An example of the intermodal freight network is presented in Figure 1. Let be a set of perishable shipments. Each perishable shipment is assumed to have an origin terminal and a destination terminal (see Figure 1). The origin and destination terminals of the perishable shipments are connected with a set of transportation routes. Each route is divided in a set of route segments. At each segment the shipping company is able to select a transportation mode (i.e., sea, air, rail, truck, etc.) out of the set of modes. It is assumed that availability of modes varies from one route segment to another (e.g., at route segments passing through sea the shipping company can transport the perishable shipment either by sea or air, while at land route segments the shipping company can transport the perishable shipment either by air or rail or truck). The transportation time (measured in hours) and associated cost (measured in USD) of a given perishable shipment at each route segment are assumed to vary depending on the transportation mode selected. For example, the transportation time of a given perishable shipment by air will be shorter as compared to truck, but it will incur a higher cost for the shipping company.

A transfer of perishable shipments from one mode to another occurs within the intermodal terminals (see Figure 1). The handling time at the intermodal terminal (measured in hours) and associated cost (measured in USD) are assumed to vary depending on the type of a perishable shipment (captured by index), route segment (captured by index), and mode selected for transport at a given route segment (captured by index). The latter allows tackling the change in handling time and cost of a given perishable shipment due to its size (e.g., increasing size of shipments will increase the handing time), mode availability depending on the route segment selected, and required resources for transfer of a cargo from one mode to the other.

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 at a given time can be estimated based on the following equation [2] , [3] , [16] , [17] :

where:

―is the quality of a perishable product in shipment at time (i.e., once it is unloaded at the destination terminal-%);

―is the quality of a perishable product in shipment at time (i.e., once it is loaded at the origin terminal-%);

―is the decay rate of a perishable product in shipment (hour⁻¹);

―is the total transportation time of perishable shipment from the origin terminal to the destination terminal (hours).

The decay rate depends on the nature of a perishable product in shipment. For example, a typical decay rate for the meat product is hour⁻¹, while a fresh vegetable product typically decays at a rate hour⁻¹ [2] . Based on Equation (1) the decay of a perishable product in shipment can be further computed as:

Note that decay function is non-linear and can be linearized using its piecewise linear approximation. Figure 2 demonstrates examples of piecewise linear approximations for a non-linear product decay function. The decay rate of a perishable product, transported in refrigerated containers was assumed to be hour⁻¹ (i.e., ≈15 ÷ 20% of the decay rate, when the product is transported in a regular container [2] ). The total transportation time was assumed to be up to 700 hours (i.e., hours ≈ 30 days). We observe that increasing number of linear segments in the piecewise linear approximation improves accuracy of function, but may incur an increasing computational time (due to increase in the number of variables in the optimization model). A tradeoff between the approximation accuracy vs. computational time will be analyzed in the numerical experiments section.

Let be a set of linear segments in piecewise linear approximation. Let if linear segment is selected to approximate the decay function for a perishable product in shipment. Let and be the transportation time value of a perishable product in shipment at the beginning and the end of linear segment respectively. Let and be the slope and the intercept of linear segment for the decay function of a perishable product in shipment respectively. Let be a large positive number. Then the approximated decay of a perishable product in shipment can be estimated using the following set of equations:

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:―is the shelf life of a perishable product in shipment (hours).

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).

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).

The numerical data for computational experiments were generated based on the academic literature and publicly available resources [19] - [30] . A total of top ten seafood perishable product types, based on the overall volumes imported to the US, were considered in this study. The top origin country and the top five US destinations (based on the overall volumes) for each perishable product type were retrieved using the data provided by NOAA [4] and are presented in Table 1.

The intermodal freight network for transport of perishable products (see Figure 3) included 5 marine terminals, 40 rail terminals, and 50 destination depots. Locations of the intermodal terminals were retrieved using the data provided by the Intermodal Association of North America [19] . Each shipment was assumed to originate at one of the marine terminals of the top exporting country. A total of 50 possible routes were considered for each origin-destination pair. The intermodal freight network was composed of 350 route segments. Since seafood perishable products are rarely transported

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 latter does not limit generality of the proposed model, and the air mode can be included when modeling transport of other perishable products (e.g., pharmaceutical products, human specimens, organs, etc.).

The transportation cost of a perishable shipment was assigned based on the unit transportation cost by mode (, USD per mile) and length of a route segment (, miles) as follows: (USD), where―is a notation used for uniformly distributed pseudorandom numbers. The handling cost of a perishable shipment was generated based on the average handling cost by mode (, USD) as follows: (USD). The transportation time along route segments of a perishable shipment was calculated based on the route segment length and the average speed by mode (,

mph) as follows: (hours). The handling

time of a perishable shipment at the intermodal terminals was computed based on the average handling time by mode (, hours) as follows: (hours). Note that the random terms (assigned using the uniform distribution) in, , , and formulas were introduced to capture changes in the transportation and handling costs/times depending on the perishable shipment type, route segment selected, and transportation mode used. The decay cost for a perishable product in a given shipment was assigned as (USD). The shelf life of a perishable product in a given shipment was generated as (hours). The quantity of perishable products in each shipment was set to (products). The quality of products in each perishable shipment at the origin terminal was assumed to be 100% (i.e.,). The decay rate of a perishable product in a given perishable shipment was assigned as: (hour⁻¹). Values of the parameters used for the input data generation are presented in Table 2.

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, [31] ) and solved using CPLEX. Piecewise linear approximations for product decay functions were developed using MATLAB 2016a [32] .

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 Table 3, including the following data: 1) shipment number; 2) perishable product type; 3) origin country; 4) US destination (randomly selected out of top five US destinations for a given product type); and 5) quantity of perishable products shipped. For example, the first shipment includes 1294 units/packages of shrimp warm-water peeled frozen and is imported from India to New York (NY).

IFNDP was solved for each one of the developed problem instances, and results are presented in Table 4, including the following information: 1) instance number; 2)

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―; 6) true average product decay estimated based on the non-linear product decay functions―; 7)

average product decay gap―; 8) objective function value estimated

based on piecewise approximations―; 9) true objective function value estimated based on the non-linear product decay functions at the solution provided by IFNDP―

; 10) objective function gap―; and 11) computation time.

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: (USD), where―is the scenario number. The generated product decay cost values for each perishable shipment and scenario are presented in Table 5. IFNDP was solved for each one of the generated product decay cost scenarios. Next this section elaborates on how increasing product decay cost affected the total miles traveled by each mode, total transportation time, decay of perishable products, and associated transportation and product decay costs.