IEx86-ProjectDescription

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IE 486/586 – COMPUTATIONAL OPTIMIZATION TERM PROJECT DESCRIPTION Deadline: June 2, 2022 SUMMARY In this term project, groups will be working on a variant of a well-known network optimization problem called Assortment Optimization Problem under Consider-then-Choose Choice Models (AOP). We are given a set of products (N) each associated with a price P i , a set of customer types (K) each associated with a consideration set C j that specifies the products the customer is willing to buy and a ranking function ࠵? ! (i.e., a permutation over products) that reflects that customer’s relative preferences. Assume we know the probability vector (࠵? " , ࠵? # , … , ࠵? $ ) where ࠵? ! denotes the respective fraction of customer time j in the population. The objective is to find an assortment ࠵? ⊆ [࠵?] of products to display which will maximize the total revenue. Groups are asked to formulate the problem in a way that is amenable to decomposition and implement a complete Branch-and- Price scheme to solve large-scale problem instances. PROBLEM DESCRIPTION In the assortment optimization problem, the decision-maker tries to pick a subset of products (A) from a catalog of N products to display to customers in a way that the total revenue is maximized. In the variant of the assortment optimization problem considered within this project, we will assume that each customer will buy at most one product from the assortment. While deciding the assortment, the decision-maker has to take into account the customers’ purchase/choice behavior. For example, if a customer does not find any product in the assortment s/he is interested in, then s/he won’t buy anything. On the other hand, if the assortment contains more than one product a customer is interested in, then, the customer will pick the one that s/he prefers more. We have K different customer types and for each customer type we are given the consideration set (a binary matrix indicating whether customer j will ever consider buying the product i), and a ranking of the products in the consideration set. For example, Customer j may have the following consideration set C j ={1,4,5,8,9} and the following ranking of those products {3,5,1,2,4}. Smaller value of a ranking means higher preference. Meaning, in this example, the customer prefers Product 5 over Product 8 over Product 1 over Product 9 over Product 4; and does not even consider to buy any other product (i.e., you can assume the ranking of the products not included in the consideration set to be a big number M). You may refer to Aouad et al. (2021) for the detailed description of this problem and to the supplementary material of the same paper (Section EC.2.1.) for a binary program formulation of the problem. Aouad, Ali, Vivek Farias, and Retsef Levi. "Assortment optimization under consider-then-choose choice models." Management Science 67, no. 6 (2021): 3368-3386. In this Project, we are looking for a reformulation of the same problem that is amenable to decomposition. This reformulation would not work for any random consideration set data, but it requires a special structure in the consideration set matrix. In Figure 1. We can call this structure a “partially decomposable structure”. Notice that there are two groups/clusters of customers and three groups of products. Specifically, the Customer Group 1 (Green) includes customer types {1,2,3}; whereas the Customer Group 2 (Blue) includes customer types {4,5,6}. Likewise, the Product Group 1 (Green) includes products {1,2,3,4} and the Product Group 2 (Blue) includes products {5,6,7}. Finally, the Product Group 3 (Gray) includes products {8,9}. Notice that products in the Product Group 1 are included exclusively in the consideration set of customer types in the

Customer Group 1. Likewise, products in the Product Group 2 are included exclusively in the consideration set of customer types in the Customer Group 2. However, products in the Product Group 3 are included in the considerations sets of customers from both Customer Groups 1 and 2. We can call this final group of products (Gray) “common products” which are typically considered by many types of customers. In the absence of such common products, we would have been able to solve the assortment optimization problem as two independent problems (one for each Product-Customer pair Groups). However, since we do have the common products, we would like to find a way to still make use of this partially decomposable structure. Figure 1: An example consideration/ranking matrix with partially decomposable structure. TASKS 1. Formulate (i.e., write the mathematical formulation in Latex) the above-described assortment optimization problem as an integer optimization problem (You may use the formulation provided in Aouad et al. (2021) for this part). 2. Write a Python code to solve the above formulation for any problem instance given the data. Illustrate your solution on the toy example given in Figure 1. 3. Reformulate (i.e., write the mathematical formulation in Latex) the problem in a way that is amenable to decomposition where there will be a subproblem for each Product- Customer pair Group. 4. Write a Python code that implements the full Branch-and-Price for any problem instance with the partially decomposable structure in the consideration matrix. Illustrate your results on the above toy example. 5. Make sure your code is as efficient as possible. You will be provided larger problem instances to test your code, if you need. You may also write your own code to generate random instances, if you want. 6. The group who can solve (via decomposition) the largest problem instance will get an additional bonus point (10%). 7. Write a short report including the problem description, details of your solution approach, your results and a short discussion your comments and further possible extensions. 8. Prepare a demo to illustrate your code. In the demo, you will be given a problem instance that you haven’t seen before, so make sure that your codes will work for any problem instance irrespective of the specific problem data. 1 2 3 4 5 6 1 1 2 4 2 5 1 3 3 2 4 5 5 1 2 6 4 7 3 2 3 8 4 1 3 1 9 2 3 3 2 1 4 Customers Products

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