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Contribution for each constrained hour, which job fills the hours first, the plan that uses them best within demand, what that plan earns over filling by contribution a job, and what one more hour is worth.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will work out each job's contribution for an hour of the scarce resource, fill the week's hours with the best job up to its demand and then the next, and measure what that plan earns over the one that ranks jobs by contribution a job. You will also find what one more hour of the scarce resource is worth.
You can name the resource that runs out first and rank two products by what they earn for an hour of it. Real weeks add a second limit: customers want only so many of each job. This lesson turns the ranking into a plan for the week that respects both limits — the hours and the demand — and then asks what one more hour of the scarce resource would be worth, which is the figure that tells an owner how much it is worth paying for overtime, a borrowed machine or a longer day.
| Term | What it means |
|---|---|
| Contribution for each constrained hour | A job's contribution divided by the hours of the scarce resource it takes. |
| Plan | How many of each job to do this week. |
| Demand | The most of a job customers will take. |
| Marginal job | The job the last scarce hour goes to: the one whose demand is not yet met. |
| Value of an extra hour | The marginal job's contribution for each hour: the most worth paying for one more. |
Fixit Mobile has 50 bench hours this week. A screen replacement contributes 32 and takes half an hour; a board repair contributes 60 and takes an hour. Customers want 80 screens and 20 board repairs — 60 hours of work for 50 hours of bench.
$$\text{contribution an hour} = \frac{\text{contribution a job}}{\text{hours a job}}$$
Screens earn 64 an hour and board repairs 60. So the procedure is:
The week earns 80 × 32 + 10 × 60 = 3160. The plan most owners make starts with the job that earns more a job: 20 board repairs take 20 hours, the other 30 hours do 60 screens, and the week earns 1200 + 1920 = 3120. Same bench, same customers, 40 dollars less.
The gap looks small here because the two rates are close. Where they are far apart it is not: Corner Bean's oven earns 36 an hour on croissants and 20 on loaves, and loaves first would cost Alexa 128 in a single week.
Another way: table
Two plans for Fixit Mobile's 50 bench hours.
| Plan | Screens | Board repairs | Hours used | Contribution |
|---|---|---|---|---|
| By the hour: screens first | 80 | 10 | 50 | 3160 |
| By the job: board repairs first | 60 | 20 | 50 | 3120 |
Both plans use every hour. Only the ranking differs.
Confirm the constraint. The ranking is for the resource that is full this week; with hours to spare, every job that contributes is worth doing and no ranking is needed.
Work each job's rate. Contribution a job over the scarce hours it takes. Turn minutes into hours first: twenty minutes is a third of an hour.
Rank, then fill. Give the hours to the top job until its demand is met, then to the next, and so on, until the hours run out.
Total the plan. Units of each job times its contribution, added.
Find the marginal job. The job that got the last hour and still has customers waiting. Its rate is what one more hour is worth.
Check the plan by counting the hours it uses: they must equal the hours available, unless every job's demand is met first. Then check it by moving one hour between two jobs in the plan: the total should not rise. If moving an hour from the lower-ranked job to the higher one raises it, the higher job's demand was not yet met and the plan stopped too soon.
The top job does not get every hour, only as many as its customers want. Past that point, more hours on it earn nothing, because nobody is waiting for the eighty-first screen. That is why the procedure has a second step, and why the lower-ranked job is still worth doing with the hours left: an idle bench earns zero, and 60 an hour is far better than zero.
It also says where to look for growth. If the top job's demand is used up with hours to spare, the bench is no longer the problem for that job; finding more customers for it would be worth 64 an hour, the best use of the bench there is.
Once the plan is made, the useful question is what an extra hour of the scarce resource would earn. It goes where the last hour went: to the marginal job, the one whose customers are still waiting. In Dan's week that is board repairs, so an extra bench hour is worth 60 — not 64, because every screen customer is already served.
That figure is the most worth paying for one more hour of the constraint. Overtime at 25 dollars an hour to keep the bench going later is worth it at 60 an hour of contribution; hiring a second bench at 70 an hour is not. A borrowed machine, an earlier start, an extra part-time pair of hands on the constraint — each can be judged the same way, against the value of an extra hour.
The value changes with the plan. When the marginal job's demand is met too, an extra hour is worth nothing until more customers are found, however much the jobs contribute. So the value of an extra hour is always worked out for a particular week, with its particular customers.
The procedure does not change with more jobs. Work each one's rate, rank them all, and fill in order: the top job up to its demand, then the second, then the third, until the hours run out. The last job to get any hours is the marginal one, and jobs ranked below it get none this week.
A sign-maker with four jobs on a full laser — plaques, shop signs, wedding sets, badges — ranks them by the hour and books them in that order. Jobs ranked last are not refused for ever: in a quieter week, with laser hours to spare, they are all worth doing.
The procedure assumes the owner knows how many of each job customers want this week. In practice, some of that demand is booked and some is a guess: walk-in repairs, customers who order on the day, a market that may or may not be busy.
The safe way to plan is to fill the scarce hours with booked work in the ranked order first, and to hold back a few hours for the top-ranked job's walk-ins, because an hour held for a job worth 64 that then goes to one worth 60 costs only 4, while an hour given away to a 60 job that turns away a 64 walk-in costs the same 4 — the arithmetic is symmetrical, but a walk-in customer turned away may not come back. So the rule of thumb is to keep a small reserve for the best job when its demand is uncertain, and to release it to the next job at a fixed time — noon on the day, say — if it has not been used.
Lower-ranked work that can be scheduled at short notice is the ideal filler: a board repair that the customer has left for the week can be picked up in any hour the screens do not need. Knowing which jobs can wait and which cannot is as much a part of the plan as the ranking itself, and it is what lets the scarce hours stay full without turning away the customers who matter most.
A print shop's large-format printer is booked solid in December. Three jobs compete for it: event banners contribute 90 and take an hour and a half; window graphics contribute 50 and take half an hour; canvas prints contribute 40 and take 40 minutes. Per printer hour: banners 60, window graphics 100, canvases 60.
The shop has 100 printer hours in December. Customers want 80 window graphics (40 hours), 50 banners (75 hours) and 60 canvases (40 hours) — 155 hours of work. Ranked by the hour, window graphics go first: 40 hours and 4,000 of contribution. Banners and canvases tie at 60 an hour, so the owner gives the remaining 60 hours to banners first because banner customers are events with fixed dates: 40 banners and 3,600. The month earns 7,600.
Under the owner's old habit — banners first, because they earn most a job — all 50 banners take 75 hours and earn 4,500, and the 25 hours left make 50 window graphics, 2,500: 7,000 in all. Ranking by the hour is worth 600 in the month.
The marginal jobs are banners and canvases, both at 60 an hour with customers waiting, so an extra printer hour is worth 60. A nearby shop offers printer time at 35 an hour; she books 20 hours of it for canvases and adds about 20 × (60 − 35) = 500. A second printer on a year's lease, at around 75 an hour of December use, she declines.
Hospitals plan their operating theaters as a scarce resource, and the same logic appears there in a different currency: which procedures to schedule in the limited theater hours, and what an extra theater session would be worth. The currency is patients treated or health gained rather than contribution, but the method — rank by what an hour of the scarce resource achieves, fill up to need, value the next hour at the margin — is the same.
The job that earns most a job goes first. Not when hours are short and the jobs take different hours.
The top job gets all the hours. Only up to its demand; then the next job gets the rest.
Leave the hours idle rather than do a lower earner. An idle hour earns nothing at all.
Split the hours evenly to be fair. Customers are served by the plan; the split only costs contribution.
An extra hour is worth the top job's rate. Only if the top job still has customers waiting; otherwise it goes to the marginal job.
Rank the salad bag: 2.50 in 0.05 hours.
$2.50 \div 0.05 = 50$
Fifty dollars a picking hour.
Rank the herbs: 1.50 in 0.025 hours.
$1.50 \div 0.025 = 60$
Sixty a picking hour: herbs first.
Fill the herbs up to their demand of 600.
$600 \times 0.025 = 15 \text{ hours}$
Fifteen of the thirty hours.
Give the other 15 hours to salad bags.
$15 \div 0.05 = 300$
Three hundred of the 400 wanted.
Total and compare with salad first.
$600 \times 1.50 + 300 \times 2.50 = 1650 \text{ against } 1600$
Ranking by the hour earns 50 more.
Rank the two jobs.
$32 \div 0.5 = 64; \quad 60 \div 1 = 60$
Screens first.
Fill the screens up to 80.
$80 \times 0.5 = 40$
Forty hours.
Give the other 10 hours to board repairs.
$50 - 40 = 10$
Ten of the 20 wanted.
Total the week.
$80 \times 32 + 10 \times 60 = 3160$
The best plan.
Find the marginal job.
$\text{board repairs: 10 customers still waiting}$
The last hour went there.
Value an extra hour.
$60$
Overtime at 25 an hour is worth paying; a second bench at 70 is not.
Rank croissants: 18 a tray in half an hour.
$18 \div 0.5 = 36$
Thirty-six an oven hour.
Rank loaves: 40 a batch in two hours.
$40 \div 2 = 20$
Twenty an oven hour.
Fill croissants up to 24 trays.
$24 \times 0.5 = 12$
Twelve of 20 oven hours.
Give the other 8 hours to loaves.
$8 \div 2 = 4$
Four of the 8 batches wanted.
Total the week.
$24 \times 18 + 4 \times 40 = 432 + 160 = 592$
Croissants first.
Total the plan with loaves first.
$8 \times 40 + 8 \times 18 = 320 + 144 = 464$
Loaves take 16 hours; the 4 left make 8 trays of croissants.
Find the difference.
$592 - 464 = 128$
Loaves first would cost Alexa 128 in the week.
Find what each earns for an oven hour.
$18 \div 0.5 = 36; \quad 40 \div 2 = 20$
Croissants first.
Fill 24 trays of croissants.
$24 \times 0.5 = 12$
Twelve hours.
Give 8 hours to loaves and total.
At Fixit Mobile, each of the screen replacements contributes $32.00$ dollars and takes $0.5$ bench hours. Complete the sentence about one of those hours.
One hour fits n of the jobs, which together contribute h dollars.
Complete the worked solution: a workshop has $26$ bench hours this week. A small job contributes $33$ dollars and takes half an hour; customers want $36$. A large job contributes $24$ dollars and takes two hours; customers want more than the bench can do. Plan the week.
Find what an hour earns on the small job.
$33 \div 0.5 =$ p
Two small jobs fit in an hour.
Find what an hour earns on the large job.
$24 \div 2 =$ q
One large job takes two hours.
Give the small jobs their hours first.
$36 \times 0.5 =$ h
They earn more an hour, up to their demand.
Turn the hours left into large jobs.
$(26 - \text{hours used}) \div 2 =$ u
Each large job needs two of the hours left.
Add the week's contribution.
$36 \times 33 + (\text{large jobs}) \times 24 =$ t
Both jobs' contributions together.
Corner Bean has $20$ oven hours this week. Each of the trays of croissants contributes $18.00$ dollars and takes $0.5$ hours, and customers want up to $24$. Each of the batches of loaves contributes $40.00$ dollars and takes $2$ hours, and customers want up to $8$. Which job should the hours go to first?
Fixit Mobile has $39$ bench hours this week. Each of the battery swaps contributes $20.00$ dollars and takes $0.25$ hours, and customers want up to $60$. Each of the data recoveries contributes $130.00$ dollars and takes $2$ hours, and customers want up to $16$. Fill in the plan that earns the most: how many of each to do, and the week's total contribution in dollars.
| Figure | |
|---|---|
| battery swaps to do | |
| data recoveries to do | |
| Total contribution |
Dan has $50$ bench hours this week. A screen replacement contributes $32$ dollars and takes half an hour; a board repair contributes $60$ and takes an hour. Customers want $80$ screens and $20$ board repairs. Mark every line of his draft plan that is sound.
This task has no paper form; do it on a device.
A tailor has $27$ sewing-machine hours this week. A hem contributes $35$ dollars and takes half an hour; customers want $44$. A suit alteration contributes $30$ dollars and takes an hour; more customers want alterations than the machine can do. A neighbor offers to lend her machine for one extra hour. On contribution alone, how many dollars is that hour worth this week?
Answer:
A sign-maker's laser cutter is booked solid. A door plaque contributes $10$ dollars and takes $20$ minutes; a shop sign $30$ dollars and $45$ minutes; a wedding set $45$ dollars and an hour; a name badge batch $12$ dollars and $20$ minutes. Fill in each job's contribution for an hour of the laser, in dollars.
| Amount | |
|---|---|
| Door plaque, dollars an hour | |
| Shop sign | |
| Wedding set | |
| Name badge batch |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Corner Bean has $20$ oven hours this week. Each of the trays of croissants contributes $18.00$ dollars and takes $0.5$ hours, and customers want up to $24$. Each of the batches of loaves contributes $40.00$ dollars and takes $2$ hours, and customers want up to $8$. The owner's usual plan fills the hours with whichever job earns more a job, up to its demand, and gives the rest to the other. How many more dollars does the plan that ranks by contribution for each hour earn this week?
Answer:
You can plan a constrained week that ranks jobs by the hour and respects demand, and value one more hour of the constraint. Tell someone why the lower-ranked job still gets the hours that are left. Next: a price that works at one volume and not at another.
16. Your turn: 20 oven hours; croissant trays 18 for half an hour, loaves 40 for two hours, step 3
$4 \times 40 + 24 \times 18 = 160 + 432 = 592$
The week's contribution.