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Relieve a constraint by a faster method, moving work off it, easing demand on it, or adding capacity; compare options by the gain each brings — counting only the extra units customers want — by the cost of each extra unit, and by the most each can cost, cheapest fixes first.
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 compare ways of relieving a constraint by the extra units each sells and the gain it brings, choose the better one, work the cost of each extra unit, stack the cheapest fixes until demand is met, and find what a relief can cost and still pay.
Once the constraint is known, the question becomes how to relieve it. There is almost always more than one way, and the most obvious — buy more capacity — is often not the best. You can already price a decision's break-even and find where the next constraint takes over; this lesson puts the two together.
| Term | What it means |
|---|---|
| Relieve | Let more units through the constraint. |
| Unmet demand | Work customers want that the business cannot currently do. |
| Cost of each extra unit | An option's cost divided by the extra units it lets through. |
| Gain | Extra units sold times their contribution, less the option's cost. |
| Most an option can cost | The extra units times their contribution: above it, the option loses money. |
Maya's throwing time is her constraint; 40 more mugs a month are wanted than she can make, and each contributes 20 dollars. Two options:
$$\text{gain} = \min(\text{capacity added}, \text{unmet demand}) \times \text{contribution} - \text{cost}$$
The assistant adds twice the capacity and loses money, because a third of it would sit idle: demand becomes the constraint at 40. Overtime is the better relief on these figures. Per extra mug, overtime costs 300 ÷ 30 = 10 and the assistant 1,000 ÷ 40 = 25, against a contribution of 20: the per-unit figure tells the same story.
Another way: table
Maya's two options.
| Option | Capacity added | Mugs sold | Cost | Gain |
|---|---|---|---|---|
| Overtime | 30 | 30 | 300 | 300 |
| Assistant | 60 | 40 | 1,000 | −200 |
Another way: steps
List the options by kind. A better method, work moved off the constraint, demand eased, then more hours or equipment. Owners usually think of the last kind first; listing all four usually finds a cheaper one.
Cap each option's extra units. The smaller of the capacity it adds, the unmet demand, and whatever the next limit allows.
Work each gain and cost per unit. Gain: extra units sold times contribution, less cost. Cost per unit: cost over extra units sold.
Stack the cheap fixes. Where options can be combined, take them in order of cost per unit until the unmet demand is met.
Check each option in two ways. Its cost per extra unit must be below the contribution for it to pay. And its gain, worked out, must agree: an option with a cost per unit below the contribution and a negative gain has an arithmetic error, usually in the capped units.
There are four kinds of relief, and they tend to get more expensive in this order:
Try the cheap ones first; they often relieve enough, and they carry little risk if demand falls later. A second cook is a monthly wage whether or not the lunches come; pre-cooked rice costs nothing when trade is slow.
Turn the question around: how much could an option cost and still pay? The extra units it would sell, times their contribution, is the ceiling. A print shop that could sell 200 more jobs at 12 dollars of contribution each can pay up to 2,400 a month for the extra capacity. A second-hand press leased at 1,500 pays comfortably; one at 2,800 does not, however fast it runs.
The ceiling is a useful figure to carry into a negotiation. An owner who knows the press is worth at most 2,400 a month to the business will not be talked into 2,800 by a salesperson's description of its speed.
Every relief is capped by the next limit, not only by demand. When the counter staff plate the salads, the cook serves more lunches — until the counter itself cannot take payment any faster. A relief plan should follow the chain: this fix lifts the kitchen to 150; at 150 the register is the next limit; so if demand grows beyond 150, the next relief is at the register, not in the kitchen. Owners who plan this way spend on each constraint only as far as the next one allows.
Sometimes the right answer is to leave the constraint alone and use it better: raise the price of the work that queues for it, or keep only the offers that earn most per constrained hour, as the focus lesson showed. A constraint that turns away 20 low-margin jobs a month may be worth more as a reason to raise prices than as a problem to spend on. Relief is one answer to a constraint; charging for it is another, and it costs nothing.
A relief that can be undone is worth more than one that cannot, for the same gain. Overtime can be stopped next month if demand falls; a second cook hired on a permanent contract, or a machine leased for three years, cannot. When two options show similar gains, the one that can be reversed cheaply carries less risk, because it does not turn a busy season into a fixed cost that must be paid through a quiet one.
This is why owners often relieve a constraint in stages. First the method changes, which cost nothing and stay. Then work moved off the constraint, which costs a few hours of someone else's time. Then overtime or temporary help, which can be stopped. Only when demand has stayed above capacity for several months, measured and not guessed, does the owner add permanent capacity: the hire, the machine, the second van. Each stage is tested before the next is bought.
Timing matters too. A constraint that binds only in the busiest six weeks of the year is best relieved with something that lasts six weeks: seasonal staff, a rented machine, a simplified menu at the peak. Buying permanent capacity for a seasonal constraint pays for idle equipment for the other forty-six weeks. The opposite mistake is to treat a constraint that binds all year as a temporary rush and keep paying overtime rates indefinitely, when a permanent hire would cost less per unit.
The cost-per-extra-unit figure makes these comparisons fair. Overtime at 15 dollars a repair against a hire at 12 dollars a repair — but only if the hire's extra repairs are all wanted — tells the owner which is cheaper at the demand the business has, and by how much. If demand falls, the hire's cost per extra unit rises, because its cost is fixed and its units shrink; the overtime's does not. Working the figure at a lower demand as well as the expected one shows how much each option depends on the demand holding.
A small dental practice's constraint was hygienist time: patients waited six weeks for a cleaning, and about 50 appointments a month went unbooked because people gave up. Each cleaning contributed about 70 dollars. The obvious fix was a second hygienist, part-time, at about 4,500 a month, adding capacity for 80 appointments.
The practice manager listed the cheaper kinds of relief first. The hygienist spent about ten minutes of each hour-long slot on paperwork and room setup that an assistant could do; moving that work off her freed about 25 appointments a month, for 900 of assistant hours. Reminder texts cut no-shows, which had been wasting about 12 slots a month, for a 60-dollar software fee. Together they added about 37 appointments at 960 a month: a gain of 37 × 70 − 960, about 1,630.
The second hygienist would have added 80 slots, but only 13 more patients were waiting once the cheap fixes were in place: 13 × 70 − 4,500, a loss of about 3,600 a month. The practice took the cheap fixes, watched the waiting list, and hired the second hygienist eighteen months later, when the list had grown back to more than 80.
People who study operations find again and again that the first gains at a constraint come from method and from moving work, not from new equipment — and that those gains are cheaper and carry less risk, because they add no fixed cost that must be paid in a slow month.
More capacity is always better. Only the capacity customers fill earns anything.
Relief means buying something. Methods and moving work are often cheaper.
Compare options by capacity added. Compare them by gain.
Relieve the constraint as far as possible. Past the next limit, relief earns nothing.
A faster machine is worth whatever it costs. It is worth at most the extra units times their contribution.
The cheapest relief is always the best. Compare what each option adds as well as what it costs; the cheapest can add too little to matter, and a more expensive one can pay for itself several times over.
A relief keeps paying forever. Once the constraint is relieved, the next limit takes over and the gain stops growing. Count the extra units only up to that next limit, and check again once it binds.
Permanent capacity for a seasonal rush. A constraint that binds for a few busy weeks is best relieved with something that can be stopped when the rush ends: overtime, a temporary hire, a rented machine. Buying for the peak leaves a fixed cost to carry through every quiet month of the year.
Relief is a one-time job. Demand and methods change, so the constraint is measured again every few months.
Read the constraint and the unmet demand.
$120 \text{ plates}; \ 150 \text{ wanted}$
Thirty a day turned away.
Take the free fix.
$\text{pre-cooked rice: } +15$
Less cook time a plate.
Take the next cheapest.
$\text{counter staff plate salads: } +15 \text{ for } 60 \text{ a week}$
Work moved off the cook.
Check the demand is met.
$120 + 15 + 15 = 150$
All the lunches wanted.
Set the dear fix aside.
$\text{second cook: } 2000 \text{ a month}$
Not needed.
Cap overtime's mugs.
$\min(30, 40) = 30$
All wanted.
Cap the assistant's mugs.
$\min(60, 40) = 40$
Twenty would sit idle.
Work overtime's cost a mug.
$300 \div 30 = 10$
Below the 20 contribution.
Work the assistant's cost a mug.
$1000 \div 40 = 25$
Above the 20 contribution.
Work each gain.
$30 \times 20 - 300 = 300; \quad 40 \times 20 - 1000 = -200$
The per-unit figures agree.
Choose the better option.
$\text{overtime}$
More profit, less capacity.
Read the unmet demand.
$200 \text{ jobs a month}$
Turned away now.
Read the contribution a job.
$12$
Price less paper and ink.
Find the most the press can cost.
$200 \times 12 = 2400$
A month.
Read the quotes.
$1500 \text{ second-hand}; \ 2800 \text{ new}$
Monthly lease.
Work the second-hand press's gain.
$2400 - 1500 = 900$
It pays.
Work the new press's gain.
$2400 - 2800 = -400$
It does not, however fast.
Find the next limit.
$\text{finishing, at 150 more jobs}$
So even the second-hand press sells only 150 until finishing grows.
Work the cost of each extra repair.
$300 \div 20 = 15$
Overtime adds 20 repairs for 300.
Compare with the contribution.
$25 - 15 = 10$
Each extra repair gains 10.
Find the month's gain.
Maya's throwing time is her constraint, and $50$ more mugs a month are wanted than she can make. Each mug contributes $25$ dollars. Weekend overtime would add capacity for $40$ mugs a month at a cost of $300$ dollars; a part-time assistant would add capacity for $50$ at a cost of $1400$. Fill in the extra mugs each option would sell and the gain, in dollars a month.
| Extra mugs sold | Gain, dollars a month | |
|---|---|---|
| Weekend overtime | ||
| Part-time assistant |
Complete the worked solution: Northside Repairs can relieve its bench constraint with Saturday overtime, adding $50$ repairs a month for $200$ dollars, or a second-hand bench, adding $70$ repairs for $700$ a month. All the extra repairs are wanted, and each contributes $38$. Compare them per repair.
Divide overtime's cost by its extra repairs.
$200 \div 50 =$ p
Cost of each extra repair.
Divide the bench's cost by its extra repairs.
$700 \div 70 =$ q
The same measure.
Take the cheaper from the contribution.
$38 - \min(\text{the two}) =$ g
What each extra repair earns at the cheaper option.
Check both pay.
$\text{each below } 38\text{?}$
An option dearer per repair than the contribution loses money.
Compare total gains too.
$\text{units times margin, each option}$
A cheaper unit on fewer units can gain less in all.
Neighborhood Kitchen's cook can serve $120$ lunches a day, and $150$ are wanted. Three fixes are on offer: pre-cooking the rice each morning adds $15$ lunches for nothing; having the counter staff plate the salads adds $15$ for $60$ dollars a week in extra hours; a second cook adds $60$ for $500$ dollars a week. Reach $150$ lunches a day while keeping the extra wages to $100$ dollars a week or less.
This task has no paper form; do it on a device.
Maya's throwing time is her constraint, and $30$ more mugs a month are wanted than she can make. Each mug contributes $40$ dollars. Weekend overtime would add capacity for $20$ mugs a month at a cost of $800$ dollars; a part-time assistant would add capacity for $50$ at a cost of $1500$. Overtime would add $0$ dollars a month; the assistant $-300$. Which option is better on these figures?
A print shop's single press is its constraint. A second-hand press would let it print $240$ more jobs a month, all of which customers want, and each job contributes $11$ dollars. For which monthly costs $x$, in dollars, would the press at least pay for itself?
This task has no paper form; do it on a device.
Bright Home Cleaning's cleaners are its constraint. Saturday overtime would let it do $50$ more cleans a month, all wanted by customers, at an extra cost of $700$ dollars. What does each extra clean cost, in dollars?
Answer:
A print shop's press is its constraint; customers want $50$ more jobs a month than it can print, and each contributes $31$ dollars. A second-hand press would add capacity for $70$ jobs at $800$ a month; a paid evening shift on the existing press would add $20$ at $500$ a month. Fill in the comparison, in dollars.
| Amount | |
|---|---|
| Jobs the press would sell | |
| Press's gain a month | |
| Jobs the evening shift would sell | |
| Evening shift's gain a month |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Maya's throwing time is her constraint, and $40$ more mugs a month are wanted than she can make. Each mug contributes $35$ dollars. Weekend overtime would add capacity for $30$ mugs a month at a cost of $800$ dollars; a part-time assistant would add capacity for $70$ at a cost of $1100$. How many dollars a month would the part-time assistant add? Write a loss with a minus sign.
Answer:
You can choose the cheapest relief that is enough. Tell someone why adding more capacity can lose money. Next: the cash that growth needs before it pays.
17. Your turn: Northside Repairs' overtime, step 3
$20 \times 10 = 200$
The overtime pays.