The company (Ramsay Ltd) investigates what product prices should be charged for the following year based on existing background information. In order to facilitate this analysis task, price elasticities of product demand should be utilised. Price elasticities of demand are based on historical records relating percentage decrease of product demand to percentage increase in product price. So, the price elasticity, E, of a product i can be defined by: E = demand decrease for product i %) price increase of product i (%) It should also be added that products P3 and P4 are somehow related, and their demand depends on each other's price as well. This is captured by cross-elasticity of demand with respect to price. So, cross-elasticity, CEik, from product k to product i can be defined by: demand increase for product i (%) CEU = price increase of product k (%) It can be assumed that effects of elasticity and cross-elasticity terms (for products P3 and P4) is linear and additive. In Table 3, the elasticities and cross-elasticities are provided. Table 3: Elasticities and cross-elasticities P1 P2 P3 P4 P3 to P4 P4 to P3 1.0 3.5 2.1 1.3 0.4 0.1 The aim here is to determine the product prices and associated demands that maximises total revenue subject to price and supply/demand constraints. In addition, there are contractual requirements that the new product demand should be at least 85% of last year's demand (see Table 2). You are required to: a) In order to develop a nonlinear program that determines the optimal production mix so as to maximise the total revenue, answer all of the following: i. What are the sets or indices in the mathematical model, if any? [2] ii. What are the optimisation variables? [3] iii. Define the objective function that maximises the total revenue and provide a mathematical expression for the revenue. [2] iv. Since the elasticity and cross-elasticity are additive, formulate the demand constraint to be the sum of these two terms which will constrain the prices through these elasticities (hint: increase of price of product i should result into a negative trend on that product's demand (elasticity) and positive trend on the demand of other products (cross elasticity)). [5] v. Formulate other constraints required for the mathematical model (supply constraints, variable bounds).