function [v_tmp] = jcr_rhs_state_value_bellman(na,nb,ntrans,V,gamma,Ra,Pa,Rb,Pb,max_num_cars_can_transfer)
% RHS_STATE_VALUE_BELLMAN - computes the right hand side of the bellman equation
%
% We have to consider the possible number of rentals at sites A/B
%                 and the possible number of returns at sites A/B
% 
% Written by:
% -- 
% John L. Weatherwax                2007-12-07
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

% the maximum number of cars at each site (assume equal): 
max_n_cars = size(V,1)-1; 

% restrict this action: 
ntrans = max(-nb,min(ntrans,na)); 
ntrans = max(-max_num_cars_can_transfer,min(+max_num_cars_can_transfer,ntrans));

v_tmp   = -2*abs(ntrans);
na_morn = na-ntrans; 
nb_morn = nb+ntrans; 

for nna=0:max_n_cars, 
  for nnb=0:max_n_cars,
    pa = Pa(na_morn+1,nna+1); 
    pb = Pb(nb_morn+1,nnb+1); 
    v_tmp = v_tmp + pa*pb* ( Ra(na_morn+1) + Rb(nb_morn+1) + gamma*V(nna+1,nnb+1) ); 
  end
end