模糊環境下的成本-數量-利潤分析模式之設計與應用

Abstract

長期以來傳統成本─數量─利潤分析模式(Traditional Cost-Volume-Profit Analysis Model)雖廣泛地為企業運用於訂價、生產規劃、利潤規劃、設備購置等決策上[Hammer, Carter and Usry(1993)]，惟此模式受其基本假設之限制，使得該模式處理各相關決策中不確定性(Uncertainty)問題時，有其缺陷存在[Jaedick and Robichek(1964)]，亦即CVP分析之問題結構若未能符合傳統CVP分析模式及產品線性規劃模式之基本假設；致無法作有效描述時，其處理過程將失真於問題本質。因此，諸多文獻主張以機率CVP分析模式(Probabilistic and Stochastic CVP Analysis Model)來解決此問題[Driscoll, Lin and Watkins(1984)]，而若能在模糊環境(Fuzzy Environment)下，將人類之推論、思考過程所運用之非二元化邏輯推廣，而將決策者於估計CVP分析相關變數之參數所隱含之模糊性因素(Fuzziness)[Siegel, Korvin and Omer(1995)]納入，則機率CVP分析模式無法衡量之模糊性因素；可進一步以模糊機率度量之。乃因傳統CVP分析模式及產品線性規劃模式未將此等模糊性因素納入模式分析所致，從而該等分析模式是否足以有效反映出CVP分析問題之實況，乃令人存疑。 本論文採用模糊決策理論(Fuzzy Decision Theory)[Bellman and Zedeh(1970)]之主張；認為實際CVP分析問題須以模糊狀態、模糊樣本訊息、模糊行動及評價函數等四項維度所構成之決策空間，方能充分描述決策者處於模糊環境下實際CVP分析時之問題情境。本論文並以模糊集合理論(Fuzzy Set Theory)以度量此決策空間內之模糊性因素，藉以作為估計CVP分析相關變數之依據，並期此估計方式能符合人類非二元化思考、推論之特性，並以模糊決策理論，最大、最小德菲法(Max-Min Delphi Method)[Iskikawa(1993)]及企業標準成本制度之整合；而提出以模糊CVP決策模式(Fuzzy CVP Decision Model)及模糊德菲線性規劃模式(Fuzzy Delphi Linear Programming Model)之解析架構，此二模式將可改善傳統CVP分析模式之限制，並提升CVP分析於實務上之可運用性。此外，再將企業標準成本制度納入模式考量後，此二模式將能確保其所擬定之產品組合或生產規劃方案，可符合企業在效率上及成本控制上之要求。 本論文之最後一般化推論，發現以下諸結果： （1） 當模糊樣本訊息 模糊狀態 存在下，若將模糊因素排除之，則產生錯誤高估CVP分析每單位售價期望值之現象。 （2） 當模糊樣本訊息 ，模糊狀態 存在下，若將模糊因素排除之，則產生錯誤高估CVP分析銷售數量期望值之現象。 （3） 當每單位售價、銷售數量為一定值，且模糊樣本訊息 模糊狀態 存在之下，若將模糊因素排除之，則產生錯誤低估其利潤期望值之現象。 （4） 令 表模糊目標歸屬函數值， 表模糊限制歸屬函數值。當 ，則模糊限制式之寬容區間 與產品組合數量期望值 間將呈正相關，故若決策者所訂模糊限制式之寬容區間因過度發散而擴大，則將錯誤高估產品組合數量期望值，且決策者將淪於一個不合理（或不存在）的產能結構下進行分析工作而不知之。 （5） 當 時，則模糊目標之寬容區間 與利潤期望值 間將呈正相關，故決策者所訂模糊目標之寬容區間因過度發散而擴大，則將錯誤高估其利潤期望值。

The traditional cost-volume-profit analysis model has long been widely used to determine price planning, production planning, profit planning, equipment purchase, etc.[Hammer, Carter and Usry (1993)]. However, because of the limitation of the assumptions, defects are shown when the traditional cost-volume-profit analysis model is employed to deal with the uncertainty in every related decision. In other words, if the problem structure of the cost-volume-profit analysis does not conform to the assumption of the traditional cost-volume-profit model and the product mix linear programming model, then the traditional cost-volume-profit analysis model and the product mix linear programming model will not be able to describe the problem effectively. Thus, it is in doubt whether the process can truly reflect the problem of the cost-volume-profit analysis. Therefore, the probabilistic and stochastic cost-volume-profit analysis model has been presented in various literature to solve this issue [Driscoll, Lin and Watkins(1984)]. If, under a fuzzy environment, the non-binary logic employed in the human deduction and thinking process in expanded by including the fuzziness implied in the decision marker’s assessing the parameters of related variable in a cost-volume-profit analysis, then the fuzziness which couldn’t be measured by the probabilistic and stochastic cost-volume-profit analysis model will be able to be measured by fuzzy probability. That is, the fuzzy factor is excluded from the traditional cost-volume-profit analysis model and product mix linear programming model. Therefore, it is in doubt whether these models are able to reflect the true problem of CVP analysis effectively. This thesis is in accordance with the assertion of the fuzzy decision theory [Bellman and Zedeh(1970)]. The problem position faced by decision makers while in fuzzy environment could be sufficiently described with the decision space which includes the fuzzy state, fuzzy sample information, fuzzy action and evaluation function. The fuzzy set theory is used to measure the fuzziness of the decision space and to forecast the related variable of CVP analysis. The forecasting is expected to be consistent with human non-binary logic deduction characteristics. The fuzzy decision theory, max-min delphi method [Iskikawa(1993)], and industry’s standard costing are combined in this thesis to propose an analysis framework of the fuzzy CVP decision model and fuzzy delphi linear programming model which will improve the constraints of traditional cost-volume-profit analysis model and upgrade the applicability of CVP analysis in practice. Moreover, the addition of industry’s standard costing will ensure that the product mix or production plan determined by these models will meet the requirements of cost and efficiency control. Further corollaries are made in this thesis, and some results are found as follows: (1)Assume stands for the fuzzy sample information, and stands for the fuzzy state. If and exist, but its fuzziness has been neglected irrationally, then the expected value of the selling price per unit will be overestimated falsely. (2)If and exist, but its fuzziness has been neglected irrationally, then the expected value of the sales quantity will be overestimated falsely. (3)If and exist,and the selling price per unit and sales quantity are fixed, but its fuzziness has been neglected irrationally, then the expected value of the profit will be underestimated falsely. (4)Assume stands for the value of membership function of fuzzy objective function, and stands for the value of membership function of fuzzy constraint function. If ，then there is a positive correlation between the tolerance interval of fuzzy constraint function and the expected quantity of product mix . If decision markers determine that the tolerance interval of fuzzy constraint function is diffused excessively and expanded, then the expected quantity of product mix will be overestimated falsely, and decision makers may have performed analysis under an unreasonable(or not existed) productivity structure without realizing it. (5)If , then there is a positive correlation between the tolerance interval of fuzzy objective function and the expected value of profit[Z]. If decision markers determine that the tolerance interval of fuzzy objective function is diffused excessively and expanded, then the expected value of profit will be overestimated falsely.