可以被替换为：图1给出了原始和修正后的模糊最小从属函数，从该图可以推断出增量趋近于0的修正函数接近于原始函数。下一节将会给出模型和各项目标描述的详尽说明。3.Model3.1 Problem formulationIn this work, the inventory planning has been addressed considering a three echelon supply network consisting of manufacturer, warehouses and retailers. Each facility is replenished by considering periodic review policies with optimal order up-to-quantities. Under this policy, when the review period for that particular facility comes and the inventory position is recorded to be below the reorder point (s), an order is issued to raise the stock up to a target level (S). Retailers are assigned to a particular warehouse and this assignment is considered during network design phase. Keeping practical constraints under consideration, holding costs reduces as level of aggregation increases with each upstream layer of the network. The decision maker seeks to formulate the inventory policy for a xed number of periods P under the forecasted demand or planned production data. The model optimizes inventory policy for the given set of periods and the forecasted data.The decision maker is required to assign target goal values, tolerances and weightages to each goal. These decision variables along with randomly initialised inventory policy variables will be used for simulating the supply chain to determine the goal values achieved. Fuzzy goal programming is used to transform these goal values into objective function for bacterial foraging algorithm. Therefore, fuzzy goal programming is used to mathematically model the aspiration of decision maker, simulation is used to determine achieved goal values for given inventory policy while the bacterial foraging algorithm optimises the inventory policy variables. The aim is to obtain the optimal combination of S and review periods for each participating facility and hence determine its inventory policy according to t