Question

Champions’ Shoes Ltd is a manufacturer of sport shoes, specialized in the production of basketball and soccer shoes. The Marketing Director of the company is about to draw the 2014/2015 plan for allocating the marketing resources to the sponsorship of basketball and soccer teams. Each soccer teams requires that 120 pairs of shoes must be provided whether sponsored, while each basketball team requires 32 pairs of shoes to be supplied. Furthermore, additional promotional and sponsorship costs include £ 150,000 for each sponsored soccer team and £ 225,000 for each sponsored basketball team. The overall monetary budget available to the Marketing Director for the sponsorship campaign is equal to £ 20,000,000. The production of promotional shoes for sponsoring teams is constrained to the availability of raw materials and in particular to the availability of a special gel used for branding shoes. Champions’ Shoes has 4,000 cm3 of gel available for the next 2014/2015 sponsorship campaign. Producing one pair of basketball shoes requires 3 cm3 of gel, while producing one pair of soccer shoes requires 1 cm3 of gel. The Marketing Director desires to find the maximum number of teams to be sponsored, constrained to the available monetary and production resources. You are required to help the Director to achieve this target by: a) Formulating the algebraic linear programming problem [35% of the marks of this Section] b) Solving the problem through the graphical method [30% of the marks of this Section] c) Formulating the linear programming problem on a spreadsheet and finding the optimal solution via the Solver [35% of the marks of this Section]

Answer 1

a) Let the number of Football teams sponsored be x

Let the number of basketball teams sponsored be y

Maximize, Number of teams = x + y

Constraints:

150000*x + 225000*y <= 20,000,000 (Budget constraint]

120*x + 32*3*y<= 4000 [constraint on gel]

120x + 96y <= 4000

b) Solving the problem through graphical contour method, we get the maxima as

x=0 and y=125/3, so the maxima exists when x=0 and y=41.6667

So, we can sponsor 41 teams with this budget

c) Solving the linear program on spreadsheet

Maximize, Number of teams = x + y

Constraints:

150000*x + 225000*y <= 20,000,000 (Budget constraint]

120*x + 32*3*y<= 4000 [constraint on gel]

120x + 96y <= 4000

using the excel solver

Note - Post any doubts/queries in comments section.