Apply p-factors for multi-TLD

[isaac-tfn]

Include perceived factors in the multi-TLD, similar to in single-area. This could be made redundant due to other improvements (i.e. implementing generalised cost over distance only).

[nhannguyen2212]

1. Background:

• current input costs = Dist[i,j] and unit for TLD measures
• original form: GC[i,j] = Time[i,j] ppm + Dist[i,j] ppk + toll[i,j] -> [a]
• [a] can be rewritten as: GC[i,j] = Dist[i,j] + Time[i,j] * ppm/ppk + toll[i,j] / ppk
• we want p-factor to somehow reflect this component (Time[i,j] * ppm/ppk + toll[i,j] / ppk) -> [b]
• so we can derive new set of costs = Dist[i,j] * p-factor, with p-factor = (Dist[i,j] + [b]) / Dist[i,j]

2. Simplified method:

   • distance-based method - cost amplification (opposite of cost damping): Cost[i,j] = (Dist[i,j] / K)^(-alpha) with K and alpha to be calibrated and/or specified, 0< alpha < 1
   • distance band-based user input method - apply adjustment P[k] (user input) to Dist[i,j,k] by distance band[k], Cost[i,j,k] = Dist[i,j,k] * P[k]
   • distance band-based automatic method -> explain in the "3. process"
   • matrix-based method - user input p-factor matrix with the same shape as the cost matrix: Cost[i,j] = P[i,j] * Dist[i,j]

3. Process

   • [1] set P[i,j,k] = 1
   • [2] calculate Cost[i,j] = Dist[i,j] * P[i,j,k]
   • [3] calibrate mu/sigma using Cost[i,j] -> produce post-gravity model matrix
   • [4] calculate TLD from post-gravity model matrix TLD_achieved[k]
   • [5] calculate P[i,j,k] = (TLD_target[k] / TLD_achieved[k])^gamma (or other function TBD)
   • [6] repeat from [2] to [5] until convergence (R2 reaches maximum or little change in R2)