This document discusses two mathematical models for business optimization problems: 1) A model for forming product catalogues to maximize customer coverage given constraints like catalogue size and number of products a customer must be interested in. It uses sets of customers, catalogues, and products. 2) A model for making optimal loan offers to customers to maximize profit while meeting constraints like budget, product growth targets, and group size. It also uses sets of customers, products, and groups. Both models can be viewed as variations of set covering problems. The models could be combined to solve a larger optimization problem by using the output of one as input to the other.

1. Analytics in Business
Surya Pavan Pynda

2. Catalogue Formation
● Problem requires formation of predetermined number of catalogues
containing various products to cover the maximum number of
customers
● Prerequisites
○ Number of catalogues T
○ Size of catalogue r
○ Threshold for coverage t
○ Customer interest database

3. Catalogue Formation
● Need to allot products to catalogues meeting the following
constraints
○ Each catalogue has size r
○ Catalogue covers a person if at least t products of his interest
are present
○ Person is covered if at least one catalogue represents him
○ Number of persons covered by each catalogue is integer
● Optimisation function is business requirement of covering the
maximum possible customers

4. Sets
● Primitive Sets
○ Customers i
○ Catalogues j
○ Products k
● Derived Sets
○ Interest Ai,k
○ Inclusion Vj,k
○ Covering Yi,j

5. Mathematical Model
● Size constraint
● Threshold constraint

6. Mathematical Model
● Representation constraint
● Integer constraint
● Objective function

7. Product Offer
● Problem requires making optimal loan offers to customers to get
maximum returns from loans
● Prerequisites
○ Number of customers N and products J
○ Number of groups G and customers in each group Ni
○ Response, profit and volume scores for customers pk,j
, rk,j
, vk,j
○ Product wise asset growth targets Tj
○ Operating overhead per product Θj
○ Targeting cost per customer

8. Product Offer
● Need to allot offers to customers meeting the following constraints
○ Cost of contacting selected customers should not exceed
budget
○ Total volume of each product must reach growth target
○ Number of offers in group should be atmost the size of group
○ Also average response probability, average expected profit and
average expected volume must be calculated from data
● Optimisation function is business requirement of maximising net
income which is total expected profit removing operating costs and
targeting costs

9. Sets
● Primitive sets
○ Groups i
○ Products j
○ Customers k
● Derived sets
○ Volume Vk,j
○ Profit Rk,j
○ Response Pk,j
○ Allotment Ai,k
○ NumGrp Ni,j

10. Mathematical Model
● Budget Constraint
● Target Constraint
● Offer Constraint

11. Mathematical Model
● Average Values
● Additional Constraints like number of people in group, allotment of
people into group and grouping exist due to integer nature of
problem

12. Use in Analytics
● The above models are variations of set covering problems with
additional constraints which differentiates them
● They can be used in conjunction to solve a bigger problem
● For example, the offers used in Model II can be obtained from
Model I if we have customer interest database to optimise the
response probability
● Model I uses binary programming while Model II uses integer
programming

13. References
● Amiri, A. (2006). Customer-oriented catalog segmentation:
Effective solution approaches. Decision Support Systems, 42(3),
pp.1860-1871.
● Gnanendran, K. and Sundarraj, R. (2006). Alternative model
representations and computing capacity: Implications for model
management. Decision Support Systems, 42(3), pp.1413-1430.