As the world becoming a global village, producing and manufacturing have been affected positively since firms are able to centralize production operations at thoroughly designed facilities that are equipped with advanced technology (i.e., automated production systems, robots, etc.) through high-capital investments. These facilities are planned to be operated effectively and efficiently aiming to produce or manufacture a high volume of outputs by achieving consistent quality so that their outputs are supplied to almost all of the customers from all around the world. Freight transportation plays an important role in the distribution of globally marketed products and, the freight carriers provide delivery service between all kind of facilities including factories, warehouses, depots for all stakeholders in the global markets such as manufacturers, wholesalers, and retailers. Briefly, today’s world economy mostly relies on the movement of freight efficiently.
Moreover, the carriers encounter new challenges eventually since the demand for fast, reliable and low-cost transportation service increases gradually. Furthermore, the carriers usually operate with low operating margins (or operating income) and aiming to maximize utilization of assets owned, due to the high ownership and operating costs in the consolidation-based transportation sector. For instance, FedEx Freight and XPO Logistics,-the first two of the Journal of Commerce’s (JOC) list of Top 50 US and Canadian LTL Trucking Companies-, publicize their operating margins as 6.9% and 4.6% in 2016, respectively [29, 18, 30]. Thus, the carriers have to design and operate their transportation networks efficiently and effectively while satisfying customer expectation perfectly to improve and sustain high operating margin.
Designing transportation networks is divided into three levels including strategic (i.e., facility location), tactical (i.e., distribution planning, allocation of demand points to facilities, resource planning), and operational planning (i.e., vehicle routing and scheduling). In this paper, we specifically focus on resource (asset) planning as well as routing and scheduling of assets in the planning horizon for a consolidation-based freight carrier. This process is known as designing of service network and accepted somewhere between the tactical and operational level of the planning process in transportation systems. From the operations research standpoint, service network design problems are the type of fixed charge capacitated multi-commodity network design problems with static or dynamic MIP formulations . Thereafter, asset positioning and balancing are also considered within service network design problems, since the carriers aim to decrease their operating costs and to increase the utilization of owned assets for operational purposes. This effort is known as ”full-asset-utilization” policy in transportation systems, which needs asset circulation continuously throughout the service network. Thus, ”design-balanced constraints” are introduced into service network design formulations to satisfy circulation of assets by imposing the number of assets entering and leaving a node must be balanced (e.g., 10, 16, 1, 3, 25).
The common assumption in the literature usually states that the current capacity of freight carrier’s assets is capable of handling all of the forecasted demand within the upcoming planning horizon (or it is assumed that an unlimited number of assets are available). As an exception to this, Crainic et al.  assumed that the number of resources is determined in advance. Besides Barnhart and Schneur , Andersen and Christiansen  and Lai and Lo  initially solved the problem with fixed resources but also proposed fleet composition as an extension. Since their formulation was able to compose the fleet mix as well when the right-hand side values of the regarding constraints were relaxed. Last, Crainic et al.  considered resource acquisition as a strategic decision due to the assumption is that resources are permanently acquired. None of the studies explicitly examined short-term resource acquisition in case of an urgent response such as a peak-demand period.
In contrast to the literature on SND problems, we consider a more realistic situation in which demand strictly exceeds the capacity of freight carrier’s assets in a certain period motivated by observations in practice. This period might be any peak season observed in pre-Thanksgiving (Black Friday), pre-Christmas and new year’s eve, and pre-school periods in the US. Regarding these facts and considered capacity assumption, we assume that the carrier takes three actions to be able to respond to peak demand. First, the carrier may choose to deliver the commodity late (demand shifting), which is observed frequently in practice. In the retailing sector, firms including Walmart, Amazon, Bestbuy, etc. offer to deliver items later than usual for free or may ask additional money to deliver them on time. Besides, we also would like to consider the early case and assume that the carrier would disperse (spread) the excess demand by picking up some of the commodities earlier than their release date (assuming it is possible). The carrier may also deliver some of the commodities later than their due date in exchange for earliness/tardiness fee such that existing assets would be able to handle all demand in the planning horizon.
Second, the carrier would reserve or immediately buy capacity from another service provider, which refers to outsourcing. The concept of outsourcing is usually observed in express shipment delivery, and air carriers prefer to pay for additional capacity from commercial passenger flights instead of operating an asset for certain routes within their transportation networks. In this paper, outsourcing is an option for the freight carriers, and there is a sufficient number of outsourced services available for any route on the network.
Third, the carrier would expand the capacity by acquiring (i.e., leasing) additional unit(s) of assets temporarily to compensate for the difference between demand and its capacity. Even though the cost of acquiring an additional unit of an asset is not low, the carriers prefer this option frequently to manage and operate their business with complete flexibility. This option corresponds to fleet sizing in the transportation sector.
Capacity scaling based on excess demand on a transportation network has not been addressed extensively. In this paper, we propose three possible actions such as demand shifting (early/late delivery), outsourcing and fleet sizing (or capacity expansion) that can be taken into account when demand strictly exceeds capacity for a short-term in the planning horizon for a consolidation-based freight carrier. All of these actions are temporary solutions for a peak demand period to get through an issue of capacity shortage. To emphasize the scope, the problem we study still belongs to the tactical level of planning for a transportation system. Among the considered actions above short-term fleet sizing and outsourcing has not been studied very well, demand shifting has never been considered. In contrast to our study, permanent resource acquisition has been addressed in Crainic et al. , however, the authors considered it as a long-term strategic decision and positioned their study such that strategic (resource acquisition) and tactical (vehicle routing and scheduling) level decisions are made jointly. Carriers usually generate a schedule weekly and repeat the same schedule for few consecutive weeks in the same month or quarter. Given that the peak period demand lasts as long as a month or five-six weeks, a carrier might need to acquire an asset temporarily for a short time period. From a decision-making perspective, it can be claimed that this acquisition is still within a tactical level of decisions, although  assumed resource acquisition as a long-term strategic decision in service network design. This paper aims to fill this gap in the literature as well as taking one of the alternative actions (i.e., demand shifting and outsourcing) against peak demand.
The contribution of this paper can be summarized in threefold. First, it recognizes explicit capacity shortage in case of observing peak demand periods ahead and introduces capacity scaling (or alternatively capacity management) in transportation networks as well as an arc-based formulation for the capacity scaling service network design (CSSND) problem. Second, two sets of valid inequalities (VIs) are generated for the proposed formulation to make the formulation more effective. The formulations with valid inequalities (VIs) outperform the regular formulation in obtaining tighter lower bounds. One set of VIs can improve the CPU time elapsed by 25% on medium-instances that can be solved optimally within the time limit. Third, we propose a multi-phase dedicate-merge-and-mix (DMaM) algorithm including a construction phase and three improving phases to solve CSSND problem.
The rest of the paper is organized as follows. Section § 2 reviews the literature and surveys of previous studies that focus on the service network design problem. In Section § 3, we introduce the capacity scaling problem in transportation networks within the service network design context and present the developed formulation. Section § 4 presents all computational experiments including solution of regular formulation, valid inequalities and a set of additional analysis. Proposed Dedicate-merge-and-mix algorithm (DMaM) that is specifically developed for the studied problem is explained and experimented on the test instances in Section § 5. Finally, we conclude this paper in Section § 6.
2 Literature Review
Service network design (SND) problems are a type of capacitated (fixed charge) multi-commodity network design problems and belong to the class of NP-Hard problems in terms of complexity. The reader can refer to a series of comprehensive reviews about further details pertaining to SND problems (e.g., 13, 16, 26). Besides, the reader should also refer to Magnanti and Wong  and Balakrishnan et al.  for extensive reviews about network design problems.
In reviewing the literature, we specifically focus on studies in which asset positioning and balancing are considered simultaneously while designing service networks. As stated in the introduction, asset balancing and ”design-balanced constraints” refer to the same problem feature, which are addressed in the literature broadly (e.g., 2, 3, 15, 21, 22, 23, 25, 27, 28).
The issue of asset management; alternatively, design-balanced constraints are one of the main topics for researchers who seek to increase the utilization of assets in transportation systems. Barnhart and Schneur , Kim et al. , Armacost et al. , and Barnhart et al.  are the earlier studies which consider asset management in service network design problems specifically in the applications of express shipment delivery. The scope of asset management in these studies includes balancing a number of fleets (or aircraft) and equipment at each node on the network. Thereafter, management of assets are generalized through design-balanced constraints, and the notation is introduced by Pedersen et al.  which studied a capacitated multi-commodity network design (CMND) problem and named it as Designed-balanced capacitated multicommodity network design (DBCMND) problem.
Among the studies which consider asset management, Pedersen et al.  developed generic arc- and cycle-based formulations for the DBCMND problem and proposed a tabu search metaheuristics to address the arc-based model for achieving computational efficiency. Andersen et al.  proposed both arc-based and path-based formulations accompanying with two more formulations by mixing arc/cycle design variables and arc/path flow variables for this problem, compare these formulations computationally. Chouman and Crainic  considered the same problem and proposed a cutting plane matheuristic with learning mechanism to combine an exact lower-bound computing method and a variable fixing procedure, which feeds a MIP solver. Bai et al.  studied service network design problem with asset balancing and proposed a guided local search metaheuristic within a multi-start framework. Bai et al.  extended this approach by studying a new neighborhood structure for getting a more effective solution strategy. Hewitt  dealt with human resources (drivers) in service network design considering daily driving limitations imposed by the Department of Transportation (DoT) or union regulations rather than scheduling equipment.
Moreover, Andersen et al.  firstly considered the multiple fleets case and fleet coordination in service network design to improve the integration of vehicle management and service network design concepts and introduced service network design with asset management and multiple fleet coordination problem (SNDAM-mFC). The authors addressed the synchronization between the collaborating services such as new (internal) and existing (external) services particularly observed in intermodal transportation where ferry services are assumed external services and fixed in terms of arrival/departure times and terminals. Likewise, Lai et al.  considered heterogeneous assets in multi-commodity network design and proposed a Tabu-search metaheuristic based on the decomposition of the problem.
proposed a three-step decomposition-based heuristic algorithm with the objective of profit maximization for a carrier. The step of the proposed heuristic includes network construction, selection of commodities and, vehicle routing and scheduling. Andersen et al. studied SNDAM problem and proposed firstly a branch-and-price (B&P) algorithm which is accompanied with a mechanism to add linear relaxation cuts dynamically and an acceleration technique for updating upper-bound. Vu et al.  presented a three-phase matheuristic in which an exact solver is combined with two heuristic methods. In the first two phases, the heuristic methods, namely tabu search and path relinking, generate as many feasible solutions as possible and reduce the problem size in the first two phases. Then, the exact solver searches on the restricted solution space and is able to solve large size of problem instances.
The researchers also specifically focused on variants of service network design problems which differentiate by transportation modes as well as applications such as express shipment delivery network design for air carriers (e.g., 20, 10, 5); rail services network design (e.g., 11, 31, 1). Lai and Lo  considered ferry service network design and ferry fleet management and applied their formulation and two-phase heuristic algorithm on the case of Hong Kong. Crainic and Sgalambro  studied urban-vehicle service network design problem as well as its variants for two-tier city logistics and proposed formulations for all considered cases. Zhu et al.  addressed the scheduled service network design problem for freight rail transportation and proposed a comprehensive formulation which integrates several core decisions belong to tactical planning process such as service selection and scheduling, car classification and blocking, train to make up and to flow of shipments.
All of the studies above presumes an unlimited number of resources available for service network planning; however, this leads to failure for the objective of maximizing resource utilization (”full-asset-utilization”) as emphasized earlier in §1. Andersen and Christiansen  decide how many assets (locomotives) to utilize in the operations of Polcorridor rail service network design. Lai and Lo  determine the number of ferries in operation that is limited by the maximum allowable fleet size for Hong Kong ferry service network design. Crainic et al.  is one of the papers which considers an SND with resource constraints based on the limited total number of resources that is determined in advance. In contrast to this, a total number of resources (or assets) utilized in the planning horizon is determined simultaneously while designing service network in our paper. Barnhart and Schneur  aimed to solve express shipment delivery network design problem for a fixed fleet, besides the proposed model is able to determine optimum fleet composition and size through releasing right-hand side values of available resource constraints. Even though fleet sizing is not addressed directly in the original problem scope, the proposed model is capable of doing that. The authors also determine the number of shipments delivered by commercial air in the same manner with the outsourcing option in that paper.
Finally, the most similar study to this paper in the literature is Crainic et al.  which considers strategic resource acquisition and allocation of resources in service network design. In this study, resources are acquired for permanently, and the acquisition cost is amortized over a series of periods. However, we assume that resources may be acquired temporarily such a leasing contract; thus the only cost of leasing incurs for the period in which the resource is acquired. The authors also take the outsourcing option into account such a way the resource is temporarily acquired from the third-party only for executing a particular service. On the other hand, the outsourcing option in our case does not include third-party’s resource acquisition and is not necessary to occupy the resource fully though, the outsourced volume of demand might be a partial load in third-party carrier’s resource. Note that the carrier in our paper cannot interfere with the schedule of third-party carrier’s resource, it assumed as given for the carrier. The authors named this problem as scheduled service network design with resource acquisition and management and proposed a cycle-based formulation as well as a matheuristic which combines column generation, slope scaling, intensification and diversification procedures, and exact optimization.
To highlight the main differences between our study and the aforementioned similar studies in the SND literature, we classified those studies based on problem features we consider in Table1.
|Paper||Excess demand||Fleet sizing||Outsourcing||Demand shifting|
|(capacity shortage)||(or acquisition)||(earliness/tardiness)|
|Barnhart and Schneur ||x||Not in problem scope||✓||x|
|Andersen and Christiansen ||x||✓||x||x|
|Lai and Lo ||x||✓||x||x|
|Crainic et al. ||x||Finite resources||x||x|
|Crainic et al. ||Implicitly||Long-term||✓||x|
To sum up, the contribution of our paper can be summarized such that we introduce capacity-demand balancing problem in service network design and specifically consider a planning horizon in which demand exceeds current capacity and the decision maker has to take action(s) among short-term resource acquisition (leasing), outsourcing or demand shifting (earliness/tardiness) to respond to the demand. To the best of our knowledge, fleet sizing and outsourcing have not been studied very well, besides demand shifting has never been considered before.
3 Problem Definition and Formulation
In service network design problems, a service can be defined as transportation of a commodity between its origin and destination with a determined capacity level and constant speed imposed by the asset (or resource in general) as well as known departure and arrival times. A set of selected services comprises a schedule for a fixed length of time (i.e., day, week, or month) and it is usually assumed that the determined schedule is cyclic meaning that it is being repeated for a certain period (i.e., season or year). To operate a service a resource is required, so an asset is assigned to a particular service during the schedule. To sum up, it can be stated that the solution of a service network design problem is briefly a schedule in which services are selected, and assets are assigned to those selected services, by doing that routes of the assets with their corresponding departure and arrival times and flow of commodities through the network within allowed time interval are determined.
Network design problems with scheduling decisions are studied on a special network that is called time-space network instead of a physical (or static) network to incorporate the time dimension into the problem compared to classical network design problems. A finite planning horizon is divided into the identical time periods, and for each period the nodes of the physical network are duplicated so that any movement on the physical network can be represented both spatially and temporally. Let and represent the set of nodes on the physical and time-space network, respectively, let denote the set of time periods. An illustration of a time-space network consists of five nodes and seven periods is given in Figure 1. Note that notation is selected in a way to be consistent with the formulations given in Andersen et al.  and Pedersen et al. .
The arcs on a time-space network represent activity and are classified into two groups, holding arcs and service arcs. A holding arc is directed from a period to another for the same location and represents only time-wise movement (see Figure 1). The time elapsed during loading/unloading of vehicles, the trailers or cars wait for a transfer to another truck or train might be represented by a holding arc. Thus, any resource is assumed busy on a holding arc within the commodity time-window (when loaded with the commodity) while it is idle when no commodity seizes it (out of commodity time-window). A service arc corresponds to transportation between two locations and the difference between periods of these locations shows the time elapsed during transportation activity. Repositioning of assets also occur via service arcs; however, no flow of commodities takes place in repositioning. In our case, the service arcs are also divided into two categories as arcs of offered services and arcs of outsourced services. Thus, let denote holding arcs, let and symbolize offered services and outsourced services arcs on the time-space network, respectively. We also denote the set of all arcs by on the time-space network, while the set includes all arcs on the physical network. Since the considered problem is capacitated, each arc except holding arcs has a capacity which is denoted by .
The distances between nodes on the physical network are defined in terms of periods and assumed to be an integer number denoted by . We also assume that distances between nodes on the physical network satisfy the triangle inequality. In addition to this, as it is discussed in Section 1 ”full-asset-utilization” policy results with circular routes which means that any asset must return to its starting node at the end of each cycle. To remind that this circulation is enforced through ”design-balanced” constraints. This issue has two results on the assumptions, the length of a route (or cycle) for any asset must be equal to the schedule length as  stated. We have the following observation for the second result:
The maximum value for the distance between any pair of nodes on the physical network must be less than or equal to half of the schedule length,; thus any asset is able to make a return trip to its starting node at the beginning of next cycle, or mathematically:
The reason behind of this observation is that an asset is supposed to be dispatched to its beginning node at the end of each planning horizon; thus any asset should complete any transportation activity in half of the time to be able to make a return-trip route in the worst case. Besides, this observation has also an effect on the frequency of a service provided between any pair of physical nodes. Given that the distance is half of the schedule length for any pair of nodes, then the service between these nodes can only occur once in each schedule.
As stated above services can be performed by a resource assuming that this is a single type asset and exactly one unit is assigned to any service consistently with the previous studies. Human resources such as crews and workers, transporting vehicles such as trucks, aircraft, rail-cars and ships, loading/unloading vehicles such as forklifts, cranes, and pallet jacks are examples of assets. In this paper, we also assume that assets are divided into two sets, owned assets and assets that can be acquired either through leasing or buying at the beginning of the planning horizon. Let represent owned assets assuming that this is a finite set from the realistic standpoint. The assets that can be acquired are symbolized by which is limited by a relatively high ; even though, there is no limit to acquire additional assets practically except a limited budget. The union of these two sets, , comprises the fleet which is assigned to services and routed through the network during the planning horizon.
Any amount of flow between any origin-destination pair demanded to transport is called a commodity and let represent the set of all commodities. Any element of is associated with a volume , an origin node and a destination node as well as a release date and a deadline in terms of the period index. Thus, let parameters and represent the origin and destination of a commodity on the time-space network. The transportation activity for any commodity may start at any time after the beginning of the period of the release date and must be completed before the end of the period of the deadline.
The option of earliness/tardiness affect all parameters regarding the commodities. We assume that any commodity can be delivered as much as one period earlier or later than its original deadline. To be able to incorporate this option into the formulation, we define two more commodities as dummy commodities which corresponds to each original commodity within . Dummy commodities represent early and late deliveries of the original commodity, and their origin and destination nodes are denoted by and on the time-space network. The time periods correspond to origin and destination of dummy commodities are and for the former and and for the latter. The combination of dummy commodities and the corresponding original commodity is called as transformed commodity and set of original commodities, , is replaced with this concept in the proposed problem formulation. Let represent a set of dummy commodities and let union set of represent the aggregated set of original and dummy commodities, namely transformed commodities. As indicated implicitly, the union set of includes three times more element than the set of original commodities, , since there are three transformed commodities per original one in the problem.
The original and transformed commodities are related to each other through an incident matrix, which equals to 1, if transformed commodity is incident to original commodity ; 0, otherwise. With consideration of transformed commodities, the volume of a transformed commodity is denoted by , and the corresponding type is represented by either as early, original or tardy.
The operating cost of such a service network is considered in three parts. The first part includes fixed cost of operating an asset in the planning horizon, for which we define and to represent the fixed cost of operating an owned asset and a leased or an acquired asset, respectively. All costs associated with the crew, depreciation, etc. are assumed to be included in the fixed cost of operating assets. The routing cost of commodities on the network is involved in the second part. Let denote variable cost of routing transformed commodity on any service arc including offered and outsourced services to cover fuel cost en route and handling at the terminals. The final part consists of penalty for delivering commodities early and tardy, a fee denoted by and incur for each unit of flow picked up early or delivered late, respectively.
In addition to routing of assets and flow of commodities, as defined earlier the capacity-demand balancing problem in service network design for a consolidation-based freight forwarder in this paper also includes the following decisions: (1) determine whether to add additional assets into the fleet, if that is the case, decide routes of them, (2) whether to pick up any commodity earlier or deliver any commodity late, in that case which commodities are delivered early/late, and (3) whether to choose an outsourced service in case of not choosing other options to handle the excess demand and decide which commodities are routed through selected outsourced services.
For the sake of readers’ convenience, all the notation given so far are also presented in Table 2.
|Physical (static) network.|
|Set of nodes on the time-space network.|
|Set of nodes on the physical network.|
|Set of holding arcs on the time-space network.|
|Set of arcs representing services on the time-space network.|
|Set of arcs representing outsourced services on the time-space network.|
|Set of all arcs on the time-space network.|
|Set of all arcs on the physical network.|
|Set of time periods.|
|Set of owned assets.|
|Set of assets that can be acquired as additional units.|
|Set of all assets such that .|
|Set of original commodities.|
|Set of dummy commodities representing early and tardy versions of|
|each corresponding original commodity.|
|Joint set of transformed commodities.|
|Distance between node and node in terms of number of periods|
|on the physical network.|
|Volume of commodity that needs to be transported.|
|Volume of transformed commodity that needs to be transported.|
|Period index of node .|
|Origin node of commodity on static network, .|
|Origin node of commodity on time-space network, .|
|Origin node of transformed commodity on time-space network,|
|Destination node of commodity on static network, .|
|Destination node of commodity on time-space network, .|
|Destination node of transformed commodity on time-space network,|
|Cost of transporting one unit of transformed commodity on arc .|
|Fixed cost of operating a unit of asset.|
|Fixed cost of acquiring/leasing and operating an additional unit of asset.|
|Capacity of service operated on arc .|
|Additional parameter defined to obtain stronger formulation.|
|Type of transformed commodity either as early, original or tardy.|
|Penalty for cost of transporting one unit of dummy commodity early/tardy.|
|1, if transformed commodity is incident to original commodity ;|
|1, if transformed commodity might be in transit in period ;|
In order to capture all decisions regarding capacity-demand balancing problem on a service network design, we propose the following sets of decision variables:
|=||Amount of flow of transformed commodity routed on arc .|
Depending on given notation and defined decision variables, capacity-demand balancing problem on service network design can be formulated as follows:
The objective function (2) accounts for the total cost over the planning horizon including the following five terms: (i) fixed cost of using owned assets, (ii) fixed cost of acquiring and using additional assets, (iii) cost of transporting commodities on offered services, (iv) cost of transporting commodities on offered services earlier or later with a penalty for earliness/tardiness and, (v) cost of transporting commodities on outsourced services respectively.
Constraints (3) are added into the formulation in this paper and do not exist in the formulations proposed in the SND literature so far. The reason for adding these depend on our observation in the preliminary analysis, and we aim to prevent an anomaly that may exist in commodity flows, which is illustrated in Figure 2. This anomaly rarely occurs in a few instances for few commodities but violates the overall feasibility of the problem literally. To refer it simply, we prefer to call it as 2-weeks anomaly. As indicated in Figure 2, the transformed commodity (original-type) shown on the left network in the figure (originated at the sixth period at node-1 and destined to the fourth period at node-2) is delivered in almost two weeks from its release date. When we add the constraint set (3), 2-weeks anomaly is eliminated from the solution and the tardy-type transformed commodity (originated at the seventh period at node-1 and destined to the fifth period at node-2) that is incident to the same original commodity is delivered instead as shown on the right network in the figure. To be able to formulate this constraint set, we define a new parameter () which takes the value of one for the period when transformed commodity would be in transit in a feasible solution based on commodity’s time-window. Thus, we are able to restrict the values of decision variable of commodity flows () to zero on an arc that takes place when the commodity is supposed to be not in transit.
Constraints (4) assure that in each period, an owned or added asset must be assigned to only one activity (holding or service) if it is utilized.  stated that this constraint imposes a maximum route-length requirement for a schedule. Even though, this constraint set is valid theoretically, the particular condition given under sum notation makes impossible to write it down mathematically due to few arcs wrapping around the planning horizon. We propose to replace constraint set (4) by (17)-(19) based on new sets defined in Table 3.
|Holding/service arcs that do not wrap around the planning horizon (regular), .|
|Holding/service arcs that DO wrap around the planning horizon (circular), .|
|Arcs of outsourced services, that do not wrap around the planning horizon (regular), .|
|Arcs of outsourced services, that DO wrap around the planning horizon (circular), .|
|Time periods which are only visited by regular holding/service arcs, i.e. .|
|Time periods which exist in circular arcs as origin nodes, i.e. .|
|Time periods which exist in circular arcs as destination nodes, i.e. .|
The balance of assets are satisfied through constraints (5) in which the incoming number of assets must be equal to outgoing assets for each node on the time-space network. Constraints (6) enforce that only one asset is assigned to a service at most. Constraints (7) ensure that at least one of the transformed commodity which is associated with the original correspondent commodity must be selected to be delivered. Since the fact that the objective is minimized, only one of a transformed commodity out of three will be selected for delivery, obviously. Inequalities (8) are flow balance constraints for the transformed commodities in case of the corresponding one is selected to be delivered. Otherwise, this set of constraints become redundant for two out of three transformed commodities that are incident to a particular original commodity. Constraints (9) and (10) are the weak and strong capacity and forcing constraints for the arcs of offered services and holding arcs, respectively. The flow on the arcs of outsourced services is restricted through constraints (11) if the corresponding arc is selected. Note that, strong version of capacity and forcing constraint (10) is redundant for MIP formulation as  stated as well. Finally, (12)-(16) are domain constraints.
3.1 Valid Inequalities
To enhance the performance of the formulation, we also propose few sets of valid inequalities which were discovered during the preliminary analysis of the problem instances. For this reason, we would like to explain the idea behind the generation of valid cuts before presenting them in this section.
To clarify the idea perfectly, we prefer to apply the procedure on a sample instance for illustration. Table 8 presents the values for commodity-related parameters of the ten original commodities (OCs) with five nodes and seven periods. The number of owned and acquirable assets are seven and five, respectively. The other details of the test instances will be explained later in Subsection of Instance Generation (§4.1).
In Table 8, the second and third column show the origin and destination node of the corresponding OC () on the time-space network. The volume of each commodity is presented in the fourth column. The origin and destination node on the physical network, and release and due date of any commodity in terms of periods are given in the fifth through eighth columns. For instance, commodity-1 is originated at node-2 on the physical network in the second period and destined to node-1 with a due date of the fifth period. The same information for these parameters is also presented regarding transformed commodities (TCs) () in Table 8 included with the type parameter of each transformed commodity in the fifth column.
The time window of each original commodity based on corresponding release times and due dates () are mapped on a week-based tabular format in Table 8 for better understanding because the main idea behind the valid inequalities comes from this mapping. Any commodity is expected to be assigned to any asset (resource) in a period where a value of one appears in the corresponding column. The tabular format is enhanced by considering transformed commodities in Table 8 where only nine transformed commodities which correspond to original commodities 1, 2, and 3, are listed in each three-lines row as a sample. In the wide top row, the first line shows the mapping of time window belongs to transformed commodity-1 which is the early version of original commodity-1 on a week. The second row (transformed commodity-2) corresponds the original commodity itself, and the third row is the tardy version of original commodity-1. The columns in which the ones are typed in bold depict the intersecting time-periods of three transformed commodities; thus the corresponding original commodity occupies a resource in these periods no matter of which one of them is selected for delivery. When we apply the same mapping for all original commodities and add all ones in the same column up, we can predict the number of assets utilized in each period to transport all commodities (see Table 8).
Based on this discussion, the additional few parameters are defined for the formulation of the valid inequalities and presented in Table 9.
|Number of assets that needs to be utilized for period within the planning horizon.|
|Minimum number of assets that needs to be utilized for the entire planning horizon.|
Regarding the idea of predicting the number of assets utilized from the period-based mapping, we propose two sets of valid inequalities based on binary decision variables and given additional parameters as follows:
In the first set of inequalities, we aim to put a lower bound on the number of assets required from the perspective of the minimum number of resources. Inequality (20) gives a loose lower bound on the number of assets that need to be utilized in the optimal solution by forcing it with the minimum predicted number of assets over all periods. The second set of valid inequalities (21) are focused on obtaining a lower bound for a total number of transportation resources required as either owned assets, acquired assets or outsourcing in each period in the same manner.
3.2 Additional Constraints for Near-optimal Solutions
In the preliminary analysis, some of the constraints we generated through the procedure that is discussed in the subsection of Valid Inequalities (§ 3.1) are not always valid, however, these constraints can be used to produce near-optimal solutions in comparably shorter computational times. We also propose these time-saving constraints in addition to valid inequalities in this paper. Additional time-saving constraints for near-optimal solutions are as follows:
Constraints (22) and (23) are proposed to obtain a tight lower bound for the total number of assets utilized in the planning horizon based on maximum predicted number of assets. Even though these constraints are valid in more than half of the valid-cut test instances separately, they do not work for few instances. However, we observe that these constraints are able to decrease the CPU time considerably while diverging reasonably from the optimal solution.
In the third set of additional inequalities (24
), we consider two binary variables representing the total number of resources utilized either as own fleet (owned and leased) or as outsourced service and, enforce them to be greater than or equal to maximum predicted number of assets over all time periods. Likewise, this inequality provides a tight lower bound for number of resources and is more successful than the former ones in hitting optimal solutions in average.
4 Computational Experiments
The performance of the proposed arc-based model is tested on randomly generated problem instances for which we dedicate the following subsection (§ 4.1) and describe the details of the generation process as well as values of problem parameters. The formulation is coded on IBM ILOG CPLEX Optimization Studio and solved with CPLEX 12.6.1. All computational experiment is run on a 64-bit PC with Intel Core i7 CPU 3.40 GHz and 16GB RAM.
In the following subsections, we first describe the instance generation and selected values for the cost parameters in § 4.1. Then, we present the optimal solutions of 60 instances as well as our comments on obtained results in § 4.2. Moreover, we implement a particular sensitivity analysis on some selected small-size instances to point out the importance of cost parameters on optimal solutions. We continue testing the problem instances with some operational restrictions that might be observed in practice in terms of demand shifting. We report the outcomes of these experiments in § 4.3 and § 4.4, respectively. Finally, we report the results of the experiment on the impact of valid inequalities by testing medium-size instances in § 4.5.
4.1 Instance Generation
The problem instances for considered CBSND problem are described through the size of the physical network (), number of periods () in the planning horizon, number of commodities (), number of owned () as well as acquirable assets (), and number of arcs () in the data. Thus, the generation process begins with determining the size of the physical (or static) network and distances between all pair of static nodes. Besides, selected values for all sets and parameters of the generated instances that are tested in the computational experiments are summarized in Table 10.
|Number of static nodes||5||6||7||10+|
|Number of time periods||7|
|Number of original commodities||10, 15, 20||20, 25, 30||30, 36, 42||72+,81+,90+|
|Number of owned assets||7||12||15||35+|
|Number of acquirable assets||5||7||10||15+|
|Number of arcs|
|Duration of service as distance|
|Capacity and Flow Parameter:|
|Service arc capacity|
|Holding arc capacity|
|Volume of flow|
|Fixed cost of owned assets||25|
|Fixed cost of acquired assets||50|
|Routing cost on offered services arcs|
|Routing cost on arcs of outsourced services|
|Routing cost on offered services arcs|
|Penalty multiplier for earliness/tardiness||1.2|
Generated instances are classified as the size of small, medium, large and very large based on the number of nodes in . The instances with five static nodes are categorized as ’small’, the ones with six nodes as ’medium’, and the instances with seven nodes as ’large’ and the instances having more than ten nodes are classified as ’very large’ in testing. The planning horizon is assumed to be week-long; thus it consists of seven identical periods, each corresponds a day in the week. As stated earlier, this is a sample period within one of the peak seasons throughout a year. The demand is defined as a number of commodities that needs to be transported in the planning horizon concerning given release dates and deadlines. We tested three different cases in which the number of commodities is increased gradually to simulate a peak season demand more effectively. The heaviest demand case in each size of the problem is limited with the assumption of only one commodity between any O-D pair. For instance, there would be 20 commodities () in small-size problems in the maximum demand case. The amount of flow for each commodity is determined as unit flow without loss of generality. To be consistent with this assumption, the capacity of all service arcs is also selected as one unit in all instances. However, to remind that we assume no capacity on holding arcs to be consistent with the literature. The capacity of outsourced services is imposed by the other service provider and able to handle all volume on that route.
The overall capacity of the carrier is defined through how many owned assets are available for the upcoming planning horizon. There is also an option of acquiring (or leasing) assets to compensate for the capacity shortage in this problem. The number of owned and acquirable assets are determined based on problem size and indicated in Table 10. The number of arcs depends on the number of nodes in the physical network, the number of periods in the planning horizon, and the number of outsourced services provided by the market.
The operating cost parameters of the assets are determined in favor of owned assets because the primary objective of the carrier is to maximize the utilization of owned resources. Any other settings for cost parameters would conflict with ”full-asset-utilization” policy that is one of the bases of this paper. The cost parameters are selected such that fixed and acquiring (leasing) cost are assumed as and , respectively, while routing one unit of flow incurs a random transportation cost of and for offered and outsourced services for each arc and commodity combination, respectively. The holding cost is assumed constant and chosen as 0.15 for all holding arcs while the penalty of delivering commodities early and tardy are assumed equal and are determined as a constant ratio of transportation cost as .
As stated earlier, we assume that a single type of asset is used for transporting commodities, so the speed of all services is the same. For this reason, the duration of each service between any pair of nodes on the physical network is interpreted as the distance between those static nodes. In addition to this, the duration of any service cannot be longer than half-length of the planning horizon based on the ”full-asset utilization” policy that requires asset balancing. To remind that, the number of assets leaving and entering a node must be balanced, and this also imposes that each particular asset must return to its starting node at the beginning of each schedule. As proposed in Corollary 1, the maximum distance value cannot be more than half of the schedule length. Since we consider a week-long planning horizon, the distance matrix might include values of 1, 2, and 3 in all generated instances.
Moreover, we also consider the overall network topology in terms of distance in our analysis. We aim to provide additional insights about how the proximity of nodes on a network affects the MIP formulation both computationally (CPU time) and operationally (total operating cost). Thus, we propose the concept of distance index and categorize generated instances based on this value either close-range (CR), medium-range (MR), or long-range (LR) instances. To illustrate this concept, we arbitrarily select ten points which spread over three distinct regions as presented in Figure 3 and 4. The ten nodes from three states in the US Southeast Region displayed in Figure3(a) (Miami (FL), Tampa (FL), Orlando (FL), Atlanta (GA), Columbus (GA), Savannah (GA), Birmingham (AL), Mobile (AL), Montgomery (AL)) compose a close-range network. The network in Figure 3(b) corresponds a mid-range network which includes another ten nodes spread over entire US Southeast Region (Miami (FL), Atlanta (GA), New Orleans (LA), Washington (DC), Louisville (KY), Charlotte (NC), Nashville (TN), Birmingham (AL), Rogers (AR), and Charleston (SC)). A long-range network may comprise of ten nodes from all around US (Miami (FL), Seattle (WA), Los Angeles (CA), Chicago (IL), Houston (TX), Boston (MA), Kansas City (MO), Atlanta (GA), Denver (CO), Anchorage (AK)) as depicted in Figure 4.
The categorization itself is a comparative fact; indeed, a close-range network in a context might be an example of a long-range network in another one, i.e., nodes selected around South Florida might be a close-range network, nodes around Florida might be a mid-range network and nodes displayed in Figure 3(a) might be a long-range network. However, the comparative feature of this concept does not prevent distance index approach from analyzing in our experiments.
Without loss of generality, we apply the following rule to categorize the generated instances properly. The range between the minimum possible total distance value (i.e., 20 for small-size instances and 42 for medium-size instances) and the maximum possible total distance value (i.e., 60 for small-size instances and 90 for medium-size instances) is divided into three intervals, each of them approximately takes one-third of the total range. Instances fall within the first interval are assumed as close-range, the ones fall within the second interval are presumed medium-range, and the instances fall within the third interval is assumed as long-range instances. The total distance value and the distance index of each instance belong to the set of small and medium size are presented in a two-fold table below (Table 11) in columns three and four. For the sake of simplicity in referring any instance across our paper, we assign a global ID number to each instance which is also shown in column two in each corresponding part.
Last, we determine a time limit of one hour for small-size instances and four hours for medium-size or above in regular model runs as well as runs with valid inequalities in all experiments.
4.2 Solutions of the Regular Model
In the first part of the computational experiment, we tested small (instances with 5-nodes physically) and medium (instances with 6-nodes physically) size instances with the regular model and presented the optimal solutions in Table 12 and Table 13, respectively. The first five columns provide information about instances’ characteristics including instance name, global instance ID across the paper, number of nodes in the static network, number of original commodities, and number of owned and acquirable (leasable) assets. The optimal solution for each instance is given starting from column-7 through column-12. The number of owned assets utilized and the number of assets leased in each corresponding instance are shown in columns 7 and 8, respectively. In columns 9 through 11, we present the number of commodities delivered on time, that of demand shifted to an earlier/a later date, and that of commodities delivered via an outsourced service on time, respectively. Finally, we present total operating cost obtained for each problem in column 12.
|Instance||ID||Problem Size||# of Assets||# of Commodities||Total|
In the first place, the number of assets utilized increases as the demand increases in both small and medium size problems, that is an expected outcome, obviously. Although increasing trend can be observed in each group of the instance, the number of assets utilized varies across the group of instances due to differences in topology on the static networks. For example, all of the owned assets are utilized in 16 out of 30 small-size instances in which the capacity shortage is overcame either by outsourcing (in instances 2, 6, 9, 17, and 26), or by leasing additional asset (in instances 21 and 30), or by both (in instances 3, 18, and 27), or only by demand shifting (in instances 5, 8, 12, 15, 16, 25). All these instances but one (instance-15) are classified either as long-range or medium-range cases based on physical networks. In one-third of medium-size instances all owned assets are completely utilized (instances 32, 33, 42, 51, 52, 53, 54, 58, 59, and 60), outsourcing (instances 53 and 56) and leasing additional asset(s) (instances 33, 53, and 54) are chosen as an option for handling excess demand. Again all of these medium-size instances belong to either long-range or medium-range categories in terms of physical network distance index.
To compare the asset need between long-range and close-range physical networks, the total number of assets utilized is as low as four assets (instance-22) in close-range cases while this value goes up to eight and nine assets in instances 3 and 18 in the long-range counterparts. The maximum total number of assets utilized is doubled, and the minimum number is not lower than six assets across all long-range small-size instances. Thus, the results confirm that asset utilization is directly affected by the distances between nodes on the physical network. Relatively less number of assets are utilized when the static nodes are getting closer, a substantial amount of resource is required to meet the transportation demand when the distances between nodes are getting higher.
To examine the optimal results in terms of response actions, the model chooses to add an asset(s) into the fleet in five out of 30 small-size and three out of 30 medium-size instances. Besides, the outsourcing option is selected in eight out of 30 small-size and three out of 30 medium-size instances for 15 and five commodities in total, respectively. The outsourcing option is mostly selected to deliver commodities in long-range problems (ten commodities); however, a few commodities are outsourced in mid-range cases (five commodities) in the set of small-size instances, as well. This outcome is consistent in medium-size instances, four outsourced commodities are included in demand of long-range networks.
|Instance||ID||Problem Size||# of Assets||# of Commodities||Total|
Moreover, the option of demand shifting seems very attractive based on current cost values, some portion of demand is shifted either at an earlier or a late period in all of the small and medium size instances. The total number of commodities shifted is greater than or equal to that of delivered on time in 23 out of 30 small size instances, whereas this statistics is increased to 25 among medium-size instances.
The similar trends are observed in total operating cost values of the test problems. It increases as the number of commodities increases as well in each instance group. The second observation is that total operating cost is strongly affected by the topology of the physical network, it is more likely to increase when the distances are getting longer since longer distances mean more resources to utilize in transportation activities.
The reason why the mathematical model mostly chooses the outsourcing option rather than adding a leased resource into the fleet is that the balance between values of the cost parameters of options in our opinion. Thus, we decided to conduct a particular test with a portion of small-size instances in § 4.3 and § 4.4 in the next two subsections to be able to illustrate that how cost parameters are effective on the choice of options to meet the excess demand. We intentionally left the discussion of computational time in §4.5 where we compare the performance of regular formulation against valid inequalities.
4.3 Impact of Cost Parameters on Solutions
As mentioned earlier, it is observed that cost parameters of the CSSND problem are very effective on optimal solutions in the way of choosing the option(s) to handle capacity shortage (or excess demand) issue in our computational experiment.
To emphasize the importance of selecting cost parameters properly as well as to prove the flexibility of the mathematical model, we conduct a particular cost analysis on some selected small-size instances in which owned assets are fully utilized. This experiment is designed as a ceteris paribus experiment and consists of two parts: (1) change in leasing (acquiring) cost and (2) change in penalty ratio of earliness/tardiness. The selected instances with global IDs as well as optimal solutions for each instance that belong to corresponding part of the experiment are shown in Table 14 and 15, respectively.
First, we tested three values of around of the fixed cost of utilizing owned assets and the cost of outsourcing option such as , and , and compared the results with the default case. The optimal solutions are presented in Table 14 and it is structured in 2x2 wide rows and columns in which optimal decisions of the problem are summarized with respect to chosen value of given on top of that column including the default settings. The first two columns identify the instance name and ID, the next three columns (columns 3 through 5) show number of assets utilized in the corresponding instance as owned, leased and sum of them in order. Columns 6 through 9 display the number of commodities delivered on time, that of demand delivered either as early/late and via an outsourced service. Finally, the total operating cost of each instance is presented in the last column. This structure is repeated in the second wide column and row, as well.
When the leasing cost is as low as utilizing an owned asset (), the model acts in favor of leasing additional assets among the three response actions. Additional assets are leased in all instances but instances 21 and 30, the number of utilized assets increases by 17% in average across ten instances compared to default case. The total number of commodities delivered by an outsourced service decreases from 15 to zero, while the number of on time deliveries increases by 29% (from 63 to 81). The total number of shifted commodities has a small change and decreases by 2.8% (from 107 to 104).
|# of Assets||# of Commodities||Total Cost||# of Assets||# of Commodities||Total Cost|