Demand of customer k for product i has been assumed Poisson with rate of λ ik. 4.
For each customer (k in {mathcal{K}}) and for each curtailable appliance (c in C_{k}), we define an electricity consumption scheduling vector as follows: varvec{x}_{k,c} = left[ {x_{k,c}^{1}, ldots,x_{k,c}^{h}, ldots x_{k,c}^{H} } right] (12).
For every subset U ⊆ S ⧹ { r } and for each customer k ∈ R, an open arc from a facility in U toward j, necessitates a directed path from r towards U. Constraints (2) can be replaced by inequality in case that c ij ≥ 0, for all ij ∈ A R. Furthermore, the same optimization problem with continuous assignment variables x ij, for all ij ∈ A R, returns an optimal ConFL solution.
Finally, for each customer (k in {mathcal{K}}), the electricity bill minimization problem for shiftable and non-shiftable appliances can be modelled as follows, which a linear programming problem and can be solved easily using existing solvers.
For each customer (k in {mathcal{K}};,triangleq,;left{ {1,2, ldots,K} right}), we denote (A_{k}) as the set of appliances in household (k).
For each customer (k in {mathcal{K}}) and each appliance (s in S_{k}), we define an electricity consumption scheduling vector: varvec{x}_{k,s} = left[ {x_{k,s}^{1}, ldots,x_{k,s}^{h}, ldots x_{k,s}^{H} } right] (7 where (x_{k,s}^{h}) represents the electricity consumption of appliance (s) at hour (h) for (k -th customer.
We calculated the sums of the purchase matrix for each customer ( k c, 0 = ∑ p M cp ( c, p ) ) and product ( k 0, p = ∑ c M cp ( c, p ) ) to estimate the isocline for the Purchase Function f ∗.
