How to Utilize DEA for Modelling Centralised Resources Allocation?

1. How to Utilize DEA for Modelling Centralised Resources Allocation? چگونه از تحلیل پوششی داده ها برای تخصیص متمرکز منابع استفاده کنیم؟ Adel Hatami-Marbini Department of Strategic Management and Marketing, Leicester Business School, De Montfort University, Hugh Aston Building, The Gateway, Leicester LE1 9BH, UK موسسهعالیآموزش و پژوهشمدیریت و برنامهریزی چهارشنبه ۱۸ مرداد ۱۳۹۶

2. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

3. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

4. Data envelopment analysis (DEA) • A powerful tool for efficiency measurement • Non-parametric approach • The relative efficiency of a set of decision making units (DMUs) with multi-inputs and multi-outputs • Introduced by Debreu (1951) and Farrell (1957) and developed by Charnes et al. (1978) • Rapid and continuous growth since 1978 • According to Emrouznejad and Yang (2017), more than 10,300DEA research studies published in journals or book chapters firms, schools, banks, etc. Emrouznejad, A., & Yang, G. L. (2017). A survey and analysis of the first 40 years of scholarly literature in DEA: 1978–2016. Socio-Economic Planning Sciences.

5. Two typical DEA models: 1- Envelopment DEA model 2- Multiplier DEA model

6. 1-Envelopment DEA model Productivity analysis needs to have a technology for assessing the performance. What is the technology? The technology is rarely known in the real world. • DEA estimates the technology from observed data and some axioms. ???

7. To define technology: T satisfies the following standard axioms: T is a closed set. Outputs cannot be produced without inputs Free disposability Convexity z returns to scale

8. Free disposability

9. Free disposability

10. Convexity • For any two points in the technology set T , the plans on the line between them are also in T Convex hull: This is the smallest convex set that contains the K observations.

11. Free disposal and convex axioms

12. Returns to scale • Scaling commonly introduced in both economics and benchmarking concerns • If some production plan is feasible, then we can

13. DEA technology

15. DMU is efficient if E=1 • DMU is inefficient if E<1 Mathematical formulation:

16. 2- Multiplier DEA model Engineering definition: • Performance indicator= output/input • For example: Rejection Rate= rejected pieces/processed pieces Inventory Turnover=Sales/inventory How to deal with multiple outputs and multiple inputs? • the sum of the weighted output divided by the sum of the weighted input Input weights Output weights

17. Under with Fixed and Known Multipliers Annual salary (1000$) Sales (Transaction) processing Times(h) Staff Service (Transaction) processing Times(h)

18. How to calculate the efficiency when the weights are unknown? Charnes et al. (1978) proposed the following model Not to exceed unity The essential idea behind this model • Each unit presents its efficiency picture in the most favorable light possible • Each DMU is allowed to choose multipliers that maximize its efficiency score

19. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

20. DEA is interpreted as ex post predictions of efficiency • Managers and organizations can use DEA • for predictive, prospective and normative purposes • In a normative setting, the evaluator controls the inputs or outputs of the DMU by setting targets and/or by allocating fixed resources (products, revenues or budgets).

21. resource allocation vs. target setting • The resource allocation: • The organization has restricted input resources or restricted output possibilities • must be allocated optimally among the units • e.g., adding extra raw material for processing among plants • The target setting: • a certain input/output value for each DMU without reference to organizational limitations (normative character) (prospective character)

22. Contributions • Main methodological contributions: • proposing a common-weights DEA method • creating a new model to show how • adequately the fixed resources (costs) can be allocated to the DMUs • adequately the target setting can be allocated to the DMUs • proposing a framework for performance evaluation • all DMUs will be efficient by incorporating the proposed model • we apply the proposed model to the numerical example to demonstrate the interesting results • Considering the impreciseness in modelling

23. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

24. Motivation Multiplier DEA model for each unit • Different weights for solving each model. • In summary: • The weight flexibility of DEA may result in the different input and output weights which induce undesirable consequences.

25. Situations to control the weights (Charnes et al. 1994): • Neglect some factors by assigning a zero (or epsilon). • The results deny the opinions of the decision maker. • The decision maker has strong preferences about the relative importance of some given factors. (e.g w1>w2) • When the number of factors is proportionately large in comparison with the number of the DMUs. (n>3(m+r)) S2 S1 S3

26. How to control the weights? • Cross-efficiency • Weight restrictions • Absolute weight restrictions • Cone ratio restrictions • Assurance regions • the common endogenous weight • etc. • since our evaluation is based on centralized organizational structure, we focus on the common endogenous weight approach

27. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

28. Common-weight DEA model • Multi-objective programming (MOFP) can be used in order to tackle the mentioned problems. • For solving the program, we consider a model based on the Goal programming (GP) method to maximize the efficiency of the DMUs simultaneously with a set of common weights

29. Aj (j=1,2,…n) presents the goal of the jth objective function : the under-achievement of the jth goal (so-called negative deviation) : over-achievement of the jth goal (so-called positive deviation) Clearly, Aj=1 the jth DMU desires to maximize the efficiency score

30. How to maximize the efficiency score of A? To minimize the sum of the total virtual gaps: by subtracting from and adding to

31. Finally, we get

32. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

33. Conventionally, DEA is used in ex post evaluation of actual performance, estimating an empirical best-practice frontier. However • DEA may also be used prospectivelyto allocate resources (costs and revenues) as well as setting targets in a given organization.

34. In the centralized situations the decision maker may desire • to allocate additional resources to the inputs of the DMUs and • to define a target for the output-level of the DMUs.

35. For instance, management of a chain store makes a decision to allocate a new product with a scarce capacity between the present stores, and management expects to attain a certain revenue with regard to selling the new product. Obviously, management of a chain store is increasingly eager to obtain 100% efficiency score. S1 Revenue New product S2 S3

36. Our aim to identify the [optimal and fair] resource allocation and target setting such that all DMUs become efficient.

37. notations Proposed method • n independent DMUs • m inputs (xij) • s outputs (yrj) • q additional resources , • p fixed outputs (expect to achieve) ,

38. Develop the following system guarantees 100% efficiency score the sum of the resources allocation the sum of the target setting • non-linear issue • alteration variables

39. We result in :

40. In order to consider effectsof the present input and output values in allocating resources and setting output targets = = We utilize and in our approach

41. There are several methods to solve the developed system such as Gauss-Jordan and Gaussian elimination. We define the extra variables to propose a LP model based on the GP concept.

43. converted to the LP problem • Theorem .There always exists a feasible solution to the above model • Theorem guarantees that all DMUs will be efficient

44. Proposed framework

45. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

46. Numerical example

47. Numerical example min worst max best worst min

48. Outline • Data envelopment analysis (DEA) • Contribution • Motivation • Common-weight DEA model • A new common-weights DEA model for allocating resources and setting target • Numerical example • Extension under impreciseness • Conclusions

49. Main idea • non-exact values of inputs and outputs • information only on intervals of input and output values • Objectives • provides the interval efficiency scores using the previous idea (CSW) and after resource allocation • The upper bound score becomes unity • the lower efficiency level is preserved or improved