Lesson video

Hi, I'm Mrs. Wheelhouse and welcome to today's lesson, which is from the unit Maths in the Workplace.

In this series of lessons we're exploring how maths is used in different careers.

Let's get started.

In today's lesson, you are going to look at some of the ways that maths is used in operational research.

Now, one of the words we're gonna be using today is logistics.

And logistics is the careful organisation of a complicated project or a set of activities so that it happens in a successful and effective way.

Let's look at how we use that word today.

Our lesson is broken into two parts and we're gonna begin by looking at packing.

Now, in order for a business to survive, its income must be greater than or equal to its expenses.

Just meeting expenses though will not allow the business to invest or improve existing resources.

Therefore, the more efficient a business is, the more likely it is that the business will make a profit.

So let's focus on preparing goods for transportation.

To make things easier, we're gonna focus on just a small number of goods.

Now obviously we know that real businesses could be dealing with thousands and thousands of goods, but we're just gonna focus on a small number just so that it's a lot easier for us to play with those numbers and get a feel of what's happening.

A company needs to send nine items to a store.

The items have to be stacked on pallets.

A pallet can not hold more than 200 kilogrammes.

So here are the masses in kilogrammes of the items. So we have 30, 70, 80, 80, 100, 110, 120 and 150.

What is the smallest number of pallets we need? Well, how might we do this? What do you think? Pause the video and have a quick think now.

Welcome back.

Let's see what you've come up with.

Well, Lucas and Izzy are gonna have a go at thinking how many pallets are needed.

Lucas says, "We only need four pallets to transport the items." Izzy thinks we need five.

Did you say either of those? Or maybe your justification is the same.

Let's find out.

Well, the total mass of all the items to send is 850.

When we divide that by 200, which is what a pallet can hold, we get 4.

25.

As Lucas correctly points out, 4.

25 rounds down to 4.

Is he right though to conclude that that's four pallets? Izzy says we need more than four pallets, so the smallest number we need is five.

And of course she's right.

Four would not be enough.

So we've got to round up in this case.

And this is one of the examples where, in a practical context, even though the maths says we should round down, we know we need to round up.

A pallet cannot hold more than 200 kilogrammes.

Here are the masses and kilogrammes of the items and it's the same masses as before.

Decide which items should be placed on which pallet.

Can you make the items fit on the five pallets? Pause the video while you have a go at this now.

Welcome back.

Did you get it to fit? Well, this is what Lucas decided to do.

He placed each item onto the first pallet that could hold it.

At this point, he's got 180 kilogrammes on that pallet, which means it can't hold any more because we don't have a 20 kilo mass left.

Now pallet two is full, in the sense of it can't take any more.

Oh, hang on, pallet three can't take any more 'cause there's nothing of 90 or less left.

Well that one goes on pallet four.

The 120 goes on pallet five.

Oh no, it doesn't fit, the 150's left over.

That didn't work.

So Lucas tries again, but this time he starts with the largest mass first.

So same idea in that he's going to put it onto the first palette that can hold it, but this time, instead of starting with the smallest mass, he's starting with the largest.

Now at this point, he's got to decide where the 80 goes.

Which pallet do you think it will go on? Well, following Lucas's rule, it's gonna go on pallet two.

The next 80 kilo mass will go on pallet three and then the 70 kilo mass goes on pallet four.

And look at that.

Now it all fits.

Did you think about putting that 30 on pallet five? Remember it's going to go on the first pallet that can hold it.

So it goes on pallet one.

He could, of course, have put it on pallet five and it would still have worked, but then he wouldn't be following the rules he set for himself.

And this is just one of the possible ways that you could stack the masses.

Now Izzy decided to do it a different way, and maybe you went for this option.

Izzy grouped the masses to create pallets that were full.

In other words, when you add up the masses on the pallet, it should make 200.

So she paired together the 80 and the 120 to make 200, and then she put together the 30, 70 and 100 kilo masses to make another 200.

At this point, she hasn't got any more full pallets, but she does her best.

And look at that, they fit.

So which of these three methods do you prefer? Maybe you did one of these, maybe you did one, but now you've seen a different method, you prefer that instead.

So I've got them on the screen for you and what I'd like you to do is just pause and have a quick discussion about which one you preferred and why.

Pause and do this now.

Welcome back.

Now maybe you said one of these and maybe this is your reasoning.

Method one was very quick.

It just didn't work for these values.

It doesn't mean it will always not work.

Sometimes it will work and it'll be really quick, which would be great.

It just didn't work in this case.

Method two was quick and it led to a similar number of items per pallet.

Can you see that four of the pallets each have two items on and it's only pallet five that doesn't? Now, method three led to pallets which we know cannot have any more items placed on them.

And you might have felt this was quite efficient.

Quick check now.

True or false? There is only one efficient way to pack items for transporting.

Is that true or false? And justify why.

Pause the video and do this now.

Welcome back.

You should, of course, have chosen false.

We saw that in our example just now, that there were two ways you could stack those pallets and both were efficient.

There are things to consider like constraints, such as the size of the items or the dimensions of any packing crates.

It's now time for your first task.

Question one: A ferry transports vehicles across a river.

The ferry has three lanes that vehicles can use and each lane is 30 metres long.

One morning, these vehicles arrive in this order.

So you can see in the table you've got the arrival order, what vehicle arrived, and the length of that vehicle.

For part A, by loading each vehicle as it arrives into the first available lane, will all the vehicles be able to cross? You need to justify your answer.

So maybe you want to say, for example, lane one, lane two, lane three, and tell me where these vehicles are going to work out if they'll all fit.

Pause the video while you do this now.

Welcome back.

For part B, by considering all the vehicles, regardless of when they arrived, can all the vehicles be loaded onto the ferry? Justify your answer.

So again, you might want to do this by telling me what vehicle goes into which lane.

And then part C, why is your answer to part B not necessarily practical? So you need to think about the context here.

Pause the video now while you work on these two questions.

Welcome back.

Question two now.

Jacob is backing up his computer.

He is copying games onto 32 gig USB sticks.

The size of each game is listed in the table.

Part A: Jacob thinks that it is possible to store all the games on five USB sticks.

Explain why he might think this and why he is mistaken.

And then part B: state the smallest number of USB sticks that will be needed and give a possible solution as to which games will be stored on which stick.

Pause the video while you work on this now.

Welcome back.

It's time to go through our answers now.

For part A, I said by loading each vehicle as it arrives onto the first available lane, will they all be able to cross? So what I did was I wrote out my three lanes and I assigned each vehicle to each lane as it arrived.

And what I discovered was the final lorry cannot make the crossing.

There's nowhere it can go because now the lanes don't have enough space for it, and you should have found that out too.

Part B, by considering all the vehicles, can they be loaded onto the ferry? And what I've done on the screen is just given a possible solution.

So the answer is can they all be loaded? Yes, they absolutely can.

And this is just one possible way that could happen.

You might have a different way and it works as well.

So part C, why is that not necessarily practical though? Well, I've said as an example, vehicles may not arrive at the same time, and if they don't, it's not necessarily possible to put one on and then the other, and so they might not be able to pass each other in the queue.

So even though, for example, you want to put the lorry into lane one, you might have to put a van there, or a car there instead.

Well done if you said something similar.

For question two, we had Jacob and the computer games.

In part A, you had to explain why Jacob thought five USB sticks would be enough and why he's mistaken.

Well, if we sum the size of all the games, we get 155.

Now when we divide that by 32, we get 4.

84375, which does round up to five.

However, when you look at the size of each game, you'll notice there are six of them with a size over half of what one USB stick can hold.

Well those games cannot be stored together, which means we're going to need at least six USB sticks.

Well done if you spotted that.

In part B, you had to state the smallest number of USB sticks that will be needed.

Well, in this case there's six and what I've done on the screen is given you a possible solution.

But remember there are other solutions too.

What was important though is that the 25, 25, 24, 22, 21 and 17 had to be on different sticks.

It's now time for part two, which is looking at planning a project.

When managing a project which has lots of activities, it can be helpful to present these on a chart.

This is especially useful when some activities cannot start until other activities have been completed.

Here's an example of such a chart.

The numbers at the top represent time, and this could be in minutes, hours, days, et cetera.

Activities are shown as boxes.

The length of the box shows how long it takes to complete an activity.

So let's do a quick check on that.

It takes four minutes to complete activity A.

How long does it take to complete activity D? And then the second question is: which two activities take the same amount of time? Pause the video now while you work this out.

Welcome back.

Let's go through our answers.

So how long does it take to complete activity D? It takes seven minutes.

You can see that because we look at the gap between the start and the end of the activity.

Then which two activities take the same amount of time? Well, that's C and E because their bars are the same length.

Now sometimes there'll be spaces between the boxes or the activities.

And this is because although one activity has finished, the next one cannot start yet.

So in this example, B has finished, but C can't start straight away.

Now each row represents the activities that one person will complete and when they will do them.

This chart therefore shows what activities two different people will be doing.

So here's a project that Aisha and Sofia are working on.

On day four, Aisha is working on activity what? Well, she's working on activity A.

And we can draw a line here to show.

So the zero to 1 represents day one, 1 to 2 represents day two, 2 to 3 represents day three.

So 3 to 4 represents day four.

And we can see Aisha clearly working on activity A.

Let's do a quick check to make sure you've got that.

On day six, what is Sofia working on? Is it activity C, activity D, or she's not working on any activity? Pause the video and make your choice now.

Welcome back.

You should have said Sofia was working on activity.

That's right.

D.

Activity C is being worked on by Aisha.

Describe what each pupil is doing on day 12.

So on day 12, are they working on an activity? And if so, which one? And if they're not working on an activity, say so.

Pause the video and do this now.

Welcome back.

Let's see what you put.

Well, on day 12, Aisha is working on activity E and Sofia is not doing an activity for the project.

Well done if you put that.

It's now time for our final task.

Izzy and Laura are working on a school project.

They have broken the project down into a series of tasks.

They create a table listing the tasks, how long it will take to complete each task, and which tasks have to be completed before a new task can be started.

They have two and a half hours to work on their project.

I'd like you to create a chart showing who will work on which tasks and when.

Can Izzy and Laura complete their project in time? And what I've done is provided a template chart for you.

Here is the table showing the activities, how long it takes to complete the activity and which activities need to be completed before this one can start.

So for example, the research and gather supplies activities, A and B, don't require anything else to be done before they can be started.

But creating templates, activity C, can't be done until activity B is finished.

In other words, you can't make the templates until you've gathered the supplies.

I've included a chart template for you to help with assigning the activities to the different pupils.

Across the top, you can see the time in minutes.

You might need to make this chart a little longer.

So do feel free to extend it if you need to.

Pause the video while you work on this now.

Welcome back.

How did you get on? Well, I'm gonna go through an example of how you can assign the activities.

Yours might be different, but you still get to the same overall answer as me.

So I'll go through this one.

So this example shows Izzy and Laura completing their project on time.

So, person one is at the top and person two is at the bottom.

And it doesn't matter if it's Izzy or Laura, whichever way around.

But I've said one of them needs to be doing the research as the other one gathers the supplies.

Then whoever gathered the supplies can create the templates.

Now at this point, one of them needs to create the artwork and one of them needs to do the writing.

Whoever's done activity A can go straight into either activity D or E.

And I've chosen it to be D, but the person who gathered the supplies and then created the templates has to wait for a bit before they can begin the writing because activity A hadn't finished.

Then I added in the checking and compiling.

And again, you could have put these around the other way if you wanted to.

I made sure though that both D and E were finished before F and G started.

And now all the activities are done and they've completed their project on time.

It is noticeable though, that one of them is having to work flat out the entire time and the other one gets some breaks.

Hmm, not exactly fair.

So maybe they wanna think about possibly changing those activities around.

Did you come up with a fairer way to do it? Well done if you did.

Let's sum up what we've learned today.

Logistics is the careful organisation of a complicated project, set of activities, so that it happens in a successful and effective way.

A solution may seem optimal, but in reality not be practical.

And we saw that a couple of times, didn't we, in some of our examples? Well done.

I hope you've really enjoyed learning about some of the ways maths can be used in operational research and how we can be a little bit more efficient in what we do.

I look forward to seeing you again for some more maths.

Goodbye for now.