An optimization model, suitable for the selection of suppliers of a single commodity, can be defined if the following data are available. Let T be the number of time periods in the planning horizon considered for the provision of a commodity; t = 1,...,T , indicates the time period in which a provision takes place.

Let V be the set of potential suppliers, from which n (1 ≤ n ≤ |V |) suppliers have to be selected; r_i, i ∈ V , is the score of the supplier i; D is the total quantity required to the suppliers (total demand) to be supplied during the planning horizon; o_it , i ∈ V,t = 1,...,T , is the minimum quantity to be eventually ordered to the supplier i at time period t; O_it , i ∈ V , t = 1,...,T , is the maximum quantity that can be eventually ordered to the supplier i at time period t; c_it, i ∈ V,t = 1,...,T , is the unit purchasing cost of supplier i at time period t;

b is the available budget for purchasing the quantity D required during the planning horizon, and α (0 ≤ α ≤ 1) is a tolerance measure, corresponding to the percentage of the ordered quantity that can be delivered late, after the end of the planning horizon, and α_i, i ∈ V , is the average percentage of ordered quantity delivered late by supplier i. Formulate the supplier selection model.

(Hint: make use of the following decision variables: x_it, i ∈ V,t = 1,...,T , representing the quantity of commodity replenished by supplier i at time period t; y_i, i ∈ V equal to 1 if supplier i is selected, and 0 otherwise, and z_it, i ∈ V,t = 1,...,T , equal to 1 if supplier i is chosen at time period t, and 0 otherwise.)