You are here:

Basic Math/Inequality word problem

Advertisement

Expert: Josh - 10/24/2009


Question
Earthquake victims in China need medical supplies and bottled water.  Each medical kit measures 1 cubic foot and weighs 10 pounds.  Each container of water is also 1 cubic foot and weighs 20 pounds.  The plane can only carry 80,000 pounds with a total volume of 600 cubic feet.  Each medical kit will aid 6 people, while each container of water will service 10 people.  How many of each should be sent in order to maximize the number of people aided?

Answer

Inequalities
Hi Anne,

Let "m" be the number of medical kits, "w" the number of water containers.

There are two separate constraints, one on total volume (volume of medical kits plus volume of water containers), another one on total weight (weight of medical kits plus weight of water containers). We express these using the following inequalities:

Weight: 10m+20w <= 80000 ...[1]
Volume: m+w <= 600 ...[2]

Finally, we want to maximize utility "U" (the usefulness of the rescue operation). Essentially, we want this to benefit the maximum number of people. Utility as a function of "m" and "w" is expressed as U(m,w) = 6m+10w (with m,w > 0) ...[3]

Goal: Find "m" and "w" to maximize U(m,w) and satisfy conditions [1] and [2].
We can also write this as

(M,W)={argmax_{m,w} U(m,w) | 10m+20w<=80000 and m+w<=600}

One approach
============
We will try to solve this by geometric construction.
Note: Please refer to the figure to make sense of this.

On the Cartesian plane, (vertical axis for "m", horizontal axis for "w"), plot the line equations and highlight the two regions corresponding to the inequalities [1] and [2].

Step 1a: For [1], plot m=-2w+8000 (see black line on diagram).
Step 1b: Since the region representing the inequality m<=-2w+8000 is either above the line or underneath it, we pick a convenient test point (say m=0,w=0) to check which is the case. With m=0 and w=0, it is easy to see that 0<=-0+8000 is true. So, indeed the correct region representing this inequality is below the black line. (shaded yellow on diagram).

Step 2a: Likewise, for [2], plot m=-w+600 (see blue line on diagram).
Step 2b: Using the same technique as in (1b), we find that the region representing m<=-w+600 is also below the blue line (shaded blue on diagram).

Step 3: Since we need to satisfy both constraints, we must look for feasible solutions where the two inequalities intersect (i.e., the green shaded region on the diagram is all the two inequalities have in common).

Step 4: Now, we focus on the utility function U(m,w)=6m+10w that we want to maximize.

(1) Observe that the solution must lie on the upper boundary of the green region. i.e., the solution (M,W) must lie on the blue line. (Reason: We want to maximize the number of people who benefit from this rescue operation after all, so we want the largest possible combination of "m" and "w" within the established bounds).
- one trivial solution is obtained letting w=600, m=0.
- but I guess both medical supplies and water are needed, so what is the best combination?
(2) One unit of water container benefits more people than one unit of medical kit. Intuitively, we want to have more "w" than "m". The values of "m" and "w" must strictly be non-negative.
(3) For a given cost "u", m=-(5/3)w-u (see [3]). This contour may be visualized as the green line on the diagram.
(4) To obtain the best trade off, we climb along a line perpendicular to this contour from the origin. The normal equation has the form m=(3/5)w ......(see red line on diagram).

Step 5: Finally, the optimal solution is given by the intersection of m=(3/5)w and the blue line m=-w+600. i.e., (3/5)w+w=600, w=(5/8)*600=375, m=225.

Josh

Expertise

When I work through problems, I like to emphasize concepts which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponential & Logarithms. Basic Calculus. Probability, Permutation and Combination. Mathematical Induction. Complex numbers. Physics problems.

Experience

I have worked as a teaching assistant in college. My hope is that more people will share knowledge without boundary, give help without seeking recognition or monetary rewards.

Education/Credentials
Bachelor degree in Engineering Science

©2012 About.com, a part of The New York Times Company. All rights reserved.