Batch size cost minimum: prove the whole-number choice

Declare the batch and its cost before choosing a size

A batch size cost minimum needs a cost rule, a unit of comparison and a set of allowed sizes. This fictional paper model supplies all three. Choose any positive whole number q as the batch size. The batch produces exactly q usable units, and its total cost is defined as 48 + q² fictional cost units. Every positive whole-number size is allowed. The task is to minimize cost per usable unit. These rules describe an invented arithmetic exercise, not observed production, actual product prices, a supplier's charges or an inventory recommendation.

Divide total batch cost by its q usable units. The resulting per-unit cost is C(q) = (48 + q²)/q = 48/q + q. The first term allocates a fixed forty-eight cost units across the batch. It falls as q grows. The second term comes from the stipulated q² batch component divided by q; it rises with q. Both components belong to the same defined total. There is no separate forecast of storage cost, demand, equipment capacity or yield hidden inside the formula. A real process would need evidence for its own cost rule.

Inspect exact costs without treating samples as a proof

At batch size six, C(6) = 6 + 48/6 = 14. At size eight, C(8) = 8 + 48/8 = 14 as well. Size seven gives C(7) = 7 + 48/7 = 97/7. Since fourteen equals 98/7, seven's model cost is lower than either neighbor by 1/7 cost unit per usable unit. That is an exact comparison on one basis. It does not compare total outlay for batches producing different numbers of units, and it does not describe a saving on any retail order.

The equal neighboring values have different component splits. At six, the rising term is six and the allocated fixed term is eight. At eight, those contributions are eight and six. At seven, the contributions are seven and forty-eight sevenths. The terms balance differently as the batch changes, so checking only the falling 48/q term would miss part of the cost. A larger batch always spreads this declared fixed term further, but that observation alone cannot determine whether the sum is lower. Both terms must be retained at every candidate size.

A short table shows four candidate costs: q = 4 gives cost 16, q = 6 gives 14, q = 7 gives 97/7, and q = 8 gives 14. These entries do not yet prove that seven is cheapest among every positive integer. The allowed set has no upper bound, and checking a few nearby values cannot exclude an unexamined lower cost. We need an argument that covers every smaller size and every larger size. The next step compares successive sizes using the exact formula, so it can cover the unbounded domain without an endless table.

Derive the difference between neighboring sizes

Write the cost at the next size as C(q+1) = q + 1 + 48/(q+1). Subtract C(q) = q + 48/q. The q terms cancel, leaving 1 + 48/(q+1) − 48/q. Combining the two fractions gives 1 − 48/[q(q+1)]. The denominator q(q+1) is positive because q is a positive integer. This expression measures how the cost changes when the size increases from q to q+1. A negative difference means the next size costs less per usable unit; a positive difference means it costs more.

For q from one through six, the product q(q+1) is at most 6 × 7 = 42. It is therefore strictly below 48. Dividing 48 by that positive product gives a value greater than one, so 1 − 48/[q(q+1)] is negative. Each step from one to two, two to three, and onward through six to seven strictly lowers cost. The argument covers all those steps through one product comparison. It does not assume that an observed trend in a table continues without a formula to support it.

For q at least seven, the product q(q+1) is at least 7 × 8 = 56, strictly above 48. The fraction 48/[q(q+1)] is then less than one, making the difference positive. Every step from seven to eight and every later one strictly raises cost. There is no positive-integer q at which this difference vanishes for the original fixed term forty-eight: products through six are below it, and products from seven are above it. That strict sign change will establish both the minimum and its uniqueness.

Turn the sign pattern into a global conclusion

Take any allowed size smaller than seven. Increasing it one step at a time until seven follows only strictly decreasing costs. Consequently its original cost is greater than C(7). Now take any allowed size larger than seven. Starting at seven and increasing to that size follows only strictly increasing costs, so that candidate also costs more than C(7). Every permitted size belongs to one of those two groups or is seven itself. Thus seven is the unique global minimizer over the stated positive-integer domain, with per-unit model cost 97/7.

The word global is justified by the complete sign argument, rather than the number of rows in the example table. The word unique is justified by strict comparisons: no earlier or later integer can tie seven. Keep those two parts visible when recording the answer. An attainable candidate and an argument excluding all better candidates are both needed for a minimum claim. Here the formula at seven supplies the attainable cost, and the successive comparisons exclude every other allowed size. No claim about continuous choices between whole numbers is needed to finish this question.

Report the comparison on the same cost basis

Against either size six or size eight, the proven reduction is 14 − 97/7 = 1/7 fictional cost unit per usable unit. Record the reference size with the reduction. Saying only that seven saves a cost unit would give the wrong amount and lose the denominator. Likewise, comparing the total cost of one six-unit batch with one seven-unit batch would compare different outputs. The optimization here deliberately normalizes each total by its own usable-unit count. It does not require the total batch outlay at the optimum to be the smallest total outlay.

The model assumes q usable units for every allowed batch. If an actual process loses output, caps batch capacity, or must deliver a fixed demand through several whole batches, its feasible choices and cost comparison need to be defined separately. None of those facts can be inferred from the formula. Keep an unexplained real input unresolved rather than assigning it the convenient value in this exercise. A proof can be exact about declared assumptions while supplying no evidence that a particular operation satisfies them. Seven is the answer to this model's question, with its own accounting boundary.

Change one input and check the possibility of a tie

For a separate version, replace the fixed forty-eight with forty-two and preserve the other rules. The new per-unit function is D(q) = q + 42/q. Its successive difference is 1 − 42/[q(q+1)]. For q at most five, the product is at most thirty, so the difference is negative. At q = 6, the product is forty-two and the difference is zero. For q at least seven, the product is at least fifty-six and the difference is positive. The new function decreases, takes one flat step, and then increases.

Both D(6) and D(7) equal thirteen: six plus seven and seven plus six. Every earlier integer has greater cost by the decreasing-step argument, and every later integer has greater cost by the increasing-step argument. This revised model therefore has two minimizing sizes, six and seven. It does not contradict the unique answer for forty-eight because the cost rule changed. Preserve both fixed-term values beside their results. The tie also explains why uniqueness should be proved for the supplied inputs rather than treated as an automatic feature of every opposing-term cost expression.

Keep the supporting references separate

The account for the Artistic Rhinolin II desk pad shows a complete dark rounded-corner pad with a calculator and pen. Those objects are unconfirmed supplied props. No Microban, antimicrobial, material, size, cleaning, eye-strain or desktop-protection result is authenticated. The pad is optional context for a paper calculation.

The account for the ComfortCorrect CO789783 desk pad shows a clear pad partly occluded by desk objects. Calendar, photograph, computer equipment, person and pen are unconfirmed supplied context. Computer labels do not identify the pad's maker. No material, fit, comfort, durability, glare or protection performance is adopted here.

The guide on minimum step cost compares largest-first choices with a complete predecessor calculation for an exact accumulated target. Its decision concerns move lists under a fixed allowed-move set. Here the variable is one positive-integer batch size and the proof uses successive cost differences, without a target-total or last-move recurrence.

The guide on the smallest positive fraction question constructs a smaller permitted rational value for every candidate. The present integer-cost model has an attained minimum because its complete difference pattern changes sign. No rational-halving argument, missing least member or general lower-bound lesson is imported into the batch calculation.

Separate fictional cost units from a transaction

If a reader independently considers ordering a desk pad through republeq.com, the actual purchase needs its own current information. The forty-eight cost units and per-unit fractions above are invented model inputs, not prices, fees, token amounts or evidence of a merchant offer. Before considering USDT, verify whether the particular checkout offers that cryptocurrency method for the intended order. A mention in this article does not establish availability. Inspect the requested asset, network, payable amount, receiving destination and order reference using the responsible checkout or service. Clarify missing or conflicting details before authorizing a transfer. Keep the selected product and quantity attached to those instructions so a response about one order does not silently become authority for another. Review current delivery information, total charges and any disclosed payment or conversion conditions. The mathematical minimum supplies no exchange rate, transaction timing or assurance of payment protection. Check applicable cancellation, return and refund terms before relying on them, including how any refund would be handled for the chosen method. Retain the acknowledgment and payment record for an action actually completed, while keeping pending steps labelled pending. If order or payment status remains uncertain, resolve that uncertainty before considering another payment. A lower value in a fictional worksheet does not authenticate a transfer destination or show that a previous transfer failed. The model can be checked entirely on paper without buying a pad, and an actual order cannot supply a missing step in the proof. Decide whether to proceed only when the relevant instructions and terms for that particular transaction are clear. Keep receipts and transaction communications separate from the batch-size worksheet and its invented cost units.

Save the function, domain and comparison together

For the original version, retain C(q) = q + 48/q, the positive-integer rule, the negative differences through q = 6 and positive differences from q = 7. Record the unique size seven, its cost 97/7, and the 1/7 per-unit reduction against six or eight. For the changed version, retain forty-two and the tied sizes six and seven at thirteen. These records let another reader check each conclusion against the exact inputs that support it, without mistaking a table sample, an allocated cost component or a real product claim for the whole proof.