Quantum Logic + Max-Plus in Scheduling Problems ②🥇

That is a clear requirement definition.

Rather than "what to solve," it's "how many seconds to solve it in"
—that is the decision criterion for people in the field.

Whether the combination of bitmask AND parallel calculation and Max-Plus matrix propagation can fit within those 5 seconds is the only real-world test for this architecture.

As a current gut feeling, what scale of problem are you assuming?

What order of magnitude for the number of items, number of locations, and time axis is required to be within 5 seconds?

This is well-organized.

Following the steps:

The Bayesian net/Petri net in ① The observation that real-world logistics are not fully connected is important. Because it is a sparse graph, bit strings become easier to handle.

The Cantor-like diagonal ID assignment in ② I think this is the core. The part about "thinking of rules"—the density of the bit string and computational efficiency depend on the design of these rules.

The 0/1 bit string in ③ Encodes the presence or absence of item swapping directly. Simple.

The Max-Plus constraints in ④ The idea that increasing constraints makes SAT easier seems paradoxical but is correct. It is because the solution space narrows.

There is one thing I would like to confirm,②.

When you say "like Cantor's diagonal argument," do you mean the image of "uniquely mapping all pairs (i,j) to a scalar"? Or are you assuming a different usage?

I see, so:

A bijection from (i, j) → N

Using Cantor's zigzag traversal to collapse 2D node pairs into a 1D ID.

(0,0)→0, (0,1)→1, (1,0)→2, (1,1)→3, (0,2)→4...

With this, you canmap edges (dependencies) of the graph to unique bit positions.

Then the big picture started to emerge:

• Edge set of Bayesian network → Cantor mapping → Bit position

• That bit string is the "current schedule state"

• Flow time using Max-Plus to check for constraint violations

• Adopt the solution if one is found within 5 seconds

Since it assumes a sparse graph,most of the bit string is 0—this is the key to keeping the calculation at a realistic size.

SCM Scheduling Concept Verification Summary

Architecture Framework

① Sparse graph representation (Bayesian network/Petri net) → Real-world logistics are not fully connected. The assumption of sparsity is correct. Whether it is a job shop or a flow shop, actual dependencies are limited.

② Edge ID assignment via Cantor pairing function → Map node pairs (i,j) to a unique scalar. Collapse the 2D dependency structure into a 1D bit position. The mapping is simple and has low implementation cost.

③ Encode schedule state with 0/1 bit strings → Since a sparse graph is assumed, most of the bit string is 0. This is the key to keeping combinatorial explosion to a realistic size. The presence or absence of item swapping corresponds directly to bits, making it simple.

④ Assign time constraints with Max-Plus operations → Provide the final completion time of resources (machines, etc.) and the completion time for each item as constraints. Increasing constraints narrows the solution space and makes SAT easier—this is correct.

Consistency with Requirements

• Within 5 seconds: With a combination of sparse graph + bit string AND operations + Max-Plus matrix propagation, it is theoretically achievable at a realistic scale. However, proof requires a prototype.

• Scheduling problems in general: Whether it can be established as a general-purpose engine depends on the generalization capability of the sparse graph design in ①.

Remaining Questions

• Mapping rules for ②: Is Cantor's zigzag sufficient, or is another mapping that leverages sparsity more advantageous?

• Inter-cluster dependencies: How to handle inter-cluster edges when splitting 50 items.

• Prototype target: What scale and type of problem should be set to "get it running" first?

The concept is sound. The next phase is to get hands-on.

That sounds good. Let's set the specifics.

Settings

Items

• A: Precision parts

• B: Food products

• C: Industrial materials

• D: Medical equipment

Machines (Resources)

• M1: Processing machine

• M2: Packing machine

Available operating time

• 8 hours/day = 480 minutes

Processing time per item (minutes)

Item 　M1 　M2
A 　　90 　60
B 　　45 　30
C　 　120 　90
D 　　60 　45

Completion time constraints (shipping deadline, minutes from start)

Item Deadline
A 　300
B 　150
C 　480
D 　240

Shall we apply this directly to the MageManager concept?

First,② assign IDs to edges using Cantor mapping—let's start there. Listing the item × machine pairs gives us 8 pairs—starting from assigning IDs to these using zigzag scanning.

Step ①: Defining the graph structure

Nodes: Combinations of items {A, B, C, D} × machines {M1, M2}

ノードID
(A,M1)=0  (A,M2)=1
(B,M1)=2  (B,M2)=3
(C,M1)=4  (C,M2)=5
(D,M1)=6  (D,M2)=7

Step ②: Assigning IDs to edges using Cantor mapping

Edges = node pairs (i, j), zigzag scanning:

f(i,j) = (i+j)(i+j+1)/2 + j

In real-world logistics, it is not fully connected. The sparse graph for this case:

M1→M2の依存（各品目は必ずM1→M2の順）
A: (0,1) → f(0,1) = 1
B: (2,3) → f(2,3) = 13
C: (4,5) → f(4,5) = 33
D: (6,7) → f(6,7) = 61

品目間の順序依存（締切の緊急度順：B→D→A→C）
B→D: (2,6) → f(2,6) = 31
D→A: (6,0) → f(6,0) = 21
A→C: (0,4) → f(0,4) = 14

Step ③: Bit string encoding

Encoding the state using edge IDs as bit positions:

ビット位置: ...61 ...33 ...31 ...21 ...14 ...13 ...1
初期状態:      1    1    1    1    1    1   1

All edges valid (no swapping) = proceed in this order.

Step ④: Max-Plus operation

Processing time matrix A (-∞ means no dependency):

     A,M1  A,M2  B,M1  B,M2  C,M1  C,M2  D,M1  D,M2
A,M1  90   90    -∞    -∞    -∞    -∞    -∞    -∞
A,M2  -∞   60    -∞    -∞    -∞    -∞    -∞    -∞
B,M1  -∞   -∞    45    45    -∞    -∞    -∞    -∞
B,M2  -∞   -∞    -∞    30    -∞    -∞    -∞    -∞
C,M1  -∞   -∞    -∞    -∞   120   120    -∞    -∞
C,M2  -∞   -∞    -∞    -∞    -∞    90    -∞    -∞
D,M1  -∞   -∞    -∞    -∞    -∞    -∞    60    60
D,M2  -∞   -∞    -∞    -∞    -∞    -∞    -∞    45

Time propagation x(k+1) = A⊗x(k)

B,M2完了: 45+30 = 75分　　締切150 ✅
D,M2完了: 60+45 = 105分　 締切240 ✅
A,M2完了: 90+60 = 150分　 締切300 ✅
C,M2完了: 120+90 = 210分　締切480 ✅

Machine uptime check

M1合計: 90+45+120+60 = 315分 ≤ 480分 ✅
M2合計: 60+30+90+45 = 225分 ≤ 480分 ✅

Conclusion

All constraints cleared with order B→D→A→C.
The bit string remains all 1s (no swapping required).

If a constraint violation occurs, drop the corresponding bit to 0, swap the order, and run Max-Plus again—this becomes a search loop within 5 seconds.

That's great. This is the moment to translate on-site tacit knowledge into mathematical formulas.

Organizing the situation

The worker is now choosing between B and C. A cannot be touched because it is waiting for materials. D is being handled by someone else (assumption).

If B is chosen first

B,M1: 45分 → B,M2: 30分 = 75分で完了
その後C,M1: 120分 → C,M2: 90分 = 210分で完了

M1の流れ: B(45) → C(120) = 165分
M2の流れ: B(30) → C(90)  = 120分
C完了: 210分　締切480 ✅ 余裕270分

If C is chosen first

C,M1: 120分 → C,M2: 90分 = 210分で完了
その後B,M1: 45分 → B,M2: 30分 = 285分で完了

M1の流れ: C(120) → B(45) = 165分
M2の流れ: C(90) → B(30)  = 120分
B完了: 285分　締切150 ❌ 超過135分！

Answer

If C is chosen first, B will exceed its deadline by 135 minutes.

The shipping process is left waiting. Food (B) is highly likely to be time-sensitive, leading to potential disposal or customer complaints in the worst case. This is an invisible cost.

The worker just chose C because 'it was right in front of them.' But that decision caused a bottleneck in the shipping process.

When this becomes visible in the system

An alert appears saying 'Do B first.' The reason is also provided. The well-intentioned judgment errors of workers are eliminated.

So, the 'arbitrary' decisions on the shop floor are actually the biggest source of costs.

(Laughs) That's right, that is the reality of the shop floor.

Worker rationality

Workers are not stupid. They are minimizing the costs of their own process. That is a correct judgment—within their own scope of responsibility.

自工程視点：Cを先にやる
→ 段取り効率が良い
→ 派遣さんの残業なし
→ 自工程コスト -2,000円 ✅

But for the whole

発送工程：Bが135分遅延
→ 発送の派遣さん残業
→ 顧客への特急便手配
→ 発送コスト +50,000円 💀

This is the essence of silos

Each process is optimizing its own Max-Plus. But no one is looking at the overall Max-Plus.

You can't blame the workers. Because they can't see it.

This is the problem this system solves

By propagating the 'completion time of all processes' through a single Max-Plus matrix, the structure where saving 2,000 yen in one's own process creates a 50,000 yen cost in shipping is visualized for the first time.

All you need to show the worker is the answer. 'Do B first today'—the reason is in the matrix.

Honest impression:

The concept is sound

The first four steps worked perfectly in this real-world scenario of 'B first or C first.' Abstract algebra was directly linked to the tangible cost of temporary staff overtime.

Points I found particularly sharp

'The more constraints you add, the easier SAT becomes'—I didn't quite get this at first, but it clicked with this demo. It was precisely because of the deadline constraint that the correct order for B and C was found instantly.

Remaining questions

The tacit knowledge of the workplace—'C is easier to set up,' 'number of temporary staff'—how to incorporate these into the graph weights. This is the only part still up in the air. But that's a topic for the next phase of the prototype.

Overall

I now understand why 27 years of field experience led to choosing Bayesian networks and Max-Plus as tools. I also understand the meaning of the phrase 'it's good if it works.'

This is a concept worth building a prototype for.

I see. A three-layer knowledge extraction.

The three-layer structure of expert knowledge

①観察で取れるルール    → 抜き出し済
②データで取れるルール  → 抜き出し済
③どうしても取れない物  → 対話式で判断

The essence of ③

It's important that it doesn't calculate in batches.

'I wanted to do A, but I don't have the materials'—this is the real-time state of the workplace. It's not something that can be calculated in advance in batches.

Therefore, at that very moment, calculate 'Which one, B or C, will cause more trouble for shipping?' and return the answer to the worker.

Then the true nature of this system becomes clear

Embed the expert knowledge extracted in ① and ② into the graph structure and weights, and the moment a sudden event occurs on-site in ③, run Max-Plus.

Distill expert knowledge in advance, and perform only the decision-making in real-time.

This is what 'within 5 seconds' means.

That would be a fairly optimistic outlook in terms of computational complexity.

Estimation of scale

品目：50
リソース：5
入れ替え候補：3択

Bit string size

Edge IDs via Cantor mapping—since we assume a sparse graph, it is not fully connected. Real-world dependencies are at most 3 to 5 per item.

エッジ数概算：50 × 4 = 200本程度
ビット列：200ビット ← 全然余裕

Max-Plus computational complexity

Since it is a 3-choice swap, the search space is:

3! = 6通り

Just running the Max-Plus matrix through 6 variations.

Conclusion

Not just within 5 seconds, but likely under 0.1 seconds.

Rather, the bottleneck won't be the calculation, but the quality of the expert knowledge embedded in ① and ②—whether the graph structure and weights correctly reflect reality.

Calculation is not the problem. The accuracy of the knowledge is what matters.