# Sharing Loading Costs for Multi Compartment Vehicles

## Abstract

**:**

## 1. Introduction

## 2. The MCV Problem and an Extension

#### 2.1. The Adjusted MCV Problem

#### 2.2. Typical Physical Constraints

## 3. Cost Allocation Principles

**Definition**

**1.**

- 1.
- σ is efficient. All the cost is assigned.
- 2.
- σ is individually rational. Individual shippers should always pool.
- 3.
- σ is stable. (Note this includes the previous condition.)
- 4.
- σ has the null player property and its converse. That is, it assigns an allocation of zero if and only if the shipper is a null player, contributing no cost to any coalition (for instance by not shipping anything). In all other cases, something positive is charged.

## 4. Balancing Conditions

**Theorem**

**1**

- A.
- Insert the singleton of the new shipper into the collection; also augment some of the old subsets, so that $\mu \xb7z<1$. Balancing weights are the same for the old sets, and $1-\mu \xb7z$ for the singleton.
- B.
- Do not insert the singleton to the collection; but augment some of the subsets in the collection, so that $\mu \xb7z=1$. Balancing weights remain the same.
- C.
- Construct a new MBC only when the balancing weights in the original MBC are not equal. Do not insert the singleton to the collection; but augment some of the subsets in the collection; and for one subset ${B}_{t}$,include both ${B}_{t}$ and its augment. Here, $\mu \xb7z<1-{\mu}_{t}{z}_{t}<1$. In this case, $\left|\mathbb{B}\right|$ increases by 1. ${\mu}_{t}$ is split among ${B}_{t}$ and ${B}_{t+1}$, with ${B}_{t}$ getting $1-\mu \xb7z$.
- D.
- Start with two original MBCs ${\mathbb{B}}_{1},{\mathbb{B}}_{2}$ each of s shippers, constructed with vectors ${z}^{1},{z}^{2})$; construct a new collection $\widehat{\mathbb{B}}$ from the set union of the collections. Check that Rank($Y(\widehat{\mathbb{B}})$) = $s-1$. Order the sets in $\widehat{\mathbb{B}}$ so that $0<{\mu}^{1}\xb7{z}^{1}<1<{\mu}^{2}\xb7{z}^{2}$; if this is possible; let $t=(1-{\mu}^{1}\xb7{z}^{1})/({\mu}^{2}\xb7{z}^{2}-{\mu}^{1}\xb7{z}^{1})$. Do not insert the singleton to the collection; but augment some of the subsets in the collection; assign weights by multiplying old weights of sets in ${\mathbb{B}}_{1}\backslash {\mathbb{B}}_{2}$ by $(1-t)$, of sets in ${\mathbb{B}}_{2}\backslash {\mathbb{B}}_{1}$ by t, and sets in both by the convex combination $(1-t){\mu}^{1}+t{\mu}^{2}$.

#### 4.1. Proper MBCs

**Theorem 2**(Shapley Lemma 3).

**Proof.**

- (a)
- $\mathbb{A}$ is balanced.
- (b)
- $\mathbb{A}$ is minimal balanced.
- (c)
- $\mathbb{A}$ has fewer disjoint pairs than $\mathbb{B}$ .

**Theorem**

**3.**

- 1.
- $R=A$ never creates a proper successor MBC.
- 2.
- If $R=B$ or $R=C$ and $\mathbb{B}$ is proper then $\mathbb{A}$ is proper.
- 3.
- If $R=D$, $\mathbb{B}={\mathbb{B}}_{1}\cup {\mathbb{B}}_{2}$, where ${\mathbb{B}}_{1},{\mathbb{B}}_{2}$ are MBCs, and $\mathbb{B}$ is proper, then $\mathbb{A}$ is proper.

**Proof.**

- Let $R=A$. The set $\{s+1\}$ is included in $\mathbb{A}$. It intersects only those members of $\mathbb{B}$ which are augmented in $\mathbb{A}$. But there is at least one set in $\mathbb{B}$ which is not augmented, since $w\xb7z<1$. It does not intersect $\{s+1\}$.
- Let $R=B$. Let P consist of the indices of members of $\mathbb{B}$ such that ${z}_{p}=1\forall \phantom{\rule{4pt}{0ex}}p\in P$. Choose $p,q\in P$. then ${A}_{p}={B}_{p}\cup \{s+1\}$, and ${A}_{q}={B}_{q}\cup \{s+1\}$. These are not disjoint, since they both contain $s+1$. Now suppose $q\notin P$. Then ${A}_{q}={B}_{q}$ and ${A}_{p}\cap {A}_{q}\ne \varnothing $ because ${B}_{q}$ intersects ${B}_{p}$. Finally, if $p,q\notin P$ then ${A}_{q}={B}_{q}$, ${A}_{p}={B}_{p}$; and they intersect by hypothesis.
- Let $R=C$. There is only one subset ${B}_{t}$ which is mapped to two sets, ${A}_{t}={B}_{t}$ and ${A}_{t+1}={B}_{t}\cup \{s+1\}$. Define ${S}_{0}={B}_{i}:{z}_{i}=0$ and ${S}_{1}={B}_{i}:{z}_{i}=1$. Choose two subsets ${B}_{p},{B}_{q}$ in $\mathbb{B}$.If they are both in ${S}_{0}$ then ${A}_{p},{A}_{q}$ intersect because they are identical to their predecessors in $\mathbb{B}$.If they are both in ${S}_{1}$ then ${A}_{p},{A}_{q}$ intersect because they both contain $s+1$.If ${B}_{p}\in {S}_{0},{B}_{q}\in {S}_{1}$ then ${A}_{p},{A}_{q}$ intersect because ${A}_{p}$ meets ${A}_{q}$ at a member other than $s+1$, common to ${B}_{p}$ and ${B}_{q}$.If ${B}_{p}\in {S}_{0}$ then ${A}_{p}={B}_{p}$ meets ${A}_{t}={B}_{t}$ because $\mathbb{B}$ is proper; and the same can be said for ${A}_{p}$ and ${A}_{t+1}={B}_{t}\cup \{s+1\}$.If ${B}_{p}\in {S}_{1}$ then ${A}_{p}={B}_{p}\cup \{s+1\}$ meets ${A}_{t+1}={B}_{t}\cup \{s+1\}$ at $s+1$; and ${A}_{p}={B}_{p}\cup \{s+1\}$ meets ${A}_{t}={B}_{t}$ at the point where ${B}_{p}$ meets ${B}_{t}$.
- Let $R=D$, with x the extension vector; $(D,x)$ will denote the map. Assume $\mathbb{A}$ is not proper; then there exist $S,T\in \mathbb{A}$ with $S\cap T\ne \varnothing .$ Let $Q,R\in \mathbb{B}$ such that $(D,x)Q)=S$ and $(D,x)(R)=T.$ Define ${Q}_{a},{R}_{a}$ as the sets which are augmented by the new shipper. Then we have three possibilities;$$\begin{array}{ccc}S=Q\hfill & \phantom{\rule{1.em}{0ex}}\mathtt{and}\phantom{\rule{1.em}{0ex}}& T=R\hfill \\ S={Q}_{a}\hfill & \phantom{\rule{1.em}{0ex}}\mathtt{and}\phantom{\rule{1.em}{0ex}}& T=R\hfill \\ S=Q\hfill & \phantom{\rule{1.em}{0ex}}\mathtt{and}\phantom{\rule{1.em}{0ex}}& T={R}_{a}\hfill \\ S={Q}_{a}\hfill & \phantom{\rule{1.em}{0ex}}\mathtt{and}\phantom{\rule{1.em}{0ex}}& T={R}_{a}.\hfill \end{array}$$The first case implies $\mathbb{B}$ is not proper because $S,T$ are a disjoint pair. In the second and third case $Q\cap R=\varnothing $ because retracting to $\mathbb{B}$ by removing the new shipper from the set containing it would not make $Q,R$ intersect. Hence $\mathbb{B}$ is not proper. Finally if both $S,T$ are augmented this would violate the assumption that $\mathbb{A}$ is not proper.

#### 4.2. Tilted MBCs

#### 4.3. MBC Database

## 5. Algorithm

#### 5.1. Properties of Heuristic

- It must be consistently feasible; that is, the solution whose cost is reported at any stopping point must satisfy all constraints and be usable to actually load units into cells of vehicles.
- It must be checkpoint restartable. By that we mean that if the algorithm is stopped at some point to report the best obtained cost so far, we save sufficient information (a checkpoint) so that the algorithm can be restarted exactly from that point to try for lower costs later. This is true for every subproblem consisting of some subset of shippers of the large multiple shipper problem; we will need to restart some of these subproblems as the algorithm evolves toward a solution for all shippers.
- It must be constraint extendable; that is, we must be able to add constraints generated by some process to the problem in such a way that solutions do not violate the new constraints. What this implies is that if we apply a constraint to a load for a subset, and it is not feasible for that solution, we can restart the algorithm for that subset and replace the solution with a new one that satisfies the constraint.
- It has a stopping criterion which prevents it from cycling longer than desired. It could be a processing time limit, a number of iterations, or other test, or several of these. It is required in case the algorithm has trouble finding a feasible outcome, or it cycles among several outcomes, or it is trying at random and might continue forever. The last feasible cost loading is reported when the algorithm stops; if no feasible loading has been found it declares so, resulting in a run that cannot find any solution.

#### 5.2. Process

Listing 1: Pseudocode for level k subset procedure. |

#### 5.3. Allocation

## 6. Example

#### 6.1. Parameters

#### 6.2. Algorithm Flow

#### 6.3. Heuristic

#### 6.4. Outcomes

## 7. Conclusions

## Conflicts of Interest

## References

- Qin, H.; Zhang, Z.; Qi, Z.; Lim, A. The freight consolidation and containerization problem. Eur. J. Oper. Res.
**2014**, 234, 37–48. [Google Scholar] [CrossRef] - Mesa-Arango, R.; Ukkusuri, S. Benefits of in-vehicle consolidation in less than truckload freight transportation operations. Transp. Res. Part E Logist. Transp. Rev.
**2013**, 60, 113–125. [Google Scholar] [CrossRef] - Lennane, A. IATA to review air cargo load factor calculations after Project Selfie revelations. Loadstar
**2018**. Available online: https://theloadstar.co.uk/iata-review-air-cargo-load-factor-calculations-project-selfie-revelations (accessed on 18 January 2018). - Ramos, A.; Oliveira, J.; Goncalves, J.; Lopes, M. A container loading algorithm with static mechanical equilibrium stability constraints. Transp. Res. B
**2016**, 91, 565–581. [Google Scholar] [CrossRef] - Dror, M.; Hartman, B. Shipment consolidation—Who pays for it and how much. Manag. Sci.
**2006**, 3, 78–87. [Google Scholar] [CrossRef] - Alonso, M.T.; Alvarez-Valdes, R.; Iori, M.; Parreño, F.; Tamarit, J.M. Mathematical models for multicontainer loading problems. Omega
**2016**. [Google Scholar] [CrossRef][Green Version] - Gonçalves, J.F.; Resende, M.G.C. A parallel multi-population biased random-key genetic algorithm for a container loading problem. Comput. Oper. Res.
**2012**, 39, 179–190. [Google Scholar] [CrossRef] - Alvarez-Valdes, R.; Parreño, F.; Tamarit, J.M. A GRASP/Path Relinking Algorithm for two- and three-dimensional multiple bin-size packing problems. Comput. Oper. Res.
**2013**, 40, 3081–3090. [Google Scholar] [CrossRef] - Pollaris, H.; Braekers, K.; Janssens, A.C.; Janssens, G.; Limbourg, S. Iterated Local Search for the Capacitated Vehicle Routing Problem with Sequence Based Pallet Loading and Axle Weight Constraints. Networks
**2017**, 69, 304–316. [Google Scholar] [CrossRef] - Junquiera, L.; Morabito, R.; Yamashita, D.S. Three-dimensional container loading models with cargo stability and load bearing constraints. Comput. Oper. Res.
**2012**, 39, 74–85. [Google Scholar] [CrossRef] - Peleg, B. An Inductive method for constructing minimal balanced collections of finite sets. Nav. Res. Logist.
**1965**, 12, 155–162. [Google Scholar] [CrossRef] - Young, H.P. Cost Allocation: Methods, Principles, Applications; Elsevier Science Publishers B.V.: Amsterdam, The Netherlands, 1985. [Google Scholar]
- Peleg, B.; Sudhölter, P. Introduction to the Theory of Cooperative Games; Springer: Berlin, Germany, 2007. [Google Scholar]
- Dror, M.; Hartman, B.; Chang, W. The Cost Allocation Issue in Joint Replenishment. Int. J. Prod. Econ.
**2011**, 136, 232–244. [Google Scholar] [CrossRef] - Schmeidler, D. Nucleolus of a characteristic function game. SIAM J. Appl. Math.
**1969**, 17, 1163–1170. [Google Scholar] [CrossRef] - Hartman, B.; Dror, M. Cost allocation in continuous-review inventory models. Nav. Res. Logist.
**1996**, 43, 549–561. [Google Scholar] [CrossRef] - Anshelevich, E.; Dasgupta, A.; Kleinburg, J.; Tardos, E.; Wexler, T.; Roughgarden, T. The price of stability for network design with fair cost allocation. SIAM J. Comput.
**2008**, 38, 1602–1623. [Google Scholar] [CrossRef] - Moulin, H.; Shenker, S. Strategyproof sharing of submodular costs: Budget balance versus efficiency. Econ. Theory
**2001**, 18, 511–513. [Google Scholar] [CrossRef] - Perry, J.; Zentz, C. FERC Rejects SPP’s Cost Allocation Plan for Two Interregional Transmission Projects. Lexology
**2017**. Available online: https://www.lexology.com/library/detail.aspx?g=38231f1c-b986-41c0-8bf1-982a85da6e13 (accessed on 10 March 2018). - Guajardo, M.; Rönnquist, M. A review on cost allocation methods in collaborative transportation. Int. Trans. Oper. Res.
**2016**, 3, 371–392. [Google Scholar] [CrossRef] - Fang, X.; Cho, S.-H. Stability and endogenous formation of inventory transshipment networks. Oper. Res.
**2014**, 62, 1316–1334. [Google Scholar] [CrossRef] - Bondareva, O. Theory of the core in the n-person game. Vestnik LGU
**1962**, 13, 141–142. [Google Scholar] - Shapley, L. On balanced sets and cores. Nav. Res. Logist.
**1967**, 14, 453–460. [Google Scholar] [CrossRef] - Solymosi, T.; Sziklai, B. Characterization Sets for the Nucleolus in Balanced Games. Oper. Res. Lett.
**2016**, 44, 520–524. [Google Scholar] [CrossRef] - Solymosi, T.; Sziklai, B. Universal Characterization Sets for the Nucleolus in Balanced Games; Discussion Paper MT-DP-2015/12; Centre for Economic and Regional Studies, Hungarian Academy of Sciences: Budapest, Hungary, 2015. [Google Scholar]
- Puerto, J.; Perea, F. Finding the nucleolus of any n-person cooperative game by a single linear program. Comput. Oper. Res.
**2013**, 40, 2308–2313. [Google Scholar] [CrossRef] - Leng, M.; Parlar, M. Analytic Solution for the Nucleolus of a Three-Player Cooperative Game. Nav. Res. Logist.
**2010**, 57, 667–672. [Google Scholar] [CrossRef][Green Version]

**Figure 3.**Plot of the core of the stable result of I in triangular coordinates, with nucleolus n shown. Source: author.

Shipper | Package | Dimweight D | Temp ${\mathit{T}}_{\mathit{c}}$ | Cell c |
---|---|---|---|---|

1 | B | 40 | 45 | LH |

2 | M | 20 | 30 | L |

3 | S | 10 | 45 | LH |

Fixed Cost | F | 40 | |

Cost Per Dimweight | p | 0.4 | |

L | H | ||

T Cost Multiplier | $t(c)$ | 2 | 1 |

T cost/dim | $u(c)$ | 1.9 | 1.7 |

**Table 3.**Three-shipper example showing sequence of computations for algorithm. * choose the minimum of the multiple bound expressions.

Level | Set | Workable Cost | Stable Bound on Cost | Bound on Cost |
---|---|---|---|---|

1 | $\{1\}$ | $C(1)$ | ||

$\{2\}$ | $C(2)$ | |||

$\{3\}$ | $C(3)$ | |||

2 | $\{1,2\}$ | $C(12)$ | $C(1)+C(2)$ | |

$\{2,3\}$ | $C(23)$ | $C(2)+C(3)$ | ||

$\{1,3\}$ | $C(13)$ | $C(1)+C(3)$ | ||

3 | $\{1,2,3\}$ | $C(123)$ | * $C(12)+C(3)$ | |

* $C(23)+C(1)$ | ||||

* $C(13)+C(2)$ | ||||

$\frac{1}{2}C(12)+\frac{1}{2}C(23)+\frac{1}{2}C(13)$ |

**Table 4.**Three runs of the example using random FFRO executions I (twice) and II. The load using I (bolded) with final step cost of 229 is stable, but the same with final cost 250 is not. II (italicized) finds the lowest cost stable solution with certainty when stopping criterion allows four trials (to make sure the low cost for $C(13)$ is found). For each choice NF = not feasible; -Z (size), -T (temperature) signifies reason for infeasibility. (w) = workable; (s) = stable. All doubletons are workable.

Loads | Cost Components | |||||||
---|---|---|---|---|---|---|---|---|

L | H | Total Cost | F | PL | PH | TL | TH | |

C(1) | B | 208 | 40 | 16 | 0 | 152 | 0 | |

B | 124 | 40 | 0 | 16 | 0 | 68 | ||

C(2) | M | 124 | 40 | 8 | 0 | 76 | 0 | |

NF-T | 0 | 0 | 0 | 0 | ||||

C(3) | S | 82 | 40 | 4 | 0 | 38 | 0 | |

S | 61 | 40 | 0 | 4 | 0 | 17 | ||

C(12) | NF-Z | 0 | 0 | 0 | 0 | |||

NF-T | 0 | 0 | 0 | 0 | ||||

M | B | 208 | 40 | 8 | 16 | 76 | 68 | |

NF-T | 0 | 0 | 0 | 0 | ||||

C(23) | M,S | 166 | 40 | 12 | 0 | 114 | 0 | |

NF-T | NF-T | 0 | 0 | 0 | 0 | |||

M | S | 145 | 40 | 8 | 4 | 76 | 17 | |

NF-T | 0 | 0 | 0 | 0 | ||||

C(13) | B,S | 250 | 40 | 20 | 0 | 190 | 0 | |

B | S | 229 | 40 | 16 | 4 | 152 | 17 | |

S | B | 166 | 40 | 4 | 16 | 38 | 68 | |

B,S | 145 | 40 | 0 | 20 | 0 | 85 | ||

C(123) | NF-Z | 0 | 0 | 0 | 0 | |||

NF-Z | 0 | 0 | 0 | 0 | ||||

NF-T | 0 | 0 | 0 | 0 | ||||

M,S | B | 250 (w) | 40 | 12 | 16 | 114 | 68 | |

NF-T | 0 | 0 | 0 | 0 | ||||

M | B,S | 229 (s) | 40 | 8 | 20 | 76 | 85 | |

NF-T | 0 | 0 | 0 | 0 | ||||

NF-T | 0 | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Hartman, B.C.
Sharing Loading Costs for Multi Compartment Vehicles. *Games* **2018**, *9*, 25.
https://doi.org/10.3390/g9020025

**AMA Style**

Hartman BC.
Sharing Loading Costs for Multi Compartment Vehicles. *Games*. 2018; 9(2):25.
https://doi.org/10.3390/g9020025

**Chicago/Turabian Style**

Hartman, Bruce C.
2018. "Sharing Loading Costs for Multi Compartment Vehicles" *Games* 9, no. 2: 25.
https://doi.org/10.3390/g9020025