Convert this primal LP to Dual LP : Decision Variables

$w_R,w_{\text{R}\text{e}\text{v}},w_{AB}$ for weights of the outputs: Rentals, Revenue, Advance Booking.

$w_F,w_E,w_S,w_M,w_A,w_C$ for weights of the inputs: Fleet, Employees, Salaries, Maintenance,

Advertising, Complaints.

Objective Function

Maximize $z,$ which represents the efficiency score of a site.

Maximize $z$

Constraints

Efficiency Constraint: This ensures that the calculated efficiency $z$ does not exceed the ratio of the

weighted sum of outputs to the weighted sum of inputs, thus adhering to the DEA's principle that no

unit can appear more efficient than it truly is$.$

$z*(w_F*F+w_E*E+w_S*S+w_M*M+w_A*A+w_C*C)w_R*R+w_{\text{R}\text{e}\text{v}}*Rev+$

$w_{AB}*AB$

Normalization Constraint: Normalizes the inputs so that their weighted sum equals $1.$ This

normalization is crucial for ensuring that all DMUs are compared on a consistent scale.

$w_F*F+w_E*E+w_S*S+w_M*M+w_A*A+w_C*C=1$

Non$-$negativity Constraint: Ensures all weights are non$-$negative, which is necessary as weights

represent the importance or contribution of each input and output in the efficiency calculation.

$w_R,w_{\text{R}\text{e}\text{v}},w_{AB},w_F,w_E,w_S,w_M,w_A,w_{C}0$

Inputs and Outputs

Inputs and outputs are the resources consumed and services or goods produced by the decision$-$

making units $($DMUs$),$ in your case, each site of the AugustMoon rental car business. The letters

correspond to specific types of inputs and outputs:

$R$ : Rentals $-$ Number of rentals handled.

Rev: Revenue $-$ Revenue generated.

$AB$ : Advance Booking $-$ Number of advance bookings received.

$F$ : Fleet $-$ Number of cars available.

E: Employees $-$ Number of employees.

$S$: Salaries $-$ Total salaries paid.

M: Maintenance $-$ Costs associated with maintaining the fleet.

A:Advertising $-$ Costs associated with advertising and promotional activities.

$C$ : Complaints $-$ Number of customer complaints $($could be used inversely as a measure of

negative output or directly to account for operational challenges$).$