# American Institute of Mathematical Sciences

July  2021, 17(4): 1809-1823. doi: 10.3934/jimo.2020047

## A note on optimization modelling of piecewise linear delay costing in the airline industry

 College of Science, Nanchang Institute of Technology, Nanchang, Jiangxi 330099, P.R. China

Received  February 2019 Revised  October 2019 Published  March 2020

Fund Project: The author was supported by Grant No. GJJ161113 (2017-2019) from the Education Department of Jiangxi Province, P.R. China

We present a mathematical model in an integer programming (I.P.) framework for non-linear delay costing in the airline industry. We prove the correctness of the model mathematically. Time is discretized into intervals of, for example, 15 minutes. We assume that the cost increases with increase in the number of intervals of delay in a piecewise linear manner. Computational results with data obtained from Sydney airport (Australia) show that the integer programming non-linear cost model runs much slower than the linear cost model; hence fast heuristics need to be developed to implement non-linear costing, which is more accurate than linear costing. We present a greedy heuristic that produces a solution only slightly worse than the ones produced by the I.P. models, but in much shorter time.

Citation: Prabhu Manyem. A note on optimization modelling of piecewise linear delay costing in the airline industry. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1809-1823. doi: 10.3934/jimo.2020047
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$S$-shaped piecewise linear curve for Cost vs Delay; the curve is convex first and concave later
Delay ranges and corresponding costs. (The letters (A) to (F) identify the various columns.)
 Range(A) $\Delta_f$ range(B) Cost (C)coefficient Delay cost(D) 1 $0 \le \Delta_f \le d_{1,f}$ $c_{1,f}$ $c_{1,f}\Delta_f$ 2 $d_{1,f}< \Delta_f \le d_{2,f}$ $c_{2,f}$ $c_{1,f}d_{1,f} + c_{2,f}(\Delta_f - d_{1,f})$ 3 $\Delta_f> d_{2,f}$ $c_{3,f}$ $c_{1,f}d_{1,f} + c_{2,f}(d_{2,f} - d_{1,f})$+ $c_{3,f}(\Delta_f - d_{2,f})$ (A) $\delta$ values (E) Requirements (F) 1 $\delta_{1,f} = 1$ and $\delta_{2,f} = \delta_{3,f}$ = 0 $\mu_{2,f} = \mu_{3,f} = 0$ 2 $\delta_{1,f} = \delta_{2,f} = 1$ and $\delta_{3,f}$ = 0 $\mu_{2,f} = (\Delta_f - d_{1,f})$ and $\mu_{3,f} = 0$ 3 $\delta_{1,f} = \delta_{2,f} = \delta_{3,f}$ = 1 $\mu_{2,f} = (\Delta_f - d_{1,f})$ and $\mu_{3,f} = (\Delta_f - d_{2,f})$
 Range(A) $\Delta_f$ range(B) Cost (C)coefficient Delay cost(D) 1 $0 \le \Delta_f \le d_{1,f}$ $c_{1,f}$ $c_{1,f}\Delta_f$ 2 $d_{1,f}< \Delta_f \le d_{2,f}$ $c_{2,f}$ $c_{1,f}d_{1,f} + c_{2,f}(\Delta_f - d_{1,f})$ 3 $\Delta_f> d_{2,f}$ $c_{3,f}$ $c_{1,f}d_{1,f} + c_{2,f}(d_{2,f} - d_{1,f})$+ $c_{3,f}(\Delta_f - d_{2,f})$ (A) $\delta$ values (E) Requirements (F) 1 $\delta_{1,f} = 1$ and $\delta_{2,f} = \delta_{3,f}$ = 0 $\mu_{2,f} = \mu_{3,f} = 0$ 2 $\delta_{1,f} = \delta_{2,f} = 1$ and $\delta_{3,f}$ = 0 $\mu_{2,f} = (\Delta_f - d_{1,f})$ and $\mu_{3,f} = 0$ 3 $\delta_{1,f} = \delta_{2,f} = \delta_{3,f}$ = 1 $\mu_{2,f} = (\Delta_f - d_{1,f})$ and $\mu_{3,f} = (\Delta_f - d_{2,f})$
Parameters used in the testing of the non-linear cost model
 Number of flights 261 arrivals, 256 departures Max. delay allowed Eight periods (four hours) Perfect capacities 20 arrivals, 20 departures (per 30-minute interval) Time intervals 30 minutes each Cost break points $d_1$ = 2 units of delay, $d_2$ = 6 units of delay Cost coefficients $c_{1,f} = c_f$, $c_{2,f} = (1.5)c_f$ and $c_{3,f} = 0.5c_f$.($c_f$ is the coefficient in the single-range model)
 Number of flights 261 arrivals, 256 departures Max. delay allowed Eight periods (four hours) Perfect capacities 20 arrivals, 20 departures (per 30-minute interval) Time intervals 30 minutes each Cost break points $d_1$ = 2 units of delay, $d_2$ = 6 units of delay Cost coefficients $c_{1,f} = c_f$, $c_{2,f} = (1.5)c_f$ and $c_{3,f} = 0.5c_f$.($c_f$ is the coefficient in the single-range model)
Results of comparing single-range versus three-range cost model (Note. The "＄" used in Rows 7 and 8 refers to a generic monetary unit, not actual amount in US dollars or Australian dollars or any other currency.)
 Model $\to$ Single-range 3-range A 3-Range B 3-Range C (1) Solution how far from optimal ($A_{\rm{S}}$ value) 1.0 $\le$ 1.36 $\le$ 1.33 $\le$ 1.32 (2) Number of delayed flights 85 117 122 122 (3) Time taken to find solution 0.18 seconds 97 mins 6.5 hours 38 hours (4) Sum of the delays of all flights (periods) 518 518 518 518 (5) Average delay (periods) over all flights 1.002 1.002 1.002 1.002 (6) Average delay (periods) only over delayed flights 6.094 4.427 4.2459 4.2459 (7) Objective function value (＄) 32960 38975 37830 37830 (8) Optimal value of the Linear Programming relaxation (＄) 32960 7762
 Model $\to$ Single-range 3-range A 3-Range B 3-Range C (1) Solution how far from optimal ($A_{\rm{S}}$ value) 1.0 $\le$ 1.36 $\le$ 1.33 $\le$ 1.32 (2) Number of delayed flights 85 117 122 122 (3) Time taken to find solution 0.18 seconds 97 mins 6.5 hours 38 hours (4) Sum of the delays of all flights (periods) 518 518 518 518 (5) Average delay (periods) over all flights 1.002 1.002 1.002 1.002 (6) Average delay (periods) only over delayed flights 6.094 4.427 4.2459 4.2459 (7) Objective function value (＄) 32960 38975 37830 37830 (8) Optimal value of the Linear Programming relaxation (＄) 32960 7762
Results of comparing the Integer Programming solution and the Heuristic solution
 Model/Solver $\to$ 3-range C (from Table 4) Algorithm 1 (Pages 10-11) Number of delayed flights 122 121 Time taken to find solution 38 hours 60 seconds Sum of the delays of all flights (periods) 518 518 Ave. delay (periods) over all flights 1.002 1.002 Ave. delay (periods) only over delayed flights 4.246 4.281 Objective function value (＄) 37830 39660
 Model/Solver $\to$ 3-range C (from Table 4) Algorithm 1 (Pages 10-11) Number of delayed flights 122 121 Time taken to find solution 38 hours 60 seconds Sum of the delays of all flights (periods) 518 518 Ave. delay (periods) over all flights 1.002 1.002 Ave. delay (periods) only over delayed flights 4.246 4.281 Objective function value (＄) 37830 39660
Results of the Heuristic solution with 791 flights and flight connections
 Solver → Algorithm 1 (Pages 10-11) 1 Number of delayed flights 130 2 Time taken to find solution 35 seconds 3 Sum of the delays of all flights (periods) 690 4 Ave. delay (periods) over all flights 0.872 5 Ave. delay (periods) only over delayed flights 5.308
 Solver → Algorithm 1 (Pages 10-11) 1 Number of delayed flights 130 2 Time taken to find solution 35 seconds 3 Sum of the delays of all flights (periods) 690 4 Ave. delay (periods) over all flights 0.872 5 Ave. delay (periods) only over delayed flights 5.308
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