In today's competitive business world, companies face the crucial task of delivering goods on time. In order to benefit from high volumes, distributors (suppliers) try to consolidate deliveries while maintaining a high percentage of on time delivery. Increasing delivery frequencies typically increases suppliers' performance, but it also increases the transportation costs. Finding a suitable middle ground is a challenge confronting many companies throughout various supply chains. This dissertation is a study about the coordination of inventory and transportation decisions in a two-stage supply chain.

This thesis explores three consolidation problems derived from the daily operations of major international airfreight forwarders. First, we study the freight forwarder's unsplittable shipment planning problem in an airfreight forwarding network where a set of cargo shipments have to be transported to given destinations. We provide mixed integer programming formulations that use piecewise-linear cargo rates and account for volume and weight constraints, flight departure/arrival times, as well as shipment-ready times. After exploring the solution of such models using CPLEX, we devise two solution methodologies to handle large problem sizes. The first is based on Lagrangian relaxation, where the problems decompose into a set of knapsack problems and a set of network flow problems. The second is a local branching heuristic that combines branching ideas and local search. The two approaches show promising results in providing good quality heuristic solutions within reasonable computational times, for difficult and large shipment consolidation problems. Second, we further explore the freight forwarder's shipment planning problem with a different type of discount structure - the system-wide discount. The forwarder's cost associated with one flight depends not only on the quantity of freight assigned to that flight, but also on the total freight assigned to other flights operated by the same carrier. We propose a multi-commodity flow formulation that takes shipment volume and over-declaration into account, and solve it through a Lagrangian relaxation approach. We also model the "double-discount" scheme that incorporates both the common flight-leg discount (the one used in the unsplittable shipment problem) and the system-wide discount offered by cargo airlines. Finally, we focus on palletized loading using unit loading devices (ULDs) with pivots, which is different from what we assumed in the previous two research problems. In the international air cargo business, shipments are usually consolidated into containers; those are the ULDs. A ULD is charged depending on whether the total weight exceeds a certain threshold, called the pivot weight. Shipments are charged the under-pivot rate up to the pivot weight. Additional weight is charged at the over-pivot rate. This scheme is adopted for safety reasons to avoid the ULD overloading. We propose three solution methodologies for the air-cargo consolidation problem under the pivot-weight (ACPW), namely: an exact solution approach based on branch-and-price, a best fit decreasing loading heuristic, and an extended local branching. We found superior computational performance with a combination of the multi-level variables and a relaxation-induced neighborhood search for local branching.

Shipment consolidation is a logistics strategy whereby many small shipments are combined into a few larger loads. The economies of scale achieved by shipment consolidation help in reducing the transportation costs and improving the utilization of logistics resources. The fundamental questions about shipment consolidation are i) to how large a size should the consolidated loads be allowed to accumulate? And ii) when is the best time to dispatch such loads? The answers to these questions lie in the set of decision rules known as shipment consolidation policies. A number of studies have been done in an attempt to find the optimal consolidation policy. However, these studies are restricted to only a few types of consolidation policies and are constrained by the input parameters, mainly the order arrival process and the order weight distribution. Some results on the optimal policy parameters have been obtained, but they are limited to a couple of specific types of policies. No comprehensive method has yet been developed which allows the evaluation of different types of consolidation policies in general, and permits a comparison of their performance levels. Our goal in this thesis is to develop such a method and use it to evaluate a variety of instances of shipment consolidation problem and policies. In order to achieve that goal, we will venture to use matrix-analytic methods to model and solve the shipment consolidation problem. The main advantage of applying such methods is that they can help us create a more versatile and accurate model while keeping the difficulties of computational procedures in check. More specifically, we employ a discrete batch Markovian arrival process (BMAP) to model the weight-arrival process, and for some special cases, we use phase-type (PH) distributions to represent order weights. Then we model a dispatch policy by a discrete monotonic function, and construct a discrete time Markov chain for the shipment consolidation process. Borrowing an idea from matrix-analytic methods, we develop an efficient algorithm for computing the steady state distribution of the Markov chain and various performance measures such as i) the mean accumulated weight per load, ii) the average dispatch interval and iii) the average delay per order. Lastly, after specifying the cost structures, we will compute the expected long-run cost per unit time for both the private carriage and common carriage cases.

This dissertation focuses on the recent supply chain initiatives, such as Vendor Managed Inventory (VMI) and Third-Party Logistics (3PL), enabling the coordination of supply chain entities; e.g., suppliers, buyers, and transporters. With these initiatives, substantial savings are realizable by carefully coordinating inventory, transportation, and pricing decisions. The impact is particularly tangible when the transporter's role and the transportation costs are explicitly incorporated into decision mechanisms that aim to coordinate the supply channel. Furthermore, expanding the perspective of channel coordination by introducing the transporter as an individual party in the channel provides tangible benefits for each member of the channel. Recognizing the need for further analytical research in the field of multi-echelon inventory and channel coordination, we developed and solved a class of integrated inventory and transportation models with explicit shipment consolidation considerations. Moreover, we examined transporter-buyer and supplier-transporter-buyer channels and solved centralized and decentralized models for these channels with the aim of investigating the impact of transporters on channel performance. In this dissertation, we also developed efficient coordination mechanisms between the transporter and the other parties in the channel.

Recent supply chain literature and practice recognize that significant cost savings can be achieved by coordinating inventory and transportation decisions. Although the existing literature on analytical models for these decisions is very broad, there are still some challenging issues. In particular, the uncertainty of demand in a dynamic system and the structure of various practical transportation cost functions remain unexplored in detail. Taking these motivations into account, this dissertation focuses on the analytical investigation of the impact of transportation-related costs and practices on inventory decisions, as well as the integrated inventory and transportation decisions, under stochastic dynamic demand. Considering complicated, yet realistic, transportation-related costs and practices, we develop and solve three classes of models: (1) Pure inbound inventory model impacted by transportation cost; (2) Pure outbound transportation models concerning shipment consolidation strategy; (3) Integrated inbound inventory and outbound transportation models. In broad terms, we investigate the modeling framework of vendor-customer systems for integrated inventory and transportation decisions, and we identify the optimal inbound and outbound policies for stochastic dynamic supply chain systems. This dissertation contributes to the previous literature by exploring the impact of realistic transportation costs and practices on stochastic dynamic supply chain systems while identifying the structural properties of the corresponding optimal inventory and/or transportation policies. Placing an emphasis on the cases of stochastic demand and dynamic planning, this research has roots in applied probability, optimal control, and stochastic dynamic programming.

Shipment consolidation is a logistics strategy whereby many small shipments are combined into a few larger loads. The economies of scale achieved by shipment consolidation help in reducing the transportation costs and improving the utilization of logistics resources. The fundamental questions about shipment consolidation are i) to how large a size should the consolidated loads be allowed to accumulate? And ii) when is the best time to dispatch such loads? The answers to these questions lie in the set of decision rules known as shipment consolidation policies. A number of studies have been done in an attempt to find the optimal consolidation policy. However, these studies are restricted to only a few types of consolidation policies and are constrained by the input parameters, mainly the order arrival process and the order weight distribution. Some results on the optimal policy parameters have been obtained, but they are limited to a couple of specific types of policies. No comprehensive method has yet been developed which allows the evaluation of different types of consolidation policies in general, and permits a comparison of their performance levels. Our goal in this thesis is to develop such a method and use it to evaluate a variety of instances of shipment consolidation problem and policies. In order to achieve that goal, we will venture to use matrix-analytic methods to model and solve the shipment consolidation problem. The main advantage of applying such methods is that they can help us create a more versatile and accurate model while keeping the difficulties of computational procedures in check. More specifically, we employ a discrete batch Markovian arrival process (BMAP) to model the weight-arrival process, and for some special cases, we use phase-type (PH) distributions to represent order weights. Then we model a dispatch policy by a discrete monotonic function, and construct a discrete time Markov chain for the shipment consolidation process. Borrowing an idea from matrix-analytic methods, we develop an efficient algorithm for computing the steady state distribution of the Markov chain and various performance measures such as i) the mean accumulated weight per load, ii) the average dispatch interval and iii) the average delay per order. Lastly, after specifying the cost structures, we will compute the expected long-run cost per unit time for both the private carriage and common carriage cases.

This thesis explores three consolidation problems derived from the daily operations of major international airfreight forwarders. First, we study the freight forwarder's unsplittable shipment planning problem in an airfreight forwarding network where a set of cargo shipments have to be transported to given destinations. We provide mixed integer programming formulations that use piecewise-linear cargo rates and account for volume and weight constraints, flight departure/arrival times, as well as shipment-ready times. After exploring the solution of such models using CPLEX, we devise two solution methodologies to handle large problem sizes. The first is based on Lagrangian relaxation, where the problems decompose into a set of knapsack problems and a set of network flow problems. The second is a local branching heuristic that combines branching ideas and local search. The two approaches show promising results in providing good quality heuristic solutions within reasonable computational times, for difficult and large shipment consolidation problems. Second, we further explore the freight forwarder's shipment planning problem with a different type of discount structure - the system-wide discount. The forwarder's cost associated with one flight depends not only on the quantity of freight assigned to that flight, but also on the total freight assigned to other flights operated by the same carrier. We propose a multi-commodity flow formulation that takes shipment volume and over-declaration into account, and solve it through a Lagrangian relaxation approach. We also model the "double-discount" scheme that incorporates both the common flight-leg discount (the one used in the unsplittable shipment problem) and the system-wide discount offered by cargo airlines. Finally, we focus on palletized loading using unit loading devices (ULDs) with pivots, which is different from what we assumed in the previous two research problems. In the international air cargo business, shipments are usually consolidated into containers; those are the ULDs. A ULD is charged depending on whether the total weight exceeds a certain threshold, called the pivot weight. Shipments are charged the under-pivot rate up to the pivot weight. Additional weight is charged at the over-pivot rate. This scheme is adopted for safety reasons to avoid the ULD overloading. We propose three solution methodologies for the air-cargo consolidation problem under the pivot-weight (ACPW), namely: an exact solution approach based on branch-and-price, a best fit decreasing loading heuristic, and an extended local branching. We found superior computational performance with a combination of the multi-level variables and a relaxation-induced neighborhood search for local branching.