## 1 Introduction  
Currently, in the state of the art, a multi-objective formulation has not been developed to address the recently proposed problem of online purchasing with multiple units per product, known as the Internet Shopping Optimization Problem with multiple item Units (ISHOP-U). An obvious scenario is presented where the objective of minimizing the cost of purchases is faced with another objective in conflict, which consists of maximizing the benefits by total spending. Credit cards and companies usually offer points or cash bonuses in proportion to the cost of purchases made. This paper proposes to formulate the problem so that the total cost of purchases is minimized while maximizing the benefits in points per purchase. The problem of optimizing online purchases with multiple units per item proves to be an NP-Hard problem [1]. Therefore, no computationally efficient algorithms can solve it classically in polynomial time in a deterministic way. The online shopping optimization challenge with multiple units of items, called ISHOP-U, seeks to find the best stores to purchase a required list of items, either in multiple units or individually, to minimize the total purchase cost. This problem, classified as NP-Hard [1, 2, 3], represents one of the most complex categories in terms of optimization; adding a second objective that conflicts with minimizing purchasing costs is another challenge. This pioneering proposal is the first to introduce contradictory objectives into the Internet Shopping Optimization Problem or its variations. The objectives in conflict are minimizing the purchase cost and maximizing the cash-back or reward points on personal or corporate credit cards, the most common payment method for online purchases. Given these conflicting objectives, a set of eight algorithms specialized in multi-objective optimization are used (AGEMOEA, AGEMOEA2, GWASFGA, MOCell, MOMBI, MOMBI2, NSGA-II [4, 5, 6], SMS-EMOA). The experimental results illustrate that NSGA-II generates the best Pareto front approximations with uniform linear dispersion for ISHOP-U instances in the current state of the art, which supports addressing it as a multi-objective optimization problem.  
Although some bi-objective formulations appear in the literature [7, 8, 9], as far as we know, this proposal is unique in the state-of-the-art because of the consideration of conflicting objectives for the online shopping optimization problem. To be more clear, the purpose of this work is to demonstrate that it is possible to formulate the ISHOP-U as a real multi-objective optimization problem where the objectives conflict and do not treat objectives as merely secondary goals or preferences as in many proposals in the literature, which claim to be multi-objective optimization proposals and by definition if the objectives do not conflict they are not multi-objective optimization problems. Problems that do not conflict with their objectives do not produce a Pareto front where the optimal solutions are nondominated between them. In a Pareto front, every solution is mathematically as good as the others. This proposal's final solutions are mathematically equivalent (nondominated between them), producing a Pareto front. The remainder of this paper is as follows: Section 1 is the introduction. Section 2 presents the mathematical formulation of the multi-objective ISHOP-U. Section 3 shows the NSGA-II implementation for the Multi-Objective ISHOP-U. Section 4 provides the experimental configuration. Section 5 gives numerical results of the experimentation. Section 6 presnetes the graphical results of the experimentation. Finally, Section 7 gives our research conclusions.