A candy company has 150 kg of chocolate-covered nuts and 90 kg of chocolate-covered raisins to b

step 1 For this problem, we are looking at a candy company. It has two different kinds of things that i

A candy company has 150 kg of chocolate-covered nuts and 90...

A candy company has 150 $\mathrm{kg}$ of chocolate-covered nuts and 90 $\mathrm{kg}$ of chocolate-covered raisins to be sold as two different mixes. One mix will contain half nuts and half raisins and will sell for $\$ 7$ per kg. The other mix will contain 3$/ 4$ nuts and 1$/ 4$ raisins and will sell for $\$ 9.50$ per kg. a. How many kilograms of each mix should the company prepare for the maximum revenue? Find the maximum revenue. b. The company raises the price of the second mix to $\$ 11$ per kg. each mix should the company prepare for the maximum revenue? Find the maximum revenue.

A candy company has 150 $\mathrm{kg}$ of chocolate-covered nuts and 90 $\mathrm{kg}$ of chocolate-covered raisins to be sold as two different mixes. One mix will contain half nuts and half raisins and will sell for $\$ 7$ per kg. The other mix will contain 3$/ 4$ nuts and 1$/ 4$ raisins and will sell for $\$ 9.50$ per kg.
a. How many kilograms of each mix should the company prepare for the maximum revenue? Find the maximum revenue.
b. The company raises the price of the second mix to $\$ 11$ per kg. each mix should the company prepare for the maximum revenue? Find the maximum revenue.

Key Concepts

Feasible Region

The feasible region is the set of all possible points that satisfy all constraints in a linear programming problem. Often represented as a convex polygon, especially in two-dimensional cases, the optimum solution typically lies at one of the vertices (corner points) of this region.

Linear Programming

Linear programming is a technique used for the optimization of a linear objective function, subject to a set of linear equality and inequality constraints. It is a fundamental method in operations research for solving resource allocation and blending problems where decision variables are continuous.

Objective Function

The objective function in a linear programming problem represents the goal that needs to be achieved, such as maximizing revenue or profit. It is expressed as a linear equation of the decision variables, and its optimization is done subject to the given constraints.

Constraints

Constraints are the limitations or restrictions expressed as linear equations or inequalities that the decision variables must satisfy. They represent resource limitations, ingredient ratios, or other conditions that must be met in the problem, shaping the possible solutions.

Sensitivity Analysis

Sensitivity analysis involves studying how changes in the coefficients of the objective function or constraints affect the optimal solution. It is useful for understanding the robustness of the solution and for making decisions when parameters, such as prices or resource availability, vary.

Transcript

00:01 For this problem, we are looking at a candy company.

00:04 It has two different kinds of things that it's selling, and we want to, if you read through the problem, we want to maximize its revenue.

00:11 This is a maximization problem.

00:16 Makes sense.

00:16 Most companies want to maximize the amount of money that they're bringing in.

00:20 So what is my revenue? well, there are two different things that they sell.

00:24 We have one mixture, which is half nuts and half raisins, $7 a kilogram.

00:29 So we're going to call this one, this x one, this is our 50 -50 split, half of one, half of the other.

00:39 And let me make this one just a little bit smaller.

00:44 And what about our second mix? well, there's one mixture that's three quarters nuts and one quarter raisin, and that's 950 per kilogram.

00:56 So 9 .50 for our second one.

00:58 And this is our 7525 split.

01:02 Okay, so that's my revenue.

01:04 If i know how much of each mixture i'm selling, i can tell you how much money i'm bringing in.

01:09 What are my constraints? well, the constraints come in the ingredients.

01:14 I only have 150 kilograms of chocolate covered nuts.

01:19 So let's look at the nuts.

01:20 I only have 150 kilograms.

01:23 So whatever i use has to be less than or equal to 150.

01:26 Well, our first mixture is half nuts.

01:29 So 0 .5 times my first mixture.

01:34 And in my second mixture, three quarters of that is nuts.

01:39 So 0 .75 times my second mixture.

01:44 That tells me how many nuts i'm using.

01:47 The other constraint are my raisins.

01:51 Well, like the nuts, i do have a limited amount of raisins, and that's 90 kilograms.

01:56 So whatever i use has to be lessen or equal to 90.

01:59 Half of the first mixture is raisins, but only a quarter of the second mixture is.

02:06 So these are my two constraints, my nuts and my raisins.

02:10 Now, i'm going to use the decimus graphing application to show what these constraints look like.

02:14 You can use a graphing application, your graphing calculator, whatever you're comfortable with.

02:19 And when you plot these, you'll see their overlapping area has four corners.

02:26 Now, our fourth corner is zero zero.

02:28 I'm not even going to consider that because that means i'm not selling anything...

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