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Treat operating modes as a connected portfolio of options, with mode-specific exergy payoffs, transition costs and path-dependent switching bands.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Map the modes
Route 2 · Work the switching loop
Route 3 · Transfer and stress
Route 4 · Respect the limits
Route 5 · Verify and continue
CHAPTER 1 OF 12 · MAP THE MODES
1. Why several operating modes form a portfolio of options
A flexible plant may burn different fuels, make different products, idle, preserve equipment or close. Each mode has its own exergy efficiency, emissions, maintenance burden and market payoff.
The decision is not simply “choose the highest present value”. Switching costs make the best action depend on the current mode and the path already taken. The collection of modes is therefore a portfolio of interacting real options.
An exergoeconomic model keeps the physical resource cost inside every mode payoff while a switching model keeps transition costs and future flexibility visible.
CHAPTER 2 OF 12 · MAP THE MODES
2. A mode graph prevents missing transitions
Represent each operating mode as a node and each allowed transition as a directed edge. Attach start-up fuel, off-spec production, labour, downtime, wear and contractual costs to the relevant edge.
The cost from A to B need not equal the cost from B to A. Some transitions may be forbidden by equipment condition, minimum run time or environmental permits.
Draw the graph before solving. A model that assumes every mode can reach every other mode instantly often invents flexibility the plant does not possess.
CHAPTER 3 OF 12 · WORK THE SWITCHING LOOP
3. Worked ledger: two modes reward different market states
Consider Mode A, which performs best in a high-price state, and Mode B, which protects value in a low-price state. The next-period net payoffs already include routine fuel and exergy-cost allocations.
| Current-period choice | High-state payoff | Low-state payoff | Immediate transition cost |
|---|---|---|---|
| Stay in Mode A | 240 | 80 | 0 |
| Operate Mode B | 170 | 140 | 18 from A; 25 from B to A |
Let q be the probability of the high state. From Mode A, staying has value (80+160q)/1.08. Switching to B has value -18+(140+30q)/1.08.
CHAPTER 4 OF 12 · WORK THE SWITCHING LOOP
4. The current mode creates the first switching threshold
Set the two values equal. Multiplying by 1.08 gives 80+160q=120.56+30q, so q=0.312.
If q is below 0.312, B’s protection in the low state repays the switch cost. Above 0.312, an operator already in A should stay in A in this one-period model.
The threshold is not a universal market probability. It belongs to the stated payoffs, discount rate, transition direction and time horizon.
CHAPTER 5 OF 12 · WORK THE SWITCHING LOOP
5. Switching back creates a different threshold
Now suppose the plant is already in Mode B. Staying in B is worth (140+30q)/1.08. Switching to A costs 25, so its value is -25+(80+160q)/1.08.
Equality gives 140+30q=53+160q, hence q=0.669. An operator in B waits for much stronger high-state evidence before paying to return to A.
The thresholds 0.312 and 0.669 create a hysteresis band. Inside it, the preferred action is to remain in the current mode.
CHAPTER 6 OF 12 · TRANSFER AND STRESS
6. Same forecast, different action: history matters
At q=0.50, an A operator values staying at 148.15 and switching to B at 125.52. A B operator values staying at 143.52 and switching to A at 123.15.
Both operators use the same forecast and physical payoffs, yet each stays put. This is not inconsistency; it is the value of avoiding a transition cost.
Ignoring current mode would recommend needless cycling and exaggerate the value of instantaneous flexibility.
CHAPTER 7 OF 12 · TRANSFER AND STRESS
7. Changed context: cheaper transitions narrow hysteresis
Suppose both direction costs fall to five while payoffs stay unchanged. Recalculate the thresholds.
| Transition rule | Original threshold | Threshold with cost 5 |
|---|---|---|
| Switch A to B when q falls below | 0.312 | 0.420 |
| Switch B to A when q rises above | 0.669 | 0.503 |
| Hysteresis-band width | 0.357 | 0.083 |
The band narrows because less evidence is needed to justify either move. If the new cost estimate excludes cycling damage or lost production, the apparent flexibility is overstated.
CHAPTER 8 OF 12 · TRANSFER AND STRESS
8. Exergoeconomic payoffs must use the same product boundary
Compare modes with consistent fuel, product and residue definitions. One mode should not receive credit for a coproduct that is ignored in another without an explicit allocation rule.
Include exergy destruction, purchased resource costs, emissions obligations and maintenance consequences where they change decision value. Separate market revenue from thermoeconomic cost formation so a reviewer can trace both.
Transient start-up and shutdown resource use belongs on transition edges, not hidden inside a steady-state mode average.
CHAPTER 9 OF 12 · RESPECT THE LIMITS
9. Multiple modes create interacting options
With three or more regimes, each switch changes the set of future actions. The value of one edge cannot always be added independently to another because exercise policies interact.
Dynamic programming compares the immediate reward plus the continuation value of every feasible next mode. Switching regions can be described by thresholds only under suitable state dynamics and payoff structure.
Check for loops with negative total cost or inconsistent subsidies. A model that earns value by cycling A→B→A without time or production has a structural error.
CHAPTER 10 OF 12 · RESPECT THE LIMITS
10. Operational constraints belong inside the policy
Minimum up/down times, ramp limits, fuel availability, equipment temperatures and staffing may block the financially preferred switch. Treat hard safety and permit limits as constraints, not soft penalties.
Forecast updates also arrive on a schedule. A policy evaluated every minute can chatter even when a daily operating decision cannot.
Stress-test the policy with delayed observations, forecast error and transition failure. A mathematically optimal threshold can be operationally unusable.
CHAPTER 11 OF 12 · VERIFY AND CONTINUE
11. Limits and common misconceptions
More modes do not always add value. Flexibility can cost capital, training, spares and efficiency, and some modes may never be optimal after those costs are included.
Hysteresis is not indecision. It is a rational response to switching friction. Conversely, a wide band may reveal inflated cost estimates or omitted deterioration from staying in the wrong mode.
Expected value does not capture every financing, safety or downside constraint. Thresholds from a one-period model do not transfer automatically to a multi-period stochastic process.
Evidence and reproducibility
Kulatilaka develops production-mode flexibility in Valuing the flexibility of flexible manufacturing systems. Pham, Ly Vath and Zhou analyse multiple regimes in Optimal switching over multiple regimes. The resource-cost connection follows Exergoeconomic analysis and evaluation of energy-conversion plants90020-9).
Archive the mode graph, feasible edges, transition timing, exergy and cost boundaries, state process, payoff table, discount convention, optimisation code and policy simulations. Reproduce every threshold from the saved inputs.
Frequently asked questions
Why can two plants choose differently under the same forecast?
Their current modes and directional switching costs can differ, placing them on opposite sides of a hysteresis policy.
Is an idle state just another mode?
It can be, if its preservation cost, restart path and physical constraints are modelled explicitly.
Should switching costs include wear?
Include measurable life consumption and maintenance consequences, while retaining hard integrity limits as constraints.
Can thresholds cross?
They can under inconsistent payoffs or costs. Treat that as a model audit signal, not an automatic policy.
What belongs in a minimum report?
Mode graph, state payoffs, directional costs, thresholds, hysteresis band, physical constraints and sensitivity results.
CHAPTER 12 OF 12 · VERIFY AND CONTINUE
12. Final transfer check and connected reading
Keep the worked payoffs and 8% discount rate, but set A→B cost to ten and B→A cost to fifteen. Recalculate both thresholds and the hysteresis-band width. Explain which physical records should support the two directional costs.
Continue through the Thermodynamics root, real-options guide, compound-options guide, abandonment-options guide, mothballing guide and expected-regret guide. Together they connect flexibility, transition friction and robust operating policy.
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