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Supplier fixed cost: the lowest line prices give the highest bill

Award each line to its cheapest quote and the total is expected to be the lowest one available. Type two items, two suppliers and one cost that attaches to each supplier engaged rather than to a line, and it stops being true. At 50 a supplier the line by line award totals 310 while the whole basket at one supplier totals 270. The crossing sits at 10, below which the split award really is cheaper, and no supplier is ranked and no line is awarded here.

Every quote is treated as the same scope from the same information on one already comparable numeric cost basis. Nothing here normalizes currency, delivery terms, tax, inspection scope or freight basis.

Live result

Line by line totals 310 while one minimizing comparison totals 270, so the stated rule breaks.
  • fixed cost for each supplier50 cost units
  • how far the rule breaks40 cost units
A cost that attaches once to each supplier is what stops the line totals from adding.
Line by line totals 310 while one minimizing comparison totals 270, so the stated rule breaks.A cost that attaches once to each supplier is what stops the line totals from adding.

The line allocation and one minimizing comparison allocation

Two cost build ups as supplier fixed cost movesAt a fixed cost of 50, the line by line allocation totals 310. The smallest computed comparison totals 270. The first fewer supplier crossing is 10.total, cost unitsfixed cost for each supplier engagedline by linecomparison allocation

Try a record

50cost units

This cost attaches once to each supplier used by an assignment, not once to every line.

Edit items, suppliers and quotes
Items
Item 1
Item 2

Two items form the smallest admitted comparison.

Suppliers
Supplier 1
cost units
cost units
Supplier 2
cost units
cost units

Two suppliers form the smallest admitted comparison.

cost units
Difference above the smallest computed total
40 cost units
Assignments examined
4
First fixed cost crossing
101=10\frac{10}{1} = 10

The rule as commonly stated

Award every line to its lowest quote and the total bill will be the lowest available total.

The smallest breaker

Two items and two suppliers are enough when a positive cost attaches once to every supplier engaged. At a fixed cost of 50, the line allocation is 90 plus 120 plus 2 times 50, which equals 310. Putting both typed lines on Supplier A is a comparison allocation totaling 100 plus 120 plus 50, which equals 270.

The restoring condition

The rule is restored when every compared cost is additive by line. In this model that means supplier fixed cost is zero, or that any such cost was validly converted into line amounts before entry and is therefore entered here as zero.

What does not break the rule

At zero supplier fixed cost the line allocation totals 210 and is a minimizing assignment, so the rule holds.

At the exact crossing of 10, the line allocation and the Supplier A basket both total 230, so the rule ties and does not break.

This adds the already comparable quote units typed by the visitor under every admitted item to supplier assignment. It awards nothing, ranks no supplier, evaluates no bid, and gives no advice to invite fewer bidders or combine awards.

Line by line

Quote sum
210 cost units
Suppliers engaged
2
Fixed cost subtotal
100 cost units
Total
310 cost units

One minimizing comparison

Quote sum
220 cost units
Suppliers engaged
1
Fixed cost subtotal
50 cost units
Total
270 cost units

Line by line

  1. Item 1 · Supplier B90 cost units
  2. Item 2 · Supplier A120 cost units

One minimizing comparison

  1. Item 1 · Supplier A100 cost units
  2. Item 2 · Supplier A120 cost units

Here is the maths

The formula, in words

For each assignment, add its selected line quotes, count the distinct suppliers used, and add that count times the fixed cost for an engaged supplier. Compare every admitted assignment with the unique lowest quote allocation formed line by line.

A worked example

The line allocation is Item 1 to Supplier B at 90 and Item 2 to Supplier A at 120. Its quote sum is 210 and it engages 2 suppliers.

At a fixed supplier cost of 50, the line total is 210 plus 2 times 50, which equals 310.

The Supplier A basket has quote sum 220 and engages 1 supplier, so its total is 220 plus 50, which equals 270.

All 4 assignments are examined. The difference between 310 and the smallest computed 270 is 40 cost units.

The exact crossing solves 210+2f=220+f210 + 2f = 220 + f, where ff is the fixed cost for an engaged supplier, so f=10f = 10. The totals tie there and the line allocation loses only above it.

How the witness stays finite

The contract admits 2 to 6 suppliers and 2 to 8 items. The largest search contains 6 to the power of 8, which is 1,679,616 assignments. Every one is examined, with no sample and no heuristic.

Every admitted item to supplier assignment is enumerated. Each total is its assigned quote sum plus the number of suppliers used times the supplier fixed cost. The page prints the exact assignment count and one minimizing comparison in typed item order. It awards nothing.

What this does not model

It does not normalize currency, delivery terms, tax, freight, inspection scope or landed cost. It does not evaluate a bid, rank a supplier, choose an award, or advise how many suppliers to engage.

Permanent refusal

Currency conversion, delivery term adjustment, landed cost normalization, supplier evaluation and award recommendation fields are outside the exact contract and are refused rather than ignored.

The calculation domain

Items and suppliers have unique plain labels. Quotes and fixed cost are whole nonnegative cost units no greater than 1,000,000,000. Every item has one unique lowest quote, because a tied lowest quote would require the page to choose a route.

Boundary

This adds the already comparable quote units typed by the visitor under every admitted item to supplier assignment. It awards nothing, ranks no supplier, evaluates no bid, and gives no advice to invite fewer bidders or combine awards.

Source anchors

  • Finite enumeration of every admitted assignment.
  • Additive arithmetic with one fixed cost for each distinct supplier used.

These two public mechanisms are stated in full because they are the independently authored witness. No private taxonomy, table, clause or quotation is reproduced.

Calculation version

1.0.0

The whole record prints. Your browser can save it as a PDF instead of printing it.

The finite enumeration and additive arithmetic are independently stated, and no private source taxonomy, table, clause or wording is reproduced. This page permanently refuses currency conversion, delivery term adjustment, landed cost normalization, supplier evaluation and award recommendation fields. The minimizing assignment is a computable comparison, not a proposed award.

Related

Lowest line fixed cost witness

Supplier fixed cost: the lowest line prices give the highest bill

Accepted inputs

Item 1
Item 1
Item 2
Item 2
Supplier 1
Supplier A
Supplier A, Item 1
100 cost units
Supplier A, Item 2
120 cost units
Supplier 2
Supplier B
Supplier B, Item 1
90 cost units
Supplier B, Item 2
150 cost units
Fixed cost for each supplier engaged
50 cost units

All results

Line by line, Total
310 cost units
One minimizing comparison, Total
270 cost units
Difference above the smallest computed total
40 cost units
Assignments examined
4
First fixed cost crossing
101=10\frac{10}{1} = 10 cost units

Record fields

Calculation version
1.0.0
Boundary
This adds the already comparable quote units typed by the visitor under every admitted item to supplier assignment. It awards nothing, ranks no supplier, evaluates no bid, and gives no advice to invite fewer bidders or combine awards.
Calculation trace
For each assignment, add its selected line quotes, count the distinct suppliers used, and add that count times the fixed cost for an engaged supplier. Compare every admitted assignment with the unique lowest quote allocation formed line by line. The contract admits 2 to 6 suppliers and 2 to 8 items. The largest search contains 6 to the power of 8, which is 1,679,616 assignments. Every one is examined, with no sample and no heuristic.
Source anchors
Finite enumeration of every admitted assignment. Additive arithmetic with one fixed cost for each distinct supplier used.

The finite enumeration and additive arithmetic are independently stated, and no private source taxonomy, table, clause or wording is reproduced. This page permanently refuses currency conversion, delivery term adjustment, landed cost normalization, supplier evaluation and award recommendation fields. The minimizing assignment is a computable comparison, not a proposed award.

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