The CCR Model and Production Correspondence Tim Schöneberg The 19th of September
Agenda Introduction Definitions Production Possiblity Set CCR Model and the Dual Problem Input excesses and output shortfalls Reference Set and imporvement Home Assignment
Introduction Until now, we have been dealing with positive input and output vectors From now on, we relax this assumption We also allow semipositive input and output vectors
Definitions Semipositive: A vector is called semipositive if
Definitions Activity A pair of semipositive input and ouptut is called activity
Production Possibility Set Production Possibility Set The set of feasible activities is called Production Possibility Set P. The activities belong to P
Production Possibility Set Properties of the Production Possibility Set belongs to P belongs to P for any positive scalar t. Constant returns-to-scale assumption : and in P Any linear combination of activities in P belongs to P
Production Possibility Set P can be definded as for any semipositve vector with and
Dual Problem Dual Optimality, no feasibility Optimality and feasibility Primal Feasibility, no optimality
CCR Model and Dual Problem CCR Model formulation for Max Subject to
CCR Model and Dual Problem Dual Problem of : Min Subject to
CCR Model and Dual Problem Correspondences Primal Constraint Dual Variable Primal Variable Dual Constraint
CCR Model and Dual Problem Observations has a feasible solution adsd,, There is no hence is semipositive due to because of Thus
CCR Model and Dual Problem Constraints and require the activity to belong to P The Objective Funktion Min tries to reduce the input vector to without leaving P
Input excesses & output shortfalls If, outperforms Regarding this property, we define two slack vectors Input excesses, Output shortfalls, # They represent the possible improvement
Input excesses & output shortfalls How to determine the slack Vectors Phase 1: Solve to get optimal objective value Phase 2: Solve following LP Max Subject to
Input excesses & output shortfalls Max-slack Solution: An optimal solution of Phase 2 is called max-slack solution Zero-slack Activity If a max-slack-solution satisfies and asddsa it is called zero-slack activity
Input excesses & output shortfalls New definition of CCR-Efficiency: An optimal solution is called CCR-Efficient if and only if Pareto-Koopmans Efficiency: A DMU is fully efficient if and only if it is not possible to improve any input or output without worsening some other input or output
Are both definitions equivalent? Old definition
Are both definitions equivalent? Proof v and u are dual multipliers corresponding to Complementary conditions between optimal solutions of and for : and
Are both definitions equivalent? Proof with case differentiation 1. If, the DMU is inefficient 2. If and is not zero-slack then because of and the elements of corresponding to the positive slacks must be zero. DMU is CCR-inefficient 3. If and zero-slack then by strong theorem of complementary must be positive
Reference Set and improvement in efficiency Reference Set of an inefficient : based on the max-slack solution
Improvement in efficiency Optimal solution can be expressed as This can be interpreted as
Improvement in efficiency Based on this formula, we can calculate the required improvement as follows Thus, we have a formula for improvement This is called CCR-Projection
Improvement in efficiency The point with coordinates is the point on the efficient frontier used to evaluate the performance of The CCR Projection identifies this point as a positive combination of other DMU s
Improvement in efficiency For an improved activity there exists an optimal solution for which satisfies and This is true because of the strong theorem of complementary, since is zero-slack
Improvement in efficiency Proposition: Any semipositive combination of DMUs in is CCR-efficient Proof: Let the combined activity be add and So satisfies and asdd,, Thus, it is CCR- Efficient by the older Definition.
Home Assignment Create the Dual Model for DMU A from the last home assignment Solve it with the 2-Phase method and compute the max-slack vectors and Include both the model and the slack vector results in your answer
Thank you for Listening!
References Lecture Material from Operations Research A, Koberstein, WS 2006/2007
Theorem of strong complementary
Theorem of strong complementary For all primal decision variables: Either the variable is zero or The corresponding dual constraint is true with equality, thus the slack is zero For all dual decision variables: Either the variable is zero or The corresponding primal constraint is true with equality, thus the slack is zero