# Multi-Location Assortment Optimization with Drone and Human Courier Joint Delivery
## 2 Literature Review
provided a summary of drone operation and drone-truck combined operation [20]. However, scant literature has analyzed product assortment under the influence of drone delivery. The assortment optimization problem has been studied for many years. MNL is one of the most wildly used models. If customers choose products according to MNL, the probability of buying a product or leaving without purchasing anything is fixed. [PERSON] and [PERSON] found that profit-ordered assortments were optimal in a single flight leg problem under the MNL model without any constraint [21]. [PERSON] et al. then first used dynamic assortment optimization under the MNL model with cardinality constraints, they developed an algorithm for static problems and derived structural properties [22]. [PERSON] et al. applied the MNL model to five classes of assortment optimization problems and satisfied a set of unimodular constraints that could be solved as a linear program directly [23]. [PERSON] and [PERSON] gave a link between the space-constrained assortment problem and the fixed-cost assortment problem [24]. Then, they derived a \(\frac{1-\varepsilon}{3}\)-approximation for any \(\varepsilon>0\) in the space-constrained problem under the MNL model. There is a well-known drawback of the MNL model called the Independence of Irrelevant Alternatives (IIA) property, which means the ratio of choice probabilities between two alternatives does not depend on what other alternatives are present. One way to deal with the IIA property of MNL is the Nested Logit (NL) model, which means the consideration set of one customer type is included in the consideration set of another. [PERSON] et al. showed the incapacitated assortment problem according to the nested logit choice model, which is polynomially solvable when the nest dissimilarity parameters of the choice model are less than one [25]. [PERSON] and [PERSON] demonstrated how to obtain an optimal assortment using the nested logit model with a performance guarantee of 2 under space constraints [26]. [PERSON] and [PERSON] studied a capacitated assortment problem when customers choose nested consideration sets [27]. They gave a fully polynomial time-approximation scheme (FPTAS) for this problem. The mixture of the multinomial logit (MMNL) model is a further generalization of MNL that can alleviate the limitations of the MNL model. There are multiple customer segments in MMNL. Each segment has a different consideration set and purchases the same product with different utilities. [PERSON] et al. gave the first polynomial time-approximation scheme (PTAS) for the assortment problem under the MMNL choice model [28]. Furthermore, they established an approximation guarantee for profit-ordered assortments, when there are multiple customer segments. [PERSON] and [PERSON] gave a fully polynomial time-approximation scheme (FPTAS) for the MMNL model when the number of mixtures or nests is constant [29]. FPTAS in the optimization problem under MNL, NL, and MMNL models were extended in [30]. [PERSON] et al. and [PERSON] et al. focused on the mixture of multinomial logit assortment problems and solved the NP-hard problem using the Column Generation algorithm and Branch-and-Cut algorithm, respectively [31,32]. Recently, the conic programming approach was also used for obtaining the optimal solution with a linear objective function subject to conic constraints and obtaining a good performance. [PERSON] developed a conic quadratic mixed-integer programming formulation with valid inequalities for a multi-location assortment problem under the MMNL model [33]. [PERSON] et al. took the logistics distribution cost and store-capacity constraints into consideration [34]. They studied the assortment problem under the MMNL model considering the geographical relationship between the distribution centers and the demands of customer orders, which will generate an additional shipping cost. Then, they developed a conic quadratic mixed-integer formulation with McCormick inequalities. [PERSON] et al.