Optimum Product Mix The Bubble Company produces a variety of bottles from recycled plastic. The

Optimum Product Mix The Bubble Company produces a variety of...

Key Concepts

Production Capacity Constraints

This concept refers to the limited availability of key resources—in this case, machine hours—that restrict the total amount of production. When a production process is limited by such a constraint, decisions about which products to produce must account for how much of the resource each product consumes and how it affects overall capacity utilization.

Contribution Margin Analysis

This involves evaluating the profit generated from each product after subtracting variable costs. By calculating the contribution margin per unit, a business can determine which products yield more profit relative to the resources they consume. In scenarios where production capacity is limited, the contribution margin often should be analyzed on a per-unit-of-resource basis.

Profitability per Unit Resource (Machine Hour) Analysis

When production is constrained by a resource such as machine time, the profitability of a product is best measured by its profit per machine hour. This metric helps in comparing different products by highlighting how efficiently each product converts limited resources into profit, thereby guiding the choice of which product to prioritize.

Optimal Product Mix Decision Making

This is the process of determining the best combination of products to produce in order to maximize overall profitability given constraints. It requires analyzing factors such as production capacity, contribution margins, and resource consumption, to decide whether to focus on a single product or a mix of products that best uses the limited resources.

Transcript

00:01 So this problem, we are looking at the murrow manufacturing company and trying to decide what allocation of television sets they should make in their plant.

00:13 Now, reading all the way through it, you can see that the question is, what should they do to maximize their profit? so that's our end goal.

00:23 This is a maximization problem.

00:25 It's always good to read the whole problem through before you start trying to make sense of the numbers.

00:30 See what the goal is.

00:32 So our goal is to maximize profit.

00:35 Well, first, let's go through and find what the profit actually is.

00:38 Well, the profit on a flex scan set is $350.

00:44 So i'm going to let x sub 1 equal the number of flex scan sets that they sell.

00:51 Great idea to mark down what the variables mean.

00:54 If you don't, it's very easy to get to the end of this problem and then not remember what the variables were that you set up.

01:00 So x sub 1 is going to be the, the number of flex scan sets we sell.

01:05 Now, what's the profit for the other one? well, the panoramic one has a $500 profit.

01:13 So this is the panoramic.

01:18 So there's my profit.

01:19 If i knew how many of each set was sold, i could tell you what my profit is.

01:23 And we want to find the maximum profit, which makes sense.

01:26 That's what most companies do want to do.

01:28 Now, there are some things that are going to be constraints, things we have to keep in mind.

01:33 First, we have an assembly line.

01:35 And i'm just going to mark what these are.

01:38 We have an assembly line.

01:40 Every flex scan takes five hours in the assembly line.

01:44 Each panoramic takes seven hours.

01:46 So this is how much time i'm going to need in my assembly line.

01:50 And at most, i have 3 ,600 hours available to make these tv sets.

01:57 Next, it has to go get a cabinet.

02:00 So in the cabinet department, the flex scans take one hour.

02:05 And the panoramics take two hours each.

02:08 And at most i can use 900 hours in the cabinet shop.

02:13 Now our last thing is our testing and packaging.

02:22 And there we're told that each, it doesn't matter which one it is, testing and packaging takes four hours apiece.

02:28 So that's four hours for the first one, four hours for the second one, and at most i can spend 2 ,600 hours in the testing and packaging area.

02:39 Now, this last one here, i can make this a little bit simpler to keep our number.

02:43 A little smaller.

02:44 If i divide everything by four, this gives me x sub 1 plus x sub 2 is less than or equal to 650.

02:52 So i'll probably use that one.

02:54 It just makes life a little bit easier having smaller numbers.

02:57 Okay, so here are my criteria, my constraints, and there's my maximum, in my profit equation that i want to maximize.

03:07 Okay, so what i have done is i've used our decimos graphing calculator here.

03:11 You can use a graphic calculator, anything that you're comfortable with.

03:15 And you can see that i have three equations.

03:20 And let me just come back out and put these in one at a time.

03:23 So this is our assembly, our cabinet making, and our testing and packaging.

03:30 And you can see there is an overlap.

03:33 So those are the corners.

03:39 Those four here.

03:40 We also have a corner at zero zero.

03:43 But if i make nothing, i have no profit.

03:45 So that is by far the smallest.

03:47 I'm not even going to going to look at that one.

03:48 I'm going to acknowledge that it is a corner point, but it makes no sense to make no tvs in this context.

03:54 I'm making something.

03:56 I want to have a profit.

03:57 So these are the four points we're looking at.

03:59 So i'm going to come over here back to our whiteboard.

04:02 This is not in any particular order.

04:04 I'm just going to write down those four points.

04:07 So 0 ,450, 300, 300, $475, 175, and $650 .0.

04:22 Those are our four points.

04:24 So what we need to do is put those points back into our profit equation.

04:30 Plug them in, see what profit we get with this allocation of our resources.

04:35 So for the first one, if i make no flex scans and 450 panoramics, my profit is going to be $225 ,000.

04:45 For my second point, 300 of each, i'll have a profit of 255 ,000...

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