Optimizing Resources with Linear Programming: A Step-by-Step Approach

[ver_mathstats]

Homework Statement

A famine relief effort is being mounted and there are three types of food bundles that can be flown out during each delivery. Bundle 1 has 4 kg. of flour, 4 kg. of sugar and 12 litres of water. Bundle 2 has 12 kg. of flour, 4 kg. of sugar and 4 litres of water and Bundle 3 has 8 kg. of flour, 8 kg. of sugar and 8 litres of water. The relief agency has 5200 kg. of flour, 3800 kg. of sugar and 6000 litres of water for each shipment. Bundle 1 can provide for 10 people between deliveries, Bundle 2 for 8 people and Bundle 3 for 11 people. How many bundles of each type should the relief agency send on each flight in order to maximize the number of people being fed.
Do this problem by setting up a linear programming problem and determining the vertices of the feasibility set.
List the number of bundles of types 1, 2, and 3, separated with commas.

Relevant Equations

Linear Programming

I am comfortable solving these types of problems, however I am having trouble setting up this problem and am unsure if I am doing this correctly so would someone be able to assist me?

I first wrote an equation for the people: 10x +8y +11z.

Then I made an equation for the flour: 4x + 12y + 8z <= 5200.
My equation for the sugar: 4x + 4y + 8z <= 3800.
My equation for the water: 12x + 4y + 8z <= 6000.
And then x>=0, y>=0, and z>=0.

Am I on the right track with this set up?

Thanks.

 

[sysprog]

I think that you're on the right track, because you've written the inequalities $-$ if you intend to solve this via the Simplex algorithm, you should first formulate the problem as a standard LP problem (all the constraints are linear) $-$ after you have determined the inequality constraints, as you have already done, you should fill out the tableau; then, to solve the LP problem, you should first establish the feasible region, then pivot on the correct points in the tableau to establish the optimal solution, and after that, you could do sensitivity analysis on the inputs, and regression to determine the vertices of the feasibiilty set.

 

[ver_mathstats]

I think that you're on the right track, because you've written the inequalities $-$ if you intend to solve this via the Simplex algorithm, you should first formulate the problem as a standard LP problem (all the constraints are linear) $-$ after you have determined the inequality constraints, as you have already done, you should fill out the tableau; then, to solve the LP problem, you should first establish the feasible region, then pivot on the correct points in the tableau to establish the optimal solution, and after that, you could do sensitivity analysis on the inputs, and regression to determine the vertices of the feasibiilty set.

Thanks so much! I ended up getting the answer using the tableau and such.