Complex Example 2: Air Cargo Loading Planning Problem

Problem Description

Business Architecture

Domain / Mathematical Model

Loading Domain

Decision Variables

$x_{cp}\in \{0,1\}$: Loading decision, dimensionless quantity, $1$ indicates placing cargo $c$ in compartment $p$.

$y_p\in \mathbb{R}-{\mathbb{R}}^{-}$: Estimated load weight, physical quantity is weight, represents the estimated load weight for compartment $p$.

$z_p\in \mathbb{N}$: Recommended load weight, physical quantity is weight, represents the recommended load weight for compartment $p$.

$u_{cp}\in \{-1,0,1\}$: Cabin adjustment decision, dimensionless quantity, $-1$ indicates removing cargo $c$ from compartment $p$, $1$ indicates placing cargo $c$ in compartment $p$.

Intermediate Values

Loading Decision

Description: For any cargo $c$ and any compartment $p$, the loading decision for whether this cargo is loaded into this compartment equals the sum of the loading decision and cabin adjustment decision.

$${\text{Loaded}}_{cp}=\{\begin{array}{ll}1,& c\in C_p^{\text{Loaded}}\\ \\ 0,& c\notin C_p^{\text{Loaded}}\end{array},\mathrm{\forall }c\in C,\mathrm{\forall }p\in P$$$${\text{Stowage}}_{cp}=\{\begin{array}{ll}{x}_{cp}+u_{cp}+{\text{Loaded}}_{cp},& \mathrm{\forall }c\in C^{SN},\mathrm{\forall }p\in P^{SN}\\ \\ {\text{Loaded}}_{cp},& \text{else}\end{array}$$

Compartment Cargo Quantity

Description: For any compartment $p$, its compartment cargo quantity is the sum of the number of cargos loaded in that compartment.

$${\text{LA}}_p^{\text{Loaded}}=|C_p^{\text{Loaded}}|$$$${\text{LA}}_p^{\text{Estimate}}=\{\begin{array}{ll}{\text{LA}}_p^{\text{Loaded}},& \mathrm{\forall }p\in (P-P^{SN})\\ \\ \sum _{c\in C^{SN}}{\text{Stowage}}_{cp}+{\text{LA}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{SN}\end{array}$$

Compartment Load Weight

Description: For any compartment $p$, its estimated compartment load weight is the sum of the weights of cargos loaded in that compartment, estimated load weight, and recommended load weight; its actual compartment load weight is the sum of the weights of cargos loaded in that compartment.

$${\text{LW}}_p^{\text{Loaded}}=\sum _{c\in C_p^{\text{Loaded}}}{W}_c$$$${\text{LW}}_p^{\text{Estimate}}=\{\begin{array}{ll}{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{\text{Unavailable}}\\ \\ \sum _{c\in C^{SN}}{W}_c\cdot {\text{Stowage}}_{cp}+y_p+{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{SN}\cap P^{PWN}\\ \\ \sum _{c\in C^{SN}}{W}_c\cdot {\text{Stowage}}_{cp}+z_p+{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{SN}\cap P^{RWN}\\ \\ \sum _{c\in C^{SN}}{W}_c\cdot {\text{Stowage}}_{cp}+{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{SN}-P^{PWN}-P^{RWN}\\ \\ y_p+{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{PWN}-P^{SN}\\ \\ z_p+{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{RWN}-P^{SN}\end{array}$$$${\text{LW}}_p^{\text{Actual}}=\{\begin{array}{ll}{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in (P-P^{SN})\\ \\ \sum _{c\in C^{SN}}{W}_c\cdot {\text{Stowage}}_{cp}+{\text{LW}}_p^{\text{Loaded}},& \mathrm{\forall }p\in P^{SN}\end{array}$$

Total Boarded Payload

Description: Total weight of currently boarded cargos.

$${\text{Payload}}^{\text{Boarded}}=\sum _{c\in C}{W}_c$$

Estimated Payload

Description: Sum of current compartment calculated load weights.

$${\text{Payload}}_d^{\text{Estimate}}=\sum _{p\in P_d}{\text{LW}}_p^{\text{Estimate}},\mathrm{\forall }d\in D$$

Actual Payload

Description: Sum of current compartment actual load weights.

$${\text{Payload}}_d^{\text{Actual}}=\sum _{p\in P_d}{\text{LW}}_p^{\text{Actual}},\mathrm{\forall }d\in D$$

Computed Total Payload

Description: For pre-loading algorithm family, computed total payload uses estimated total payload; for full loading algorithm family, computed total payload uses real-time total payload.

$${\text{Payload}}^{\text{Computed}}=\{\begin{array}{ll}{\text{Payload}}^{\text{Plan}},& \text{Predistribution}\\ \\ {\text{Payload}}^{\text{Boarded}},& \text{FullLoad}\end{array}$$

Estimated Total Payload

Description: Sum of current compartment calculated load weights.

$${\text{Payload}}^{\text{Estimate}}=\{\begin{array}{ll}{\text{Payload}}^{\text{Computed}},& \text{FullLoad }\text{ \& }\text{ Predistribution}\\ \\ \sum _{d\in D}{\text{Payload}}_d^{\text{Estimate}},& \text{otherwise}\end{array}$$

Actual Total Payload

Description: Sum of current compartment actual load weights.

$${\text{Payload}}^{\text{Actual}}=\{\begin{array}{ll}{\text{Payload}}^{\text{Boarded}},& \text{FullLoad }\text{ \& }\text{ RecommendWeight}\\ \\ \sum _{d\in D}{\text{Payload}}^{\text{Actual}},& \text{otherwise}\end{array}$$

Center of Gravity Domain

Airworthiness Safety Domain

Center of Gravity Optimization Domain

Soft Safety Domain

Cargo Timeliness Domain

Loading/Unloading Efficiency Domain

Payload Maximization Domain

Recommended Weight Equalization Domain

Redundancy Domain

Code Implementation

Complete Implementation Reference:

• Kotlin