以虛擬差額模型為基衡量含混合數據的決策單位

Abstract

從(Charnes, Cooper, & Rhodes, 1978)發展CCR模型，做為多項投入與多項產出綜合績效的評估方法，奠定資料包絡分析法(Data Envelopment Analysis, DEA)之基礎，隨後DEA發展逐漸成熟後，根據不同的改善方式則有輻射型(radial)和非輻射型(non-radial)模型兩大類，前者是依投入或產出擇一改善，後者為投入和產出可以同時改善。這些模型在使用上均限制投入與產出的數值為非負值，在實際應用時數值可能為零、負值及正值，或是三者的混合，模型的計算遭遇了困難。 本研究先將混合的數據分成：正與負、全負值、正負零、負與零、正與零，共五種類型。本研究以差額變數為基的評量法(Virtual-gap-based Measurement，VGM) ，建立了四個數學規劃模型作為五種數據類型在不同情況下解析的工具，命名為Mixed Virtual-gap-oriented Model-x (MVGOM-x)模型，x=a, b, c, d。為處理不同的混合數據類型，其中c型又分為c1, c2, c3, c4四個子模型， d型又分為d1, d2, d3, d4四個子模型。本研究建立處理混合數據的流程，此流程中包含MVGOM的四個模型及四階段的數據轉換。我們對於數據為「零」的處理有深入的分析，突破了文獻所討論的困難。

CCR (Charnes et al., 1978) model is the method to evaluate multi-inputs and multi-outputs, which lay the foundation of the DEA researches. There are two major DEA methods: radial and non-radial models. Radial models improve either the inputs or outputs with the same ratio. Non-radial models measure the improvement slack of each input and output simultaneously. The radial models require all inputs and outputs values need to be non-negative. The non-radial models require all inputs and outputs values need to be positive. However, values could be zero, negative and positive, or a mixture of them in many practical applications. Only several literatures introduced the methods to deal with values with mixed positive and negative. A data set encounters zero value still unsolvable. This study deals with five possible data types: positive and negative values mixed, all negative values, positive, zero and negative values mixed, negative and zero values mixed, and positive and zero values mixed. We develop an analytical procedure to solve the problem. We employ virtual-gap assessment method (Virtual-gap-based Measurement, VGM) to establish four mathematical programming models for the five data types in different situations, named Mixed Virtual-gap-oriented model-x (MVGOM-x) model, while x = a, b, c, d. Models -c and -d even have four sub-models c1, c2, c3, c4 and d1, d2, d3, d4 to deal with variety data types. The procedure consists of four phases of data conversions.