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Using AHP for resource allocation problem

European Journal of Operational Research 80 (1995) 410-417. Using AHP for resource allocation problem. R.Ramanathan, L.S. Ganesh. Industrial Engineering and Management Division, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India. Agenda.

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Using AHP for resource allocation problem

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  1. European Journal of Operational Research 80 (1995) 410-417 Using AHP for resource allocation problem R.Ramanathan, L.S. Ganesh Industrial Engineering and Management Division, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India

  2. Agenda • Introduction of AHP • Multi-criteria resource allocation problems • Methodology of evaluation • Evaluation for direct criteria • Evaluation for inverse criteria • The most general case • Summary and conclusions

  3. Introduction of AHP(1/9) • AHP (Analytic Hierarchical Process) 分析層級程序法 (AHP) 是Thomas, L. Saaty 在l971 年發展出的一種多屬性決策方法，能支援個人或群體的決策。分析層級程序法主要是用來協助群體決策的制定，是最常用來協助決策者找出最佳策略方案的工具，例如，資訊系統的評估與選擇，旅遊方案的選擇。 • AHP的執行步驟可分成三階段： 1.建立各項因素的階層架構 2.由決策者與專家填表以主觀決定相關因素的權重 3.計算出最佳的結果與建議

  4. Introduction of AHP(2/9) • AHP與ANP (Analytic Network Process)的差別 分析網路程序法（ANP）於1996年由Saaty所提出，此方法藉由早期之分析階層程序法（Analytic Hierarchy Process, AHP）所衍生而來並加以結合網路系統型態所呈現。主要優勢乃是將分析階層程序法結合一回饋（feedback）之機制而加以闡述及發展之。 過去AHP所求出的是主觀的量化結果，忽略對於準則及方案之間的相互回饋之關係特性，因此運用ANP法來所得之量化結果，可提通作為群體決策及評估結果更具理論及實用基礎之信賴度。其ANP法目前應用的範圍大多在解決研發方案之選擇、資訊系統方案選擇等方面的問題。

  5. Introduction of AHP(3/9) • AHP的流程 ( Satty, 1971) ： Step1.定義問題及解決目標： Step2.建立整個問題的階層架構 首先建立階層來決定不同層級間的隸屬關係，從最上層的目標經過中階層級的重要評估標準到最低層級的所有可行方案 Step3.建立各層級因素間的相關矩陣 各層級因素間權重的取得為透過決策者對兩兩標準間的相對重要性進行成對比較 (pair wise comparison)，一般使用9點量尺來評比因素間的比重 Step4.建立標準化矩陣來計算各評估標準的相關權重 將評比矩陣中的每個輸入項除以所在欄位輸入項之加總值，以取得一個新的標準化矩陣A’=[aij’] ，然後再計算標準化矩陣中的每一列之平均值以求得相關權重W=[wk]。

  6. Introduction of AHP(4/9) Step5.計算一致性比率 為了確認決策者所給予因素間重要性的一致程度是在有效的範圍內，則使用一致性比率（Consistency Ratio, CR）來測量整體評斷的一致性。CR的計算方式為先求一致性指標（Consistency Index, CI），再除以一個與矩陣大小相對應之隨機一致性指標（RI），其CI與CR之計算方式如下： 若CR<0.1，則各個標準間的主觀評比可以被接受 Step6.重複執行步驟3-5，直到完成整個架構中各層級因素的權重計算。 Step7.選擇最佳的決策方案

  7. Introduction of AHP(5/9) • 舉例說明 (李秀琴等人，2002) Step1.定義問題及解決目標： 利用AHP分析工具，能協助消費者評估旅遊方案以解決其線上採購所面臨重要資訊獲取與選購策略抉擇的瓶頸。 Step2.建立整個問題的階層架構

  8. Introduction of AHP(6/9) Step3.建立各層級因素間的相關矩陣 階層一之相關權重矩陣 Step4.建立標準化矩陣來計算各評估標準的相關權重 以人為因素為例，其相關權重的計算方式為（0.545 + 0.600 + 0.533 + 0.444）/ 4 = 0.531。

  9. Introduction of AHP(7/9) Step5.計算一致性比率 一開始先計算權重加總矩陣 然後再將計算的結果個別除以相關權重元素，以得到下面的結果： 接著取這些值之平均數來求得，並進一步求出CI。 最後依據標準化矩陣之大小（階數）從隨機一致性指標表（下表）中選擇合適的隨機一致性比率 RI，來計算出一致性比率CR： CR<0.1，整個重要性評斷仍可被接受，否則整個相關因素比重評斷需要重新檢查與調整， 直到一致性比率位於接受的範圍內。

  10. Introduction of AHP(8/9) Step6.重複執行步驟3-5，直到完成整個架構中各層級因素的權重計算。 在最後層級的計算上，依三個旅遊建議地點，泰國普吉島、印尼巴里島與馬來西亞綠中海來進行評估。以安全性為例，我們可以得到一個地點偏好之比重矩陣。 接著再由地點偏好比重的資料中，求出在安全性標準下之組內標準化矩陣與相關權重值(下表)。以此程序來求出其他標準對於三個地點的相關權重。 人為因素之下各評估標準的相關矩陣 人為因素評估標準標準化矩陣與相關權重值

  11. Introduction of AHP(9/9) Step7.選擇最佳的決策方案 我們在此舉一個計算層級架構中的一條路徑（從目標->人為因素->安全性->普吉島）權重值的例子來說明普吉島於整體安全性考量所佔的比重： 依此方式計算出普吉島在所有11項因素（安全性、人員、服務、飲食、住宿等）各佔的權重並予以全部加總，可獲得普吉島這個旅遊方案最後的總分，在此為0.328。而由下表可知綠中海為AHP分析中的最佳的旅遊選擇。

  12. Multi-criteria resource allocation problem(1/3) • Multi-criteria resource problem (MCRA) problems involve allocation of limited resources to different activities keeping in mind many conflict criteria. • The AHP has emerged as a useful decision making technique for solving MCRA problems (Wedley, 1990) • The proposed approach is to use AHP priorities as coefficients in a single objective, maximization-type LP problem.

  13. Multi-criteria resource allocation problem(2/3) • There are two approaches can be identified from a detailed study of the literature. • Expected Priority (EP) approach (Saaty and Mariano, 1979;Liberatore, 1987; Weiss and Rao, 1986): －Using AHP priorities as coefficients in a single objective, maximization-type LP problem • Benefit-cost ratios (B/C) approach (Saaty and Kearns, 1985): －the benefit-cost ratios of each activities are separately obtained, in the form of priorities, using two different AHP models

  14. Multi-criteria resource allocation problem(3/3) • A mathematical statement of the EP approach is : where Pi is the priority, and xi is the allocation to be made to the activity I • A mathematical statement of the benefit-cost approach is : where bi is the priority of benefits, and ci is the priority of costs

  15. Methodology of evaluation

  16. Methodology of evaluation

  17. Evaluation for direct criteria(1/2) • The EP approach Theorem1. (1) and (9) provide the same optimal solution Proof:

  18. Evaluation for direct criteria(2/2) • The BC approach Theorem2. (5) and (9) provide the same optimal solution Proof:

  19. Evaluation for inverse criteria(1/3) • Both the EP and BC approach do not always provide optimal solutions when inverse criteria are used. • The EP approach

  20. Evaluation for inverse criteria(2/3) • The BC approach Theorem2. (7) and (1) provide the same optimal solution Proof

  21. Evaluation for inverse criteria(3/3) • The suggested approach 因為在inverse criteria下，EP與BC都無法求出optimal solution，所以作者自己提出一個在inverse criteria下approach一定可以求出optimal solution。

  22. The most general case • A modified approach : 1. Consider all the direct criteria separately and synthesize final priorities (pi) 2. Consider all the inverse criteria separately and synthesize final inverse priorities (p’i) 3.The double objective function problem will be obtained:

  23. Summary and conclusions(1/2) • Two approaches using AHP for resource allocation problems have been examined in this paper. • Both two approaches have been proved to give correct results when only direct criteria considered, but do not include inverse criteria. • It is concluded that both the approaches are not appropriate for resource allocation problems when considering direct and inverse criteria.

  24. Summary and conclusions(2/2) • A new and simple approach has been suggested to obtain correct results for dealing with inverse criteria.

  25. Future work • On the basis of this new approach, a double objective function methodology has been proposed for the case involving many, mixed criteria. • The double objective function problem can be solved using any multi-objective mathematical programming approaches, such as goal programming.

  26. Reference • 李秀琴、林孟郁、黃木榮、2002年，「運用智慧型代理人與分析階層程序(AHP)於商品選購策略─以旅遊行程規劃為例」，Journal of Information, Technology and Society，pp.1-12 • Liberatore, M.J., 1987, An extension of the analytic hierarchy process for industrial R&D project selection and resource allocation, IEEE transactions on Engineering Management 34,12-18 • Satty, T.L.,1980, The analytic Hierarchy Process, New York: McGraw-Hill • Satty, T.L., and Kearns, K.P.,1985, Analytical Planning: The organization of systems, Pergamon, Oxford. • Satty, T.L., and Mariano, R.S.,1970, Rationing energy to industries: Priorities and input-output dependence, Energy Systems and Policy 3,85-111 • Saaty, T. L.,1996, Decision making with dependence and feedback: The analytic network process. Pittsburgh: RW Publications • Wedley, W.C.,1990, Combining qualitative and quantitative factors - An analytic hierarchy approach, Socio economic Planning Sciences 24, 57-64 • Weiss, E.N., and Rao, V.R.,1986, AHP design issues for large scale systems, Decision Sciences 18, 43-61

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