Question

The Granary Restaurant chain has bought out a competing restaurant chain. However, it now has too many restaurants in close proximity to each other in certain cities. In the Ottawa area, the chain has 6 restaurants, and it does not want any restaurants closer than 5 km from each other. The following are the monthly revenues (in $1,000s) from each restaurant and a map showing the general proximity of the restaurants. Restaurants within 5 km of each other are boxed with the dotted lines. Restaurant # 1 2 4 Monthly Revenue ($1,000s) $127 $83 $165 $96 $112 $88 2 3 4 5 6 5 6 Develop a binary programming model to determine which restaurants the Granary chain should keep open in the Ottawa area, while maximizing its monthly revenue. ONLY FORMULATE THE ALGEBRAIC MODEL, DO NOT SOLVE

Answer 1

Variable definition:

Xi is a binary variable such that

Xi = 1 when restaurant is open

Xi= 0 when restaurant is closed

Where i = 1, 2, 3, 4, 5, 6

Objective function:

The granary restaurant chain wants to maximize their monthly revenue.

Therefore we can write that,

Maximize Z = 127*X1 + 83*X2 + 165*X3 + 96*X4 + 112*X5 + 88*X6

Constraints:

As restaurants 6 and 3 are in 5 km radius, we want to select only 1 of these restaurants.

X6 + X3 = 1

As restaurant 5 and 4 are in 5 km radius, We want to select only 1 of these restaurants.

X5 + X4 = 1

As restaurants 1, 2, & 4 are in 5 km radius, we want to select only 1 of these restaurants.

X1 + X2 + X4 = 1