16. Allocation Models Juha Saloheimo 17.1.27 S steemianalsin Optimointiopin seminaari - Sks 27
Content Introduction Overall Efficienc with common prices and costs Cost Efficienc S steemianalsin Revenue Efficienc Profit Efficienc Profit Ration Efficienc New Cost Efficienc Under Different Unit Prices Illustrative Eample Formulation of a New Production Possibilit Set Optimointiopin seminaari - Sks 27
Content New Technical Efficienc New Cost Efficienc Theorem 8.3 New Revenue Efficienc New Profit Efficienc Summar Home Assignment S steemianalsin Optimointiopin seminaari - Sks 27
Introduction(1 The models covered so far focus on situations where unit prices and costs are unknown or too variable to take into consideration Allocation models analse the efficienc of DMUs when unit prices and unit costs are known the total costs of imputs and total revenue from selling outputs are known for each DMU Allocation models concentrate on mone used on inputs and recieved from outputs not on phsical amount of outputs and inputs S steemianalsin Optimointiopin seminaari - Sks 27
Introduction(2 Total costs for DMU k: n i1 c ik ik where c ik unit cost of input i Total revenues for DMU k: S steemianalsin s j1 p jk where p jk jk unit price of input j Optimointiopin seminaari - Sks 27
Introduction (3 There are two different situations: 1. Unit prices and costs are the same for all DMUs 2. Unit prices and costs differ from DMU to DMU S steemianalsin Optimointiopin seminaari - Sks 27
1. Overall Efficienc With Common Prices and Costs S steemianalsin Optimointiopin seminaari - Sks 27
S steemianalsin Optimointiopin seminaari - Sks 27 Technical Efficienc: Allocative Efficienc: Overall Efficienc: (Overall Eff. = Allocative Eff. Technical Eff. 2 2 1 1 k c c Overall Efficienc With Common Prices and Costs Cost Efficienc A B C D E 2 1 O F Figure 1. Five DMUs producing one unit of output using two inputs 1 ( ( E O d F O d 1 2 2 1 1 k c c R OPTIMAL SOLUTION 1 ( ( F O d R O d 1 ( ( E O d R O d ( ( ( (. ( ( E O d R O d E O d F O d F O d R O d
Overall Efficienc With Common Prices and Costs Cost Efficienc The optimal point can be obtained as the optimal solution the following LP (Farrell(1957: [Cost] s. t. where c E C c (c X Y 1 min c... c c (Cost Efficienc c n is unit input - price vector of S steemianalsin Optimointiopin seminaari - Sks 27
Overall Efficienc With Common Prices and Costs Revenue Efficienc [Revenue] p ma p s. t where X Y L e U p (p1... ps is unit output-price vector Additional canstraint on scale scale assumptions We can define Revenue Efficienc S steemianalsin L e E R U p p copes with various return to Optimointiopin seminaari - Sks 27
[Profit] s.t. Overall Efficienc With Common Prices and Costs Profit Efficienc p - c Yλ L λ Xλ eλ U ma p c -The LP problem finds a profit-maimization mi in the production possibilit set - Based on an optimal solution ( the profit efficienc can be defined as E P (Profit Efficienc p p c c S steemianalsin Optimointiopin seminaari - Sks 27
Overall Efficienc With Common Prices and Costs Profit Ration Efficienc We introduce a model for maimizing the ration [Profit Ration] s.t. ma Yλ L λ Xλ p c eλ U revenue epense S steemianalsin Optimointiopin seminaari - Sks 27
S steemianalsin Optimointiopin seminaari - Sks 27 The previous problem can be transformed into following LP Overall Efficienc With Common Prices and Costs Profit Ration Efficienc. t t where 1 c s.t. p ma t λ Ut e λ Lt t λ Y t λ X
Overall Efficienc With Common Prices and Costs Profit Ration Efficienc The revenue/cost efficienc of DMU can be calculated as E RC p c p c ERC S steemianalsin Optimointiopin seminaari - Sks 27
2. New Cost Efficienc Under Different Unit Prices S steemianalsin Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unti Prices Illustrative Eample(1 We appl the formulated model with common prices and costs to a situation where two DMUs A and B have different cost vectors DMUs A and B have the same amounts of inputs and outputs ( A B and ( A B The unit cost of DMU A is twice that of DMU B for each input ( ca 2cB As the DMUs A and B have the same inputs and outputs the have the same technical efficienc ( A B We appl introduced LP for cost efficienc for DMU A (and DMU B S steemianalsin Optimointiopin seminaari - Sks 27
s. t. New Cost Efficienc Under Different Unit Prices Illustrative Eample(2 min c A ( A X ( 2c B B Y DMU A and B have the same optimal solution The also have the same overall cost efficienc since caa 2cBB cbb A c 2c c A A B B B B A B DMUs A and B have the same cost and allocative efficienc but the cost of DMU B is half that of DMU A The model doesn t work under different unit prices B S steemianalsin Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unit Prices New Production Possibilit Set This shorcoming of the model can be cured b modifing the production possibilit set P (cost-based in this case: P {( X Y } P C {( X Y } where X ( 1... n with j ( c 1 j 1 j... c mj mj T The new production possibilit set costs c for each DMU takes account of input In production possibilit set P C DMU A is more efficient than DMU B if costs (dollars/euros for A for producing the same amount of output are less than for DMU B P C S steemianalsin Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unit Prices New Technical Efficienc Based on the new production possibilit set P C a new measure for technical efficienc is the optimal solution for the following LP problem: [NTech] s. t θ min X Y S steemianalsin Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unit Prices New Cost Efficienc Optimal input-mi in the production possibilit set optimal solution of the following LP: P C is the [NCost] e min e s. t X Y The new cost efficienc is defined as e e S steemianalsin Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unit Prices Theorem 8.3 Theorem 8.3 If and c A B and A B A B A B A c B. Furthermore strict inequalities hold if ca c B Proof. Since A B and it can be seen from the [Ntech] A B that A B. Inequalit holds if c Optimal solution of [Ncost] depends onl on DMUs have a common optimal solution e e A B. ea eb A c B. Since A B Strict inequalit holds if S steemianalsin ca c B. Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unit Prices New Revenue Efficienc New revenue-based production possibilit set P R {( where Y ( A new revenue efficienc model can be defined as: [Profit Ration] s. t S steemianalsin X Y } 1... n with j ma e X Y L e U ( p 1 j 1 j... p sj mj T Optimointiopin seminaari - Sks 27
New Cost Efficienc Under Different Unit Prices New Revenue Efficienc The new revenue efficienc measure is e e. S steemianalsin Optimointiopin seminaari - Sks 27
S steemianalsin Optimointiopin seminaari - Sks 27 U e L Y X. ma e e [NProfit] s t e e A new profit efficienc model can be defined as: New Profit Efficienc New Cost Efficienc Under Different Unit Prices New Profit Efficienc e e e e
Summar In allocative models both input costs and output prices are known for all DMUs The models concentrate on minimizing the mone spent on outputs and maimizing mone recieved from outputs This is a fundamental difference between allocative models and models covered before Possible differences in unit prices and costs have to be taken into account in production possibilit set The models under common prices are special cases of the models under different unit prices S steemianalsin Optimointiopin seminaari - Sks 27
Home Assignment S steemianalsin DMU1 DMU2 DMU3 DMU4 1 1 7 2 1 2 1 3 8 1 3 4 2 2 c1 2 3 5 4 c2 2 1 1 4 There are 4 DMUs two inputs with different unit prices and one output Caclucate for DMU4 new technical efficienc (3p new cost efficienc (4p and allocative efficienc (3p. Optimointiopin seminaari - Sks 27