## 1 Introduction

_{2}emissions from fuel combustions (IEA 2019). Reports state the allocation of 14% of the GHG for this sector. A further increase by up to 50% until 2050 was projected (Edenhofer et al. 2014). This development is unique compared to the other significant industries, which often have already realized substantial reductions of harmful emissions in the recent past. Furthermore, other sectors have reliable agreements on how to reduce their emissions in the next decades.

## 2 Sustainability and pricing in the container shipping industry

## 3 Coordination of mutually agreed demand-oriented rates and market rates in transportation

^{opt}multiplied with the corresponding rate DPF(q

^{opt}). Therefore, we understand the DPF as a representative for a demand-based freight rate determination scheme. However, the existence of a market rate disrupts the exploitation of a DPF.

## 4 Literature on model-based pricing in the transport sector

## 5 Decision problem of coordinating short-term and long-term freight rates

### 5.1 The decision task

_{p}. A midterm sales planning aims at selling this capacity to the shippers. We collect all shippers in the set C. The carrier aims to identify a rate (per TEU) f

_{i}for each shipper i that is fixed for all periods so that the total sum of revenues gained by the carrier over all periods and from all shippers is maximal. Each shipper can overrule f

_{i}in a period p when the short-term rate SR

_{p}in period p beats the long-term rate, i.e., SR

_{p}<f

_{i}.

### 5.2 Mathematical pricing model

_{i}as well as the short-term (market) rate SR

_{p}. The model enables the comparison of both revenue sums and identifies the rate with the higher revenue as the applicable rate for the shipper in the considered period.

#### 5.2.1 Determination and comparison of revenues for both rates

_{p}applies for shipper c in period p since this market rate is less than the agreed long-term rate \(f_{c}\) (2). Analogously, the corresponding indicator variable \(\mathrm{SEL}_{cp}^{FR}\) is enforced to “1” if the contracted rate is less than the spot market rate (3). Precisely one of the two cases above takes place in every period and for every shipper (4).

#### 5.2.2 Period- as well as shipper-specific capacity allocations

_{c}known that is period-invariant. It maps the number of TEU allocated for a shipper to the rate per TEU. If k∈K represents the number of TEU, then the associated customer-specific rate is DPF

_{c}(k). If and only if we select contingent k for shipper c in period p the binary decision variable \(YY_{\mathrm{kcp}}^{FR}\) equals “1” and the decision variable \(q_{cp}^{FR}\) stores the contingent k (11). Exactly one contingent is selected for every shipper in each period (12). Similarly, we determine the contingent \(q_{cp}^{SR}\) for the situation when the spot market rate applies in period p for shipper c (13). Either the associated rate (between 0 and the maximal demand) is determined, or it is that the spot-market rate exceeds the maximal willingness to pay of shipper c in period p. In the later mentioned case, the indicator variable \(ZZ_{cp}^{\mathrm{STR}}\)is set to “1” (14). In order to avoid allocating a TEU number larger than the available capacity, we split the determined contingent into two parts. The first part \(q_{cp}^{SR,FF}\) represents the realizable fraction but the second part \(q_{cp}^{SR,NF}\) cannot be served (15). In the case that the spot market rate exceeds the maximal willingness to pay of shipper c in period p the contingent 0 is allocated for this shipper in this period (16). In all other cases, we set the corresponding binary decision variables \(\mu _{\mathrm{kcp}}^{SR,FF}\) appropriately (17)–(18).

#### 5.2.3 Demand-price-functions

#### 5.2.4 Limitations of capacity and demand

## 6 Experimental results

### 6.1 Scenario description

^{max}(c) for their customers (the shippers) and the maximal demand \(D_{cp}^{\max }\) for each shipper in each period.

_{c}(q) associated with the contingent q allocated for shipper c. We use the factor \(F_{v}\left(q\right)\) to define the price-sensitivity of shipper c. This sensitivity mainly depends on the value of the parameter v. For shipper c =1, we assume v =0.97. The resulting DPF is under proportional. For shipper 2 applies v =1.00, but shipper c =3 deploys v =1.01. Fig. 1 prints the resulting three differently-shaped DPFs. The three DPFs induce different rates for a given contingent, but the same rate requires different contingents allocated to the three shippers.

^{max}(1) = 1000 €/TEU (maximal demand equals 600 TEU), but shipper 2 pays not more the r

^{max}(2) =600 € for a TEU (max. demand equals 800 TEU). The third and last shipper is willing to pay at most r

^{max}(3) =400 € for a TEU (max. demand equals 1000 TEU). In all three cases, the number of TEUs to be purchased by one shipper increases if the rate falls.

_{c}(q) with different maximum points (marked by the vertical bars). The optimal allocation for shipper 1 is 184 TEUs, and the associated freight rate per TEU is 395.86 €, but shipper 2 maximizes its payments if 400 TEU are sold with a freight rate per TEU of 300 €. The optimal freight rate for shipper 3 is 280 € if the carrier reserves 624 TEUs for this shipper. In order to realize these revenues, the shipper must provide 184 TEUs + 400s TEUs + 624 TEUs = 1208 TEUs. Over the 26 considered periods, these rates enable the carrier to gain the total revenue sum of 9557 TEUR carrying 31,408 TEU at most.

### 6.2 Simulation results

#### 6.2.1 Scenario A: stable SR/stable shipper rate

_{p}(p∈P)). We investigate changes in the capacity utilization as well as the revenues for the considered vessel service for the values SR∈{200; 250; 300; 350; 400}.

SR (EUR) | Rates f _{i} (in EUR) | Contingents q _{i} (in TEU) | CU (%) | Revenues (in TEUR) | SR (EUR) | Rates f _{i} (in EUR) | Contingents q _{i} (in TEU) | CU (%) | Revenues (in TEUR) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\(f_{1}\) | \(f_{2}\) | \(f_{3}\) | \(q_{1}\) | \(q_{2}\) | \(q_{3}\) | \(f_{1}\) | \(f_{2}\) | \(f_{3}\) | \(q_{1}\) | \(q_{2}\) | \(q_{3}\) | ||||||

400 | 396 | 301 | 280 | 4784 | 10,400 | 16,224 | 50 | 9557 | 400 | 396 | 301 | 280 | 4784 | 10,400 | 16,224 | 100 | 9557 |

350 | 349 | 301 | 280 | 5382 | 10,378 | 16,198 | 51 | 9539 (−0%) | 350 | 345 | 310 | 285 | 5447 | 10,068 | 15,940 | 100 | 9536 (−0%) |

300 | 299 | 301 | 280 | 6084 | 10,376 | 16,198 | 52 | 9482 (−1%) | 300 | 300 | 301 | 300 | 6036 | 10,378 | 15,024 | 100 | 9438 (−1%) |

250 | 249 | 250 | 250 | 6890 | 12,142 | 17,784 | 59 | 9188 (−4%) | 250 | 250 | 250 | 250 | 1586 | 12,116 | 17,758 | 100 | 7865 (−18%) |

200 | 200 | 142 | 200 | 7800 | 19,465 | 19,942 | 76 | 8318 (−13%) | 200 | 200 | 200 | 200 | 25 | 538 | 15,659 | 52 | 3244 (−66%) |

#### 6.2.2 Scenario B: volatile SR/stable shipper rate

_{c}of a shipper, the shipper has the power to overrule the contracted rate, and the corresponding SR of the current period applies. Furthermore, we assume that the shipper-specific DPF

_{c}determines the associated number of sold TEUs. The resulting TEU-values for the three shippers are 5939, 10,906, and 16,609 (total sum 33,454). These quantities let the shipper-specific revenue sums become 1,817,520 €, 3,093,740 €, 4,510,500 € leading to a total revenue sum of 9,421,760 TEUR for the carrier. The maximal carrier capacity must be 1526 TEU (in period 26). On average, the capacity utilization rate is 84%, but it varies between 79 and 100%.

_{1}, f

_{2,}and f

_{3}climb. However, the contribution of the revenues from the three shippers is kept more or less stable. However, the distribution of the total vessel capacity to the three shippers is only slightly varied. In some situations, the carrier does not fix a long-term competitive rate (“–”) but uses only the SR to serve shipper 1. Finally, for the smallest vessel capacity, no freight rate discrimination is useful anymore, and the revenue sum collapses by up to 40%.

CAP (TEU) | Rates f _{i} (in EUR) | TEU (contribution) (in TEU) | CU | % of revenue | Revenues (in TEUR) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\(f_{1}\) | \(f_{2}\) | \(f_{3}\) | 1 (%) | 2 (%) | 3 (%) | Min. (%) | Max. (%) | Avg. (%) | 1 (%) | 2 (%) | 3 (%) | ||

1526 | 390 | 300 | 280 | 17 | 33 | 50 | 79 | 100 | 84 | 19 | 33 | 48 | 9422 (–) |

1500 | 390 | 300 | 280 | 17 | 33 | 50 | 81 | 100 | 86 | 19 | 33 | 48 | 9416 (−0%) |

1400 | 390 | 300 | 280 | 17 | 33 | 50 | 86 | 100 | 91 | 19 | 33 | 48 | 9369 (−1%) |

1300 | – | 300 | 280 | 17 | 33 | 50 | 93 | 100 | 97 | 19 | 33 | 48 | 9241 (−2%) |

1200 | 398.02 | 305.25 | 297.52 | 17 | 33 | 50 | 100 | 100 | 100 | 19 | 33 | 48 | 8980 (−5%) |

1100 | 390 | 328.50 | 320 | 20 | 32 | 48 | 100 | 100 | 100 | 21 | 32 | 47 | 8546 (−9%) |

1000 | – | 349.50 | 340 | 18 | 31 | 51 | 100 | 100 | 100 | 19 | 31 | 50 | 7945 (−16%) |

900 | 390 | 380 | 352.36 | 19 | 32 | 49 | 100 | 100 | 100 | 20 | 32 | 48 | 7231 (−23%) |

800 | 391.56 | 390 | 360 | 17 | 36 | 47 | 100 | 100 | 100 | 17 | 36 | 47 | 6449 (−32%) |

700 | 390 | 390 | 386.85 | 18 | 31 | 50 | 100 | 100 | 100 | 18 | 32 | 50 | 5666 (−40%) |

_{0}) and contingents allocated to the three shippers (also in relation to the initially assigned contingents in period 0). During the first SR-peak between period 3 and period 7, there is a stable distribution of the available capacity to the shippers. As soon as the SR starts to change from period to period, we can observe a contingent adjustment from period to period. Significant and extreme contingent variations occur in both types of SR-changes. During phases of SR-increase as well as phases of SR-decrease, the contingents assigned to the shippers are unstable. This behavior might be problematic since the shippers might demand a stable contingent.