Question

ProblemYou work for a large conglomerate of 100 associated companies. In order to avoid antitrust issueswith the DOJ, you need to divide the 100 companies into 4 dierent groups such that:1. Companies within the same group cannot do business with each other.2. Companies in dierent groups can do business with each other.What organization of the 100 companies into the 4 groups maximimizes the number of businessopportunities?a) Start by labeling the groupsX,Y,Z, andW. Letx=the number of companies assignedto groupXand so on for the other groups. Construct an equation inxfor the numberof business opportunities for a company in groupX— i.e., how many companies can thatcompany do business with?b) Now build an equation inxfor the total number of business opportunities for all companiesin groupX.c) Do likewise for the remaining groups and construct a functionf(x,y,z,w)that gives thetotal number of business opportunities across all the groups.d) What is the constraint onx,y,z, andw?e) Introduce a Lagrange multiplierλand determinefx=λgx, wheregis the function con-structed from the above constraint.f) Do likewise forfy,fz, andfw, and combine them with the constraint so that you have 5equations in 5 unknowns.g) Use substitution to determine a value forλ.h) Substitute the value forλinto the other equations to determine the optimal distributionof companies into the 4 groups.1

Answer 1