Optimum Cruise Alt formula

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You asked for it (B737-800W):

The FMC generates optimum and maximum altitude data to assist the flight crew in choosing the best cruise altitude for a given trip. The calculated maximum altitude is used to ensure that the predicted path of the airplane remains within its performance capabilities. This maximum altitude is the lower of the thrust-limited altitude and the limits of the VNAV operational envelope.

The FMC calculates an economy cruise optimum altitude in one of two ways depending on the length of the planned trip. A "short trip" optimum altitude is calculated to minimize trip costs when the choice of cruise altitude is constrained by the climb and descent performance of the airplane. On longer trips, where cruise altitude choice is constrained only by the airplane's initial cruise altitude capability, a "long trip" optimum altitude is computed to minimize cost.

STOA
The short trip optimum altitude (STOA) is that altitude which yields the minimum trip cost when cruise altitude is constrained to be lower than the long trip optimum altitude by either the intersection of the climb and descent profiles, or minimum cruise time considerations. The STOA calculations are based on the standard trip profile as below. The STOA is displayed by the FMC as the optimum cruise altitude when the equivalent trip distance is less than 400 nautical miles.

SEA LEVEL TAKEOFF, CLIMBOUT TO 1500 FT
FLAP RETRACTION, ACCELERATION TO VREF40 +70 KCAS CLIMB AT 250 KCAS TO 10,000 FT
ACCELERATION TO ENROUTE CLIMB SPEED
CLIMB AT ENROUTE CLIMB CAS/MACH
CRUISE
DESCENT AT MACH/CAS SCHEDULE
DECELERATION TO 250 KCAS AT 10,000 FT
DESCENT AT 250 KCAS TO 1500 FT
DECELERATION AND FLAP EXTENSION AT 1500 FT APPROACH AND LANDING AT SEA LEVEL AIRPORT

The optimum altitude determined for each short trip has a corresponding optimum cruise time which is directly related to the computed equivalent trip distance. If the operator desires additional cruise time, the STOA may be decreased to make this accomodation. The minimum cruise time considered may be varied according to airline policy from one minute (FMC default) to twenty minutes.

Several parameters affect the short trip optimum altitude calculation. These include trip distance, cruise time, gross weight, temperature, wind, and airport elevations. Airplane speed effects are negligible. The altitude data for a baseline condition is used as a function of trip distance and cruise time. Approximate linear corrections are provided to account for the other variables. Wind effects are the most difficult to generalize. The FMC wind model is a linear variation in wind magnitude from zero on the ground to the input value at the input cruise altitude.

LTOA
The economy cruise long trip optimum altitude (LTOA) is used if the calculated equivalent trip distance is in excess of 400 nautical miles. The economy cruise LTOA is that altitude which the airplane should fly for lowest cost operation when flying in economy cruise mode at a given gross weight. The data stored in the FMC is computed by maximizing the inverse of the economy cruise cost function.


Function Maximized = VG / ((100 * CI) + WF)
where:
VG = ground speed in nautical miles per hour,
CI = cost index with units of dollars/hour, cents/pound
100 = cents to dollars conversion, WF = fuel flow in pounds per hour.

This function relates ground speed to the equivalent fuel flow "cost" of flying at that speed. The equivalent fuel flow consists of a pseudo fuel flow (100*CI) representing time costs, plus the actual fuel flow (WF) representing fuel costs. Therefore, the function which is maximized has the units of nautical air miles per pound (nam/lb). For the economy cruise mode the LTOA occurs where the result of this expression is maximized. This results in an optimum altitude yielding the minimum fuel burn at the specified speed.

The FMC stores the economy cruise LTOA as optimum gross weight/delta (W/δ) vs cost index/gross weight (CI/GW). The computation of optimum altitude includes an estimate of the climb fuel burn between the current altitude and optimum altitude, unless the airplane is above the optimum altitude. If the current altitude is above the optimum altitude, the computations are performed at the current gross weight. The economy cruise optimum altitude may be determined in the following manner:

1. Calculate CI/GW by dividing the cost index entered in the FMC by the current gross weight, in pounds.
2. Determine the optimum W for the CI computed in step 1
3. Determine the altitude corresponding to the optimum W/δ from step 2 and the current gross weight, in pounds.
4. Use the altitude determined in step 3 and the current altitude to estimate a climb fuel burn value using a linear climb fuel burn estimate.
   The climb fuel burn can be estimated by the equation:
   CLB FUEL = (A + B*WSOC/1000)*(HTOC - HSOC)
   where:
   A = -0.012243 (LB/FT),
   B = 0.000657 (1000/FT),
   WSOC = Gross weight at start of climb pressure Altitude (LB),
   HTOC = Top of climb pressure altitude (FT),

   HSOC = Start of climb pressure altitude (FT).
5. Subtract the value of CLB FUEL from the current gross weight to get the estimated TOC gross weight. Repeat steps 1 through 3 to get the final approximation of the LTOA.

Note: Charts are required to determine LTOA as per above formula, but these are proprietary to the manufacturer, sorry these cannot be included here.