1957年學者Farell提出「效率衡量」的觀念後，開啟了以生產函數計算效率的研究。不同投入產生最大產量的生產投入組合位於生產前緣上，其效率值當作1。然而真實的生產函數無法得知，所以估計生產函數成為一項挑戰。資料包絡分析(data envelopment analysis, DEA)作為一種衡量決策單位(decision making unit, DMU)生產效率的方法，其原理與估計生產函數相同。在不同的假設下，DEA模型建構生產可能集合，其前緣為生產效率前緣，代表未知的生產函數，用以計算受測DMU的效率。在短期生產的環境中，生產函數一般具有拉長S型，其生產可能集合為非凸性。本論文提出一接近拉長S型的生產前緣建構方法，從Banker, Charnes, Cooper (BCC)模型參考凸性組合與參考集合的觀念，並擴展自由處置組合體Free Disposal Hull (FDH)模型提出一個三階段數學模型，此模型目前探討多投入單產出情形。三階段產出模型首先將所有DMUs經由FDH模型判定為有效率或無效率DMU，然後讓有效率DMUs執行凸性組合涵蓋受測DMU，並且參與凸性組合的有效率DMUs建立受測DMU之生產前緣，用以計算受測DMU的產出效率。三階段模型解決BCC模型因凸性生產可能集合假設，將一些無支配DMUs應該判定為有效率DMUs卻判定為無效率DMUs的問題。FDH模型的生產前緣為階梯型，較無法表現出生產函數的對應關係，而三階段模型的生產前緣較FDH模型平滑。從生產可能集合來觀察發現BCC模型的生產可能集合包含三階段模型，而後者包含FDH模型，可以得知面對相同DMUs，FDH模型的產出效率大於等於三階段模型大於等於BCC模型。三階段產出模型以五種投入因子與單產出因子的例子測試，得出合理的結果。

This paper proposes a model for constructing the production frontier by extending the data envelopment analysis (DEA) model. The model refers to the concept of convex combination and reference set from the Banker, Charnes, and Cooper (BCC) model, and extends the Free Disposal Hull (FDH) model, which is called a three-stage mathematical model. The threestage model judges all decision-making units (DMUs) as either efficient or inefficient DMUs through the FDH model, then which lets the efficient DMUs perform convex combination to span the assessed DMU. The efficient DMUs participating in the convex combination
establish the production frontier of the assessed DMU, which is used to calculate the output efficiency of the assessed DMU. The three-stage output model is tested using a data with five input factors and a single output factor to obtain reasonable results. The three-stage model solves the problem that some non-dominated DMUs should be judged as efficient DMUs but judged as inefficient DMUs due to the assumption of the convex production possibility set in the BCC model. The production frontier of the FDH model belongs to the type of stairs-shape, which is less able to show the corresponding relationship of the production function, while the three-stage model is smoother than the FDH model.

陳巧蓁（2008），運用生產力指數法評估全球前24大國際貨櫃港埠經營績效之研究。國立臺灣海洋大學航運管理學系碩士論文
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