Delivery order: compare six routes and retain both ties

A delivery order check adds the directed journey costs for every allowed stop sequence, including the return to the depot. Use this independent three-stop card to calculate all six routes and preserve both routes tied for the minimum.

Updated: 18 October 2026.

What is the delivery order check for?

A delivery order check compares the sequence of required stops using a fixed table of directed travel costs. The small exercise here has one depot, three stops and one vehicle returning to the depot. Your output is a complete six-row comparison, including every route tied for the lowest total. You do not need routing software to inspect this particular finite problem.

The English practice card uses its own data and has two equally short routes. W is a fictional depot and P, Q and R are fictional stops, not locations on a real map. Every number is a supplied teaching input in abstract travel minutes. Nothing here is a live traffic estimate, a measured route or a promise about when a customer will receive a delivery.

Keep the decision distinct from assigning staff or equipment. The task-assignment exercise chooses eligible people, while a resource-conflict check inspects overlapping reservations. Here the vehicle and stops are fixed; only their visiting order varies.

Write the closed-route contract before calculating

Start at W, visit P, Q and R exactly once each, and return to W. There is one vehicle and no extra stops, reloads or repeated customer visits. Use the table entry directly for each journey. Rows identify the origin and columns identify the destination. Do not reverse those roles when reading a value.

W to P costs three, but P to W costs five. Q to R costs seven, whereas R to Q costs two. The matrix is directed, meaning that reversing a journey may change its cost. The zero diagonal represents staying at a point; none of our six candidate routes uses a self-transition.

All directed journeys are supplied and remain constant across calculations. We are not finding shorter journeys through intermediate points or assuming these numbers are shortest-path distances in a road network. The exercise has no capacity restriction, service duration, driver break or customer opening window. These omissions define a small teaching model, not a sufficient model for dispatching real deliveries.

The critical-path example answers a different timing question about predecessor tasks. Do not treat the rows of this travel matrix as a project dependency table. A directed journey cost tells you what one movement contributes; it does not describe a prerequisite relationship between jobs.

List all six distinct stop orders

There are three choices for the first stop, two remaining choices for the second and one for the last. Multiplying gives six possible orders. W remains fixed at both ends, so it does not add another choice. Each candidate must include all three customer identifiers exactly once, plus the starting and returning depot.

1. W → P → Q → R → W
2. W → P → R → Q → W
3. W → Q → P → R → W
4. W → Q → R → P → W
5. W → R → P → Q → W
6. W → R → Q → P → W

Do not collapse a route and its reverse into one entry. With directed costs, the reverse can have a different total. Do not remove the final W either. An open journey ending at the last customer is not the closed journey specified here. Its cheaper-looking sum would answer another question.

This complete list is the reason a minimum claim will be possible within the written model. Comparing only two promising orders cannot rule out a better omitted candidate. For a larger stop list, the number of possible orders grows quickly. This page does not offer an efficient general-purpose optimisation algorithm or claim that manual enumeration scales to a normal delivery operation.

Read four directed legs for every candidate

For W-P-Q-R-W, read three from W to P, eight from P to Q, seven from Q to R and three from R to W. The total is 21. For W-P-R-Q-W, the entries are three, two, two and four, totalling 11. Keep the four contributions alongside each total so someone else can audit the direction of every lookup.

Now compare 21, 11, 11, 22, 24 and 15. The minimum is 11, achieved by two routes: W-P-R-Q-W and W-Q-P-R-W. List both. Choosing whichever appears first in a spreadsheet would hide the tie and introduce an unstated preference. A separate owner-approved rule would be needed to select one for an actual decision.

All six allowable stop orders have been evaluated, which establishes the minimum for this exact fixed matrix and closed-route contract. It does not establish a fastest real-world trip. Missing constraints or changed journey values would define a new problem and require a fresh comparison.

Keep the input and result version together. The version-approval guide explains why a decision must refer to particular content. If a journey entry changes later, the old minimum belongs to the old matrix; a new number should not silently inherit the old review.

Separate a nearest-next rule from a complete comparison

A nearest-next rule selects the cheapest unvisited destination at each step. In this English card, it starts W-P because three is the lowest outgoing value from W. From P, it chooses R at two rather than Q at eight. Q is then the only remaining stop, followed by the return to W. The total is 3 + 2 + 2 + 4 = 11.

That happens to match a minimum here, but it identifies only one of the two tied routes. Its success on this card is not a general guarantee. For a concrete counterexample, consider a separate fictional card with depot D and stops A, B, C: D-A=2, D-B=5, D-C=6; A-D=7, A-B=2, A-C=4; B-D=3, B-A=8, B-C=10; C-D=2, C-A=3, C-B=1.

The nearest-next rule on that complete directed card chooses D-A-B-C-D and totals 2 + 2 + 10 + 2 = 16. The permitted alternative D-A-C-B-D totals 2 + 4 + 1 + 3 = 10. One better permitted alternative is enough to show that 16 is not a minimum. The comparison is an arithmetic counterexample, not evidence of actual driver savings.

If a required journey value is unknown, keep it unknown. Zero would mean a supplied zero-cost journey, not missing information. Routes using that value remain unresolved. Unless all omitted or unresolved candidates can be excluded under a stated rule, do not claim a global minimum from the routes you happened to finish.

Do not round individual contributions to force a preferred tie or ignore the depot return. If real costs have decimals, preserve their supplied precision for calculation and explain any display rounding. In this exercise, all entries are integers and the two English minima are exactly equal, not an artefact of rounding.

Save a small route-comparison record

For the suggested fifteen-minute core on Windows, search Start for Notepad, open the application and create a new document. Copy the matrix, six orders and six totals into it. The first step produces the fixed input contract. The second produces a complete candidate list. The third produces four checked contributions per row. The final step records the minimum and every tie. These are inspectable outputs rather than vague tasks such as “improve delivery”.

V1 | fictional fixed directed minutes | one vehicle
W starts/ends; visit P,Q,R once each
Totals: PQR21; PRQ11; QPR11; QRP22; RPQ24; RQP15
Minimum11: W-P-R-Q-W AND W-Q-P-R-W
Nearest-next finds one minimum here, not a general guarantee
No traffic, service times, capacity or time windows
Missing required value: unresolved, never silently zero

Press Ctrl+S, choose Documents and save as delivery-order-V1.txt. Close the editor. Open File Explorer, go to Documents and double-click the saved file. Check that both minimum routes remain present and every candidate ends at W. A useful separate inventory record is the warehouse-location lookup; locating a parcel inside a warehouse is not the same as ordering customer visits outside it.

Anar’s published Azerbaijani reflection on perfection and effectiveness describes projects delayed by chasing perfection. The modest application here is to finish one auditable six-row record before expanding the model. It is not evidence that he tested these routes, used a routing product or measured a client improvement. The short practice duration is a suggestion, not a measured participant result.

Google’s TSP documentation presents a depot and matrix-based tour example. Its VRP documentation distinguishes broader vehicle routing and additional capacity or time-window constraints. Those are the narrow concepts attributed here. No Google solver was installed or executed; our finite candidate totals were independently checked. Actual dispatch would require the missing operating constraints and reliable journey inputs before either tied route could become an approved plan.