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The amount of perishable products transported via the existing intermodal freight networks has significantly increased over the last years. Perishable products tend to decay due to a wide range of external factors. Supply chain operations mismanagement causes waste of substantial volumes of perishable products every year. The heretofore proposed mathematical models optimize certain supply chain processes and reduce decay of perishable products, but primarily deal with local production, inventory, distribution, and retailing of perishable products. However, significant quantities of perishable products are delivered from different continents, which shall increase the total transportation time and decay potential of perishable products as compared to local deliveries. This paper proposes a novel optimization model to design the intermodal freight network for both local and long-haul deliveries of perishable products. The objective of the model aims to minimize the total cost associated with transportation and decay of perishable products. A set of piecewise approximations are applied to linearize the non-linear decay function for each perishable product type. CPLEX is used to solve the problem. Comprehensive numerical experiments are conducted using the intermodal freight network for import of the seafood perishable products to the United States to draw important managerial insights. Results demonstrate that increasing product decay cost may significantly change the design of intermodal freight network for transport of perishable products, cause modal shifts and affect the total transportation time and associated costs.

Many of products transported via intermodal freight networks are perishable in their nature. Perishable products (such as agricultural products, meat, fish, shellfish, pharmaceutical products, etc.) are sensitive to a wide range of different factors, which include but are not limited to temperature, barometric pressure, humidity, air composition and transportation time [

Refrigerated containers (a.k.a., “reefers”) are generally used for transport of perishable food products. Reefers are able to maintain a certain temperature and decrease physiological, microbiological, and physical changes in the perishable product [

Implementation of refrigerated containers and RFID technology for transporting perishable products and tracking their quality allows decreasing waste of perishable products, but does not completely eliminate it. Mismanagement of operations within supply chains may cause from 20% to 60% of wasted agricultural products in a given country [

Therefore, there is a need for more comprehensive models to design the intermodal freight network that would allow selection of the appropriate transportation routes and transportation modes for shipments with perishable products. This paper aims to fill the existing gap in the state-of-the-art and proposes a novel optimization model for efficient management of supply chains with perishable products that can be used for both local and long-haul deliveries. The objective of the model aims to minimize the total cost associated with transportation and decay of perishable products. A set of piecewise approximations are applied to linearize the non-linear decay function for each perishable product type, and the resulting mixed integer linear problem is solved using CPLEX. Numerical experiments are performed using the intermodal freight network for import of the seafood perishable products to the US. The rest of the manuscript is organized as follows. The next section provides a detailed problem description, while the third section presents the mathematical model and the solution methodology. The fourth section describes a set of numerical experiments conducted to evaluate performance of the adopted solution methodology and reveal important managerial insights using the developed mathematical model. The last section summarizes the study findings and proposes directions for the future research.

This section of the paper focuses on description of the main problem features, including the following: 1) network elements; 2) cargo transfer within intermodal terminals; 3) perishability modeling; 4) shelf life of perishable shipments; and 5) decisions.

An example of the intermodal freight network is presented in

A transfer of perishable shipments from one mode to another occurs within the intermodal terminals (see

The quality of perishable products within each shipment is assumed to deteriorate over time. Increase in the total transportation time (which includes the total transportation time along the route segments of the selected route and the total handling time at the intermodal terminals) negatively affects freshness of products in each shipment. Each perishable shipment is assumed to be homogenous (i.e., each shipment is composed of perishable products of the same nature, which deteriorate at the same rate over time). Based on the available literature, the quality of a perishable product in shipment

where:

^{−1});

The decay rate ^{−1}, while a fresh vegetable product typically decays at a rate ^{−1} [

Note that decay function ^{−1} (i.e., ≈15 ÷ 20% of the decay rate, when the product is transported in a regular container [

Let

Constraints set (3) indicates that only one segment of the piecewise function should be selected for approximation of the decay function for a perishable product in a given shipment. Constraints sets (4) and (5) define the range of the total transportation time values, when a given linear segment should be used to approximate the decay function for a perishable product in a given shipment. Constraints set (6) computes the approximated decay of a perishable product in a given shipment at the destination terminal.

As discussed in the introduction section of the paper, perishable products should be delivered to their destinations before the end of their shelf lives in order to be of an acceptable quality for the consumers. This study captures the latter operational aspect by imposing the following constraints set for each perishable shipment

where:

Constraints set (7) ensures that each perishable shipment will be delivered to its destination before the end of its shelf life.

In this problem, the shipping company needs to make the following two major decisions: a) select a route for transportation of each perishable shipment; and b) choose a transportation mode at each segment of the selected route for each perishable shipment. Both decisions should account for a number of factors such as: 1) transportation mode availability at a given route segment; 2) increasing transportation cost for selection of faster transportation mode (e.g., transportation time by air will be smaller than by sea, but will incur higher transportation costs); 3) handling time at the intermodal terminals depending on selected mode (e.g., loading containers on a vessel may take longer as compared to loading containers on a train); 4) increasing product decay costs due to increasing total transportation time; 5) decay rate of a perishable product in a given shipment (e.g., higher decay rate will require the shipping company to select faster transportation modes at route segments to ensure that the products will be delivered to their destination terminal before the end of their shelf life).

This section presents notations that will be further used throughout the paper and a mixed integer mathematical model for the intermodal freight network design problem with perishable products.

Sets

Decision variables

Auxiliary variables

Parameters

The mixed integer mathematical model for the intermodal freight network design problem with perishable products (IFNDP) can be formulated as follows:

IFNDP:

Subject to:

In IFNDP, the objective function (8) aims to minimize the total cost associated with transportation of perishable shipments, handling of perishable shipments at the intermodal terminals, and decay of perishable shipments throughout the transportation process. Constraints set (9) ensures that only one route should be selected for transport of a given perishable shipment. Constraints set (10) indicates that the route for transport of a given perishable shipment should be selected only from the routes available for transport of that particular shipment. Constraints set (11) ensures that a given perishable shipment should be transported along all the segments of the selected route. Constraints set (12) ensures that the mode for transport of a given perishable shipment should be selected only from the modes available for transport of that particular shipment at a given route segment. Constraints set (13) estimates the total transportation time of a given perishable shipment. Constraints set (14) ensures that each perishable shipment will be delivered to its destination before the end of its shelf life. Constraints set (15) indicates that only one segment of the piecewise function should be selected for approximation of the decay function for a perishable product in a given shipment. Constraints sets (16) and (17) define the range of the total transportation time values, when a given linear segment should be used to approximate the decay function for a perishable product in a given shipment. Constraints set (18) computes the approximated decay of a perishable product in a given shipment at the destination terminal. Constraints sets (19)-(21) define the nature of IFNDP variables and parameters.

Application of the piecewise linear approximations for the product decay functions allows formulating IFNDP as a mixed integer linear mathematical model, which can be solved using commercial optimization solvers (e.g., CPLEX) within an acceptable computational time even for large size problem instances (as will be discussed in the numerical experiments section).

This section of the paper describes a set of numerical experiments that were performed to assess efficiency of the proposed solution methodology and reveal important managerial insights using the developed mathematical model.

The numerical data for computational experiments were generated based on the academic literature and publicly available resources [

The intermodal freight network for transport of perishable products (see

a/a | Product | Top Exporting Country | Top US Destinations |
---|---|---|---|

1 | SHRIMP WARM-WATER PEELED FROZEN | INDIA | NEW YORK, NY; LOS ANGELES, CA; MIAMI, FL; SAVANNAH, GA; HOUSTON-GALVESTON, TX |

2 | TILAPIA (OREOCHROMIS SPP.) FILLET FROZEN | CHINA | BALTIMORE, MD; BOSTON, MA; BUFFALO, NY; CHARLESTON, SC; CHICAGO, IL |

3 | SALMON ATLANTIC FILLET FRESH FARMED | CHILE | MIAMI, FL; LOS ANGELES, CA; NEW YORK, NY; HOUSTON-GALVESTON, TX; DALLAS-FORT WORTH, TX |

4 | CATFISH (PANGASIUS) FILLET FROZEN | VIET NAM | BALTIMORE, MD; BOSTON, MA; CHARLESTON, SC; CHICAGO, IL; CLEVELAND, OH |

5 | SALMON ATLANTIC FRESH FARMED | CANADA | SEATTLE, WA; PORTLAND, OR; DETROIT, MI; BUFFALO, NY; OGDENSBURG, NY |

6 | TUNA ALBACORE IN ATC (OTHER) NOT IN OIL OVER QUOTA | THAILAND | BALTIMORE, MD; BOSTON, MA; CHARLESTON, SC; CHICAGO, IL; DALLAS-FORT WORTH, TX |

7 | SHRIMP FROZEN OTHER PREPARATIONS | THAILAND | LOS ANGELES, CA; TAMPA, FL; NEW YORK, NY; SAVANNAH, GA; NORFOLK, VA |

8 | CRAB SNOW FROZEN | CANADA | PORTLAND, OR; DETROIT, MI; SAINT ALBANS, VT; OGDENSBURG, NY; BUFFALO, NY |

9 | GROUNDFISH COD NSPF FILLET FROZEN | CHINA | NORFOLK, VA; BOSTON, MA; SEATTLE, WA; NEW YORK, NY; LOS ANGELES, CA |

10 | SHRIMP BREADED FROZEN | CHINA | LOS ANGELES, CA; TAMPA, FL; MIAMI, FL; NEW YORK, NY; NORFOLK, VA |

by air, a total of three transportation modes were considered: 1) road; 2) rail; and 3) sea (i.e., a set of modes is

The transportation cost of a perishable shipment was assigned based on the unit transportation cost by mode (

mph) as follows:

time of a perishable shipment at the intermodal terminals was computed based on the average handling time by mode (^{−1}). Values of the parameters used for the input data generation are presented in

All numerical experiments were conducted on a Dell Intel(R) Core^{TM} i7 Processor with 32 GB of RAM. IFNDP mathematical model was coded in General Algebraic Modeling System (GAMS, [

As discussed in section 2.3 of the paper, increasing number of segments in the piecewise approximation increases accuracy of estimating the product decay values and the objective function itself, but may increase the computational time required to solve IFNDP mathematical model. A total of 25 problem instances were generated using the retrieved data, described in section 4.1, to analyze the latter tradeoff by changing the number of perishable shipments to be transported (from 2 to 10 shipments) and the number of linear segments in the piecewise approximation (from 10 to 100 segments). Detailed information regarding each shipment is provided in

IFNDP was solved for each one of the developed problem instances, and results are presented in

Parameter | Value | References |
---|---|---|

Unit transportation cost by mode― | [3.0; 2.0; 0.5] | [ |

Average handling cost by mode― | [400; 450; 500] | [ |

Average speed by mode― | [60; 40; 20] | [ |

Average handling time by mode― | [0.8; 0.9; 1.0] | [ |

Decay cost for a perishable product in a given shipment― | N/A | |

Shelf life of a perishable product― | N/A | |

Quantity of perishable products― | N/A | |

Decay rate of a perishable product―^{−1}) | [ |

Shipment | Product Type | Origin | Destination | Quantity |
---|---|---|---|---|

#1 | SHRIMP WARM-WATER PEELED FROZEN | INDIA | NEW YORK, NY | 1294 |

#2 | TILAPIA (OREOCHROMIS SPP.) FILLET FROZEN | CHINA | BOSTON, MA | 1012 |

#3 | SALMON ATLANTIC FILLET FRESH FARMED | CHILE | MIAMI, FL | 1253 |

#4 | CATFISH (PANGASIUS) FILLET FROZEN | VIET NAM | BALTIMORE, MD | 1470 |

#5 | SALMON ATLANTIC FRESH FARMED | CANADA | DETROIT, MI | 1694 |

#6 | TUNA ALBACORE IN ATC (OTHER) NOT IN OIL OVER QUOTA | THAILAND | CHARLESTON, SC | 1352 |

#7 | SHRIMP FROZEN OTHER PREPARATIONS | THAILAND | TAMPA, FL | 1329 |

#8 | CRAB SNOW FROZEN | CANADA | BUFFALO, NY | 1640 |

#9 | GROUNDFISH COD NSPF FILLET FROZEN | CHINA | SEATTLE, WA | 1498 |

#10 | SHRIMP BREADED FROZEN | CHINA | NORFOLK, VA | 1922 |

Instance | #Shipments | #Segments | #Variables | , 10^{6} USD | , 10^{6} USD | CPU, sec | ||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 1-2 | 10 | 2228 | 39.5733 | 39.5551 | 4.61E−04 | 6.6377 | 6.6360 | 2.52E−04 | 0.170 |

2 | 30 | 2268 | 39.5587 | 9.02E−05 | 6.6364 | 5.02E−05 | 0.172 | |||

3 | 50 | 2308 | 39.5571 | 5.04E−05 | 6.6362 | 2.76E−05 | 0.183 | |||

4 | 70 | 2348 | 39.5557 | 1.61E−05 | 6.6361 | 9.40E−06 | 0.197 | |||

5 | 100 | 2408 | 39.5556 | 1.28E−05 | 6.6361 | 6.99E−06 | 0.198 | |||

6 | 1-3 | 10 | 3340 | 39.6599 | 39.6411 | 4.75E−04 | 10.8365 | 10.8338 | 2.49E−04 | 0.199 |

7 | 30 | 3400 | 39.6432 | 5.34E−05 | 10.8341 | 2.81E−05 | 0.221 | |||

8 | 50 | 3460 | 39.6423 | 3.01E−05 | 10.8340 | 1.38E−05 | 0.224 | |||

9 | 70 | 3520 | 39.6417 | 1.57E−05 | 10.8339 | 8.34E−06 | 0.225 | |||

10 | 100 | 3610 | 39.6411 | 6.04E−07 | 10.8338 | 6.31E−08 | 0.242 | |||

11 | 1-5 | 10 | 5564 | 40.8238 | 40.8042 | 4.80E−04 | 19.5961 | 19.5912 | 2.45E−04 | 0.287 |

12 | 30 | 5664 | 40.8056 | 3.56E−05 | 19.5916 | 1.70E−05 | 0.308 | |||

13 | 50 | 5764 | 40.8052 | 2.47E−05 | 19.5915 | 1.18E−05 | 0.342 | |||

14 | 70 | 5864 | 40.8047 | 1.38E−05 | 19.5914 | 6.85E−06 | 0.358 | |||

15 | 100 | 6014 | 40.8046 | 9.37E−06 | 19.5913 | 4.50E−06 | 0.371 | |||

16 | 1-7 | 10 | 7788 | 42.6822 | 42.6633 | 4.43E−04 | 29.4200 | 29.4133 | 2.31E−04 | 0.390 |

17 | 30 | 7928 | 42.6642 | 2.04E−05 | 29.4136 | 1.02E−05 | 0.393 | |||

18 | 50 | 8068 | 42.6640 | 1.63E−05 | 29.4135 | 7.45E−06 | 0.396 | |||

19 | 70 | 8208 | 42.6639 | 1.32E−05 | 29.4135 | 6.74E−06 | 0.417 | |||

20 | 100 | 8418 | 42.6639 | 1.23E−05 | 29.4135 | 6.59E−06 | 0.418 | |||

21 | 1-10 | 10 | 11,124 | 42.2643 | 42.2482 | 3.81E−04 | 42.9171 | 42.9089 | 1.92E−04 | 0.440 |

22 | 30 | 11,324 | 42.2489 | 1.62E−05 | 42.9092 | 7.64E−06 | 0.463 | |||

23 | 50 | 11,524 | 42.2489 | 1.55E−05 | 42.9092 | 6.99E−06 | 0.473 | |||

24 | 70 | 11,724 | 42.2486 | 9.33E−06 | 42.9091 | 5.07E−06 | 0.486 | |||

25 | 100 | 12,024 | 42.2482 | 2.11E−07 | 42.9089 | 6.58E−07 | 0.490 |

number of shipments; 3) number of linear segments in the piecewise approximations; 4) total number of variables in IFNDP mathematical model; 5) average over all shipments product decay estimated based on piecewise approximations―

average product decay gap―

based on piecewise approximations―

We observe that increasing the number of segments in the piecewise function from 10 to 100 segments on average reduces the product decay and objective function gaps by 98.46% and increases the computational time only by 17.24%. Furthermore, the computational time over all the generated problem instances did not exceed 0.49 sec. The latter results demonstrate efficiency of the proposed solution approach, considering the fact that relatively large size problem instances were analyzed with up to 12,024 variables. The computational time may increase for larger intermodal freight networks. Application of the developed mathematical model for larger intermodal freight networks can be one of the future research directions of this study. Piecewise approximations with 100 segments will be further adopted for analysis of the managerial insights.

This section of the paper demonstrates how the developed optimization model can be used to draw important managerial insights. A total of 10 scenarios were developed for the problem instance with 10 perishable shipments by increasing the product decay cost as follows:

The total miles traveled (TMT) by each mode were calculated for each one of the considered product decay cost scenarios, and results are presented in

Shipment\Scenario | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Shipment #1 | 55 | 127 | 203 | 271 | 340 | 410 | 482 | 545 | 613 | 682 |

Shipment #2 | 56 | 124 | 198 | 262 | 333 | 412 | 472 | 541 | 613 | 689 |

Shipment #3 | 54 | 134 | 194 | 265 | 333 | 402 | 475 | 550 | 611 | 688 |

Shipment #4 | 54 | 123 | 198 | 266 | 344 | 401 | 473 | 554 | 619 | 691 |

Shipment #5 | 53 | 132 | 203 | 272 | 338 | 408 | 478 | 542 | 621 | 682 |

Shipment #6 | 63 | 120 | 203 | 260 | 334 | 408 | 479 | 542 | 617 | 685 |

Shipment #7 | 62 | 125 | 197 | 272 | 344 | 407 | 474 | 553 | 610 | 682 |

Shipment #8 | 56 | 130 | 196 | 264 | 343 | 400 | 481 | 547 | 624 | 692 |

Shipment #9 | 57 | 129 | 194 | 264 | 333 | 410 | 481 | 552 | 617 | 685 |

Shipment #10 | 58 | 122 | 191 | 267 | 330 | 413 | 481 | 551 | 615 | 684 |

Average | 57 | 127 | 198 | 266 | 338 | 406 | 477 | 547 | 616 | 686 |

Throughout the numerical experiments the average total transportation time

The scope of numerical experiments also included a detailed analysis of IFNDP cost components. The objective function and its components were estimated using the proposed mathematical model for each one the generated product decay cost scenarios. Results are presented in

Furthermore, numerical experiments show that the total decay cost is still increasing from one scenario to the other despite decrease in the actual decay of perishable pro-

ducts (see Section 4.3.2 for more details). The latter can be explained by marginal in- crease in the unit decay cost for each product (e.g., decrease of the product decay by 10% and increase of the unit product decay cost by 30% will still increase the total decay cost). It was found that the total cost associated with both transportation and decay of perishable products may be significantly affected with the product decay cost. Hence, decisions that have to be made by the shipping company will be substantially influenced depending on how the shipping company perceives the value of perishable products to be transported. In conclusion, the proposed mathematical model and solution methodology can serve as efficient practical tool in design of the intermodal freight network for both local and long-haul deliveries of perishable products and understanding of important tradeoffs.

The amount of perishable products transported via the existing intermodal freight networks significantly increased over the last decade. Due to operations mismanagement within supply chains with perishable products drastic losses associated with the product decay have been reported. Moreover, published to date mathematical models primarily optimize supply chain processes that deal with local production, inventory, distribution, and retailing of perishable products. Nevertheless, many perishable product types are imported from different continents, which increases the total transportation time and decay potential as compared to local deliveries. Unlike previous models in the literature, this paper proposed a novel mathematical model to design the intermodal freight network for both local and long-haul deliveries of perishable products. The objective aimed to minimize the total cost associated with transport and decay of perishable products. A set of piecewise approximations were adopted to linearize the non-linear decay function for each perishable product type. CPLEX was used to solve the problem. Numerical experiments, conducted using the intermodal freight network for import of the seafood perishable products to the United States, demonstrated efficiency of the adopted solution methodology in terms of solution quality and computational time. Furthermore, it was found that decisions that have to be made by the shipping company in design of the intermodal freight network were significantly dependent on how the value of perishable products was perceived. The developed mathematical model can serve as an efficient practical tool to manage both local and long-haul deliveries of perishable products.

The scope of future research may include the following extensions: 1) apply the proposed mathematical model for larger intermodal freight networks; 2) consider different types of perishable products (e.g., agricultural products, meat, pharmaceutical products, human specimens, etc.); 3) model decay of perishable products due to other factors (e.g., temperature, humidity, barometric pressure, air composition); 4) deployment of alternative cost functions for inland transport of perishable products (e.g., which capture changes in the unit transportation cost for a given mode depending on the distance traveled and shipment weight); 5) account for uncertainty in product decay throughout the transportation process; 6) quantify reliability associated with transportation of perishable products by a given mode; 7) consider the effects of economies of scale; and 8) consider the effect of real-time delay and congestion.

Dulebenets, M.A., Ozguven, E.E., Moses, R. and Ulak, M.B. (2016) Intermodal Freight Network Design for Transport of Perishable Products. Open Journal of Optimization, 5, 120-139. http://dx.doi.org/10.4236/ojop.2016.54013