Designing fee tables for retail delivery services by third-party logistics providers

Manufacturers are increasingly relying on third-party logistics service providers to distribute their products to retail stores. Fee tables, specifying how much to pay for each delivery based on weight and distance, are commonly used as the basis for compensating distributors for their delivery services. This article proposes and solves an optimization model to help a large building products manufacturer design an appropriate fee table for payments to its distributors for delivering products from regional distribution centers to retail stores. Given the distance and the distribution of shipment weights to each store served by every distribution center, the model selects the weight and distance ranges of the fee table and sets the fees for each combination of ranges to minimize total distribution costs while satisfying fee structure requirements and ensuring adequate total compensation for each distributor. Since the problem is difficult to solve using commercial solvers, we develop a tailored approach to obtain near-optimal solutions quickly by adding valid inequalities to strengthen the model formulation and using an optimization-based procedure to generate a heuristic solution. When applied to actual data from the building products manufacturer, our Composite solution method, combining cutting planes and heuristic, was effective (yielding solutions that are within 1% of optimality) and generated substantial savings (of nearly 10%) over the current fee table.


Introduction
As retail supply chains have grown and evolved, manufacturers are increasingly relying on third-party logistics (3PL) providers to deliver product replenishments to customer locations. This trend is evident in the dramatic growth of the 3PL industry in the U.S. from $56 billion in 2000 to an estimated $142.2 billion in 2012. In 2010, more than 82% of Fortune 500 companies used 3PL services (Armstrong and Associates, 2012). Such partnerships permit manufacturing firms to focus on their core competencies while assuring high service levels to retail stores. These collaborative arrangements raise the important issue of how much the manufacturer should pay the 3PL operators for their delivery services. This article stems from a project to address this issue for a leading building products manufacturer.
The manufacturer sells its products through national hardware and home improvement retail chains that operate thousands of stores nationwide and require periodic deliveries of product replenishments, with short lead times, direct to each store. To meet these requirements, the manu- * Corresponding author facturer established several Regional Distribution Centers (RDCs) nationwide and partnered with regional 3PL firms (one for each RDC), who we call distributors, to deliver the products from RDCs to their assigned stores. Following industry practice, the distributors are paid for each store delivery they make based on a fee table that divides the possible delivery distances and weights into a discrete set of ranges and specifies the delivery fee to be paid in each distance and weight range. The manufacturer wanted to develop an appropriate fee table that takes into account the nature and scope of its delivery needs (e.g., scheduled periodic deliveries, relatively short distances, less-thantruckload shipments) and, consistent with its long-term commitment to distributors, ensures that each distributor is able to cover costs and make profits. This article proposes an optimization model to address this problem, which we call the Fee Table Design problem; develops an effective solution methodology; and applies the model to actual data from the manufacturer's distribution network.
The academic literature related to compensation for 3PL delivery services is relatively sparse and largely focuses on conceptual or qualitative discussions of contracting and best practices but does not provide methods to determine the actual fees to be paid for these services. Several papers examine factors that drive the viability and sustainability of these alliances and the characteristics of effective contractual arrangements with 3PL service providers. For instance, Lambert et al. (1999) and Logan (2000) examined 3PL partnerships using various strategic theories, listed some important criteria for the partners to consider, and emphasized the need to design outcome-based rather than behaviorbased contracts. Van Hoek (2000) observed the prevalence of two broad types of contract relationships-"detailed and fixed" versus "open-ended and broad"-in 3PL relationships and delineated the factors that lead to the use of each type of contract. Using a case study, Halldorsson and Skjoett-Larson (2006) highlighted the importance of using contracts that are dynamic (i.e., reviewed periodically) for the success of long-term partnerships. In our problem setting, the fee table scheme that the manufacturer used to compensate distributors incorporates many of the desirable contract attributes discussed in these papers. For instance, this scheme is outcome-based (fees depend on actual deliveries), detailed (specifying an unambiguous pricing mechanism), and dynamically adaptable (permitting fee adjustments to reflect changes to operations and costs over time). Industry experts (Cram, 1996;Carter, 1998) have supplemented academic theory on 3PL compensation by offering practical tips and guidance on commonly used pricing mechanisms. These and several other papers in the 3PL literature have focused on applying empirical models and methods to identify desirable attributes of 3PL contracts (Maloni and Carter, 2006;Selviardis and Spring, 2007;Marasco, 2008;Lukassen and Wallenburg, 2010) but do not offer specific decision support models or methodologies.
Another stream of literature describes model-based approaches to assist commercial carriers in pricing shipments and deciding tariffs. Brotcorne et al. (2000) proposed and solved a bi-level optimization model to set truckload tariffs on various lanes that a carrier operates, taking into account the response of shippers to these lane tariffs. King and Topaloglu (2007) developed a model for dynamic fleet management and lane pricing, considering revenues from price-dependent lane demands and costs for repositioning vehicles on the lane network. Using a bi-level optimization framework, Yano and Newman (2007) developed a model and method to decide day-of-the-week and speed-of-service pricing for an express package delivery service, incorporating the relationship between customers' shipping patterns and prices. These models do not directly apply to the problem of setting fees for periodic delivery services to multiple stores from a distribution center.
To set delivery fees for the retail distribution setting, Balakrishnan et al. (2000) formulated a linear program that determines appropriate fee values for a given fee table structure; i.e., for given distance and weight ranges. We consider the higher-level problem of determining both the structure of the table and its fee values over multiple periods. For this new Fee Table Design problem, we propose a novel mixed-integer programming model and develop a tai-lored methodology to effectively solve the problem. Among the methodological contributions, we identify several new classes of valid inequalities (and a variable elimination method) to strengthen the problem formulation, develop a cutting plane approach to iteratively add these inequalities, and propose an optimization-based heuristic algorithm to generate good quality feasible solutions. Computational tests using actual data from the manufacturer's distribution network demonstrate that our methodological enhancements significantly improve performance and the model generates fee tables that are superior to current practice.
The rest of the article is organized as follows. Section 2 elaborates on the Fee Table Design problem and formulates it as a mixed-integer program. Section 3 develops valid inequalities to strengthen the problem formulation. Section 4 outlines our cutting plane procedure to incorporate these inequalities, and describes our heuristic procedure. Section 5 reports on the application of our model and methodology to actual data for the manufacturer's retail delivery operations and validates the economic benefits of using our approach. Section 6 offers concluding remarks.

Problem description
Two important trends have characterized the transformation of the retail industry: the growth of large nationwide retail chains and manufacturers' increasing reliance on 3PL service providers for retail delivery services. In the home improvement sector, large retailers such as Home Depot and Lowes have become dominant by offering a wide assortment of products at competitive prices; they operate thousands of stores, each carrying tens of thousands of products. To ensure high product availability while controlling store inventories, these retailers impose stringent delivery requirements in the form of frequent direct-tostore deliveries in small lot sizes. The building products manufacturer we worked with relied on such retail chains for most of its sales of residential products. Faced with the challenges of providing responsive and cost-effective service to stores, the manufacturer set up a nationwide distribution network with 13 RDCs, and partnered with independent regional 3PL operators or distributors to deliver products from RDCs to stores. Each RDC serves an exclusive geographic region and is responsible for fulfilling the replenishment orders from all the retail stores assigned to it. On average, RDCs were assigned around 150 stores each, located from tens of miles to a few hundred miles from the distribution center. Figure 1 shows the histogram of delivery distances for stores served by a particular RDC.
Each retail store places and receives orders from its assigned RDC and requires one or two deliveries per week at scheduled dates and times. Stores can order products in small quantities (e.g., cartons rather than pallet loads) and are permitted to place their orders up to 24 hours before the scheduled delivery time. Thus, the total quantity ordered by each store varies from week to week but is often significantly less than a truckload. Figure 2 shows a histogram of the mean weekly shipment weight across all stores assigned to an RDC. Over 30% of the stores have mean shipment weight of less than 2500 pounds per week, and shipments rarely exceed 20 000 pounds, whereas a truck can carry up to 43 000 pounds. (Throughout the article, "pounds" refers to the dimensional weight, adjusted for product density, applicable to the manufacturer's products.) For each RDC, the chosen distributor is responsible for picking up the ordered items from the RDC and delivering them to the respective stores at the scheduled times. These distributors are small regional trucking firms (versus national carriers), and the manufacturer is often their largest client. They each own and operate their own trucks and plan their delivery routes. The manufacturer pays for actual deliveries based on a fee table that specifies the fee payable for weights and distances that lie within specified ranges. Table 1 shows a sample fee table with nine distance ranges covering 0 to 500 miles and seven weight ranges spanning 0 to 20 000 pounds. We refer to the specification of the ranges (their respective starting values and widths) along the two dimensions, distance and weight, as the fee table structure and to each combination of distance range and weight range as a block. The table specifies the fee value for each block of the table; this value represents the payment to be made to the distributor for a delivery whose store-to-RDC distance and shipment weight falls within that block. For instance, if a distributor makes a delivery of 2200 pounds to a store that is 140 miles from the RDC, this delivery falls within the block defined by the fifth distance range and the third weight range in the fee table shown in Table 1; so, the distributor receives a fee of $375.25 for this delivery. Observe that the distance of a store from its RDC is fixed, but the shipment weights to the store can vary for each delivery. Hence, the amount the manufacturer pays to the distributor for delivering to the same store can also vary by delivery.
The fee table mimics the table-based tariff structure, with discrete distance and weight ranges, used by commercial carriers. For the building products delivery context, this type of payment scheme was preferred (over other compensation methods such as fixed payments or per mile rates) because it is well accepted in the industry, is unambiguous and easy to administer, and is outcome based (i.e., payments are based on actual deliveries). Commercial Less-Than-Truckload (LTL) freight rates (e.g., Middlewest Motor Freight Bureau, SMC 3 ) provide some guidance on the desired characteristics of delivery fees. However, neither the specific ranges of commercial LTL rate tables nor the quoted rates were considered directly applicable to the manufacturer's delivery operations due to the distinctive characteristics of these operations. For instance, published LTL rates are typically applicable to one-off and/or longhaul shipments (from hundreds to thousands of miles) by nationwide carriers, do not reflect any discounts for overall shipment volume (across stores) or long-term commitments, and do not necessarily apply to scheduled deliveries with tight time windows. In contrast, for deliveries to its retail customers, the manufacturer relies on small independent and regional distributors whose cost structure and capacity-sharing opportunities differ from those of longhaul carriers. Delivery distances are relatively short, and since all store shipments originate at the RDC and are LTL, the distributor can deploy a single truck to deliver to multiple stores on each trip (versus point-to-point movements). The manufacturer commits to using a distributor for a year or more and exclusively assigns over 100 stores, essentially establishing a long-term partnership with the distributors rather than simply viewing them as contract carriers for ad hoc shipments. For these reasons, the firm wanted to develop a customized fee table that accounts for the distribution of shipment weights and delivery distances to stores within each region and the distributors' costs but broadly conforms to commercial rates.
To maintain some continuity in the compensation practices, the manufacturer wished to keep the fee table structure (distance and weight ranges) fixed for 3 years or so but permitted annual (or more frequent) changes to the fee values to account for store openings and closings as well as changes in demand and costs. The duration for which the fee table structure is fixed is our planning horizon. This horizon is divided into a set of periods (e.g., years) at which the fee values can change. We next discuss the design characteristics and restrictions on the fee table structure and fee values that the manufacturer wanted to incorporate in order to be consistent with industry norms and gain acceptance by distributors.
First, the fee values must be non-decreasing with distance, weight, and time. Second, as illustrated in the fee table shown in Table 1, the widths of the ranges must be non-decreasing in distance and weight. Fee tables commonly used in the industry have this property; for instance, the LTL base rates specified by SMC 3 have the following weight ranges (in pounds): 0-500, 501-1000, 1001-2000, 2001-5000, 5001-10 000, 10 001-20 000, 20 001-30 000, 30 001-40 000, and more than 40 000, whose widths increase as the weight increases. A similar characteristic applies to distance ranges. For instance, LTL rates published by national carriers are specified for zones with increasing radial width. Accordingly, planners at the building products company felt that the chosen table should have increasing range widths for both weight and distance to be consistent with industry practice and ensure the credibility of the fee table. We refer to this requirement as the increasing range width requirement. Third, the fee values for different distance/weight combinations must lie within certain specified upper and lower bounds that are obtained, for instance, from benchmark (e.g., commercial carrier) rates. Fourth, the fee table should not have too many or too few ranges, both to ensure that the table is easy to use and to provide flexibility in fee setting. Our model treats the needed number of distance and weight ranges in the table as user-specified input parameters. These fee structure and value requirements collectively govern the shape of the fee value function. The following final requirement deals with the aggregate compensation to each distributor.
The distributors are small firms largely dedicated to delivering the manufacturer's products to stores. For the fee table to be acceptable to distributors, it must provide adequate total revenue to cover their costs and yield some profits. Distributors incur both fixed costs and variable costs (that vary with distances and weights) for their operations. The fixed costs include investments in trucks, costs to hire and pay drivers, and recurring costs for maintenance, depreciation, and insurance. Moreover, their actual distanceand weight-dependent costs are determined by how they combine store deliveries into single trips originating at the RDC. Rather than match the fee paid for each delivery to the cost associated with that delivery distance and weight (which is difficult to determine due to allocations of various fixed costs), the firm wanted to ensure that distributors' costs over all deliveries are covered in aggregate. That is, for each distributor and every period, the total anticipated payments made for all deliveries must at least cover the distributor's cost plus a profit margin. We refer to the desired total payment (equal to the estimated cost plus profit margin) to a distributor in each period as the compensation target for that distributor and incorporate this requirement as a constraint in our fee table design model.
In the building products retail distribution setting, the manufacturer was able to set this target realistically and gain the distributors' acceptance of this approach for the following main reasons. First, based on information from the distributors on their underlying costs and through industry benchmarking, the manufacturer was able to develop good estimates of the total cost for each distributor in every year of the planning horizon to meet the anticipated store demands in their respective coverage areas. Specifically, the trucking industry has developed well-accepted estimates for resource costs (equipment, personnel, insurance, fuel, etc.). Furthermore, since each distributor serves a fixed set of stores and makes periodic deliveries to each store, the manufacturer can determine, using commercial software (e.g., TransCAD), the resources needed and total travel distance to complete all deliveries. It is, therefore, possible to determine each distributor's monthly operational costs for each period quite accurately. Thus, the manufacturer does not rely solely on the distributor for such information and can independently validate any resource or cost information that the distributor provides. The manufacturer shares its cost estimates with the distributor, and may make some adjustments to get their buy-in. Second, successful logistics partnerships, such as those between the manufacturer and its distributors, require trust and collaboration. The distributors (who are modest-sized regional operators) rely on the manufacturer for a significant portion of their revenues; conversely, the manufacturer depends on the distributor to provide good customer service. Thus, the manufacturer and distributors view their relationship as a long-term partnership, with both having mutual interest and benefit from each other's success. Accordingly, the manufacturer commits to using a distributor for multiple years (subject to satisfactory delivery performance) to serve an exclusive coverage area (i.e., distributors are assigned the set of stores they will serve) and needs to ensure their financial viability. In turn, the distributors willingly share information and provide operational transparency and are open to improvements based on best practices and benchmark comparisons (similar cooperative arrangements arise, for instance, in partnerships between automobile manufacturers and their suppliers; e.g., Liker and Choi, 2004). Finally, due to these close partnership arrangements and mutual trust, the distributors agreed to the manufacturer's approach of setting fees that meet the compensation targets, knowing that the manufacturer would take into account any concerns and feedback from the distributors. This approach was already adopted and in use for several years, albeit using manual and intuitive fee setting methods, before this project began. The manufacturer wished to develop an optimization framework/model and solution method for delivery compensation planning not only to formalize the fee setting process and support the related decisions, but also because such a model enhances the credibility of the fee table and facilitates the process of reaching agreement with distributors. By incorporating requirements such as the minimum and maximum fee values and the minimum target compensation for each distributor, the model that we developed provides many parameters that the manufacturer can vary to generate acceptable fee values. Thus, the model plays a valuable role in the periodic review and discussion of delivery compensation with distributors. Indeed, using our model, the manufacturer was able to successfully introduce a fee table that the distributors felt was fair and equitable.
In summary, the problem of designing a fee table entails selecting a fee table structure for the planning horizon and deciding the fee values for each block of this table for every period of the planning horizon to satisfy the fee monotonicity, bounds, and increasing range width requirements and ensure that the total expected payment to each distributor equals or exceeds the annual compensation target for that distributor. To ensure equity among distributors and to avoid adverse comparisons, the manufacturer wanted to design a common fee table that min-imizes the excess payments to distributors over their respective compensation targets. This objective also serves to minimize the total payments that the firm makes to cover all of its retail delivery needs during the planning horizon. We refer to this problem as the Fee Table Design (FTD) problem.
Manually deciding the structure of the fee table and the fee values for each period during the planning horizon is difficult both because of the vast number of choices and the various requirements. The optimization literature has not addressed (modeled or solved) the FTD problem or related problems. We propose a novel approach for modeling this problem as a mixed-integer program that incorporates all of the fee table design requirements.

Modeling approach
One of the main challenges in modeling the FTD problem is to express the expected fee payments to each distributor as a function of the decision variables. Recall that the fee paid for a particular delivery depends on the distance/ weight ranges in which the delivery falls and the fee value for that block, both of which are decision variables. Moreover, since shipment weights are random, the fees for delivering to the same store can vary by shipment, so computing the expected payment requires taking into account the probability distribution of shipment weights. Given these complexities, it is not possible to model the decisions on range limits as continuous variables without imposing restrictive distributional assumptions and introducing nonlinearity into the problem. Instead, our modeling approach discretizes each dimension (delivery distance and shipment weight) into small intervals, permitting us to formulate the FTD problem as a (linear) mixed-integer program without assuming particular shipment weight distributions. Specifically, we divide the distance axis into N intervals that span all delivery distances from zero to the maximum delivery distance. Similarly, we consider M intervals that cover all shipment weights from zero to the maximum shipment weight over all stores. Together, the distance and weight intervals create an M × N interval grid. Given this grid, the range selection decisions correspond to combining or merging the intervals in each dimension to form the desired number of ranges (see Fig. 3). Accordingly, we define binary variables to decide which contiguous intervals are aggregated into each range; these variables determine the "structure" of the recommended fee table. By using (continuous) decision variables to represent fee values in each cell of the interval grid and imposing the requirement that cells belonging to the same ranges must have the same fee value, we are able to develop a model with a linear objective function and constraints that readily incorporates the practical requirements of fee monotonicity, bounds, and increasing range widths.
Illustration of modeling approach on interval grid.

Notation and problem formulation
Let T denote the number of periods in the planning horizon, with periods indexed as t = 1, 2, . . . , T. The structure of the fee table (i.e., ranges) remains the same for the entire horizon, but the fee values in the table may change (increase) from period to period to accommodate changes in delivery requirements and costs. Let S denote the set of all retail locations to which the manufacturer's products need to be delivered, and let R be the set of regional distributors that the firm uses to deliver products. Each regional distributor r ∈ R has an exclusive geographic coverage area and is responsible for deliveries to the stores within that coverage area. Due to new store openings and closures in each period, the set of stores assigned to a distributor can change from one period to the next. Let (r, t) denote the set of stores that are assigned to distributor r in period t. For every store s ∈ (r, t), let q s be the store's delivery distance from its assigned distribution center to the store location, f st the number of deliveries per period, and st (•) the probability distribution of shipment weights (per delivery) to the store in period t. Since store demands change over time, we permit the delivery frequency and shipment weight distribution to vary by period t. Table 2 summarizes our notation. Let C rt be the compensation target (minimum expected total payment needed) for distributor r in period t. For the given M × N interval grid, let i = 1, 2, . . ., M and j = 1, 2, . . ., N, respectively, denote the indices of the ith weight interval and jth distance interval in increasing order of weights and distances. Distance interval j corresponds to delivery distances from d j- The intervals need not be uniformly spaced; i.e., their widths (w i -w i-1 ) or (d jd j-1 ) can vary with i or j.) Every pair of weight and distance intervals forms a cell <i, j> in the interval grid. For each cell <i, j> and every period t, we are given the minimum and maximum permissible fees, l ijt and u ijt , for any delivery that falls within that cell. Given the fee monotonicity requirements (with distance, weight, and time), these lower and upper bounds are non-decreasing with i, j, and t. Let K (< M) and H (< N) respectively denote the desired number of distance and weight ranges for the fee table.
With this framework, the decisions on the table structure separate into two sets of range selection choices, one for weight and the other for distance. For 1 ≤ j 1 ≤ j 2 ≤ N and k = 1, 2, . . . , K, we define a distance range selection binary variable X k j 1 , j 2 that takes the value of one if the kth distance range begins at interval j 1 and ends at interval j 2 (inclusive) and is zero otherwise. Similarly, for 1 ≤ i 1 ≤ i 2 ≤ M and h = 1, 2, . . . , H, we define a weight range selection binary variable Y h i 1 ,i 2 that has the value of one when the hth weight range spans the intervals from i 1 to i 2 (including both i 1 and i 2 ) and is zero otherwise. The curved arrows in Fig. 3 illustrate these range selection decisions for the weight and distance axes. To model the fee value decisions, we define continuous variables P ijt denoting the delivery fee in each cell <i, j> for period t.
We next discuss how to express the expected fees that a distributor r receives in every period t in terms of our decision variables. For each store s, let j(s) denote the index of the distance interval corresponding to the RDC-to-store delivery distance q s to this store; i.e., j(s) is the interval Interval grid cell corresponding weight interval i and distance interval j Expected number of deliveries by distributor r in period t that fall in weight interval i and distance interval j Frequency of deliveries to store s in period t q s Delivery distance for store s from its assigned distributor Set of stores covered by distributor r in period t Decision variables P ijt Fee value for cell <i, j> in period t X k j1, j2 Range-indexed distance range selection; has a value of one if the kth distance range starts at j 1 and j 2 and is zero otherwise Range-indexed weight range selection; has a value of one if the hth weight range starts at i 1 and i 2 and is zero otherwise the cumulative distribution of shipment weight to store s in period t, the probability that the weight of a shipment to this store lies in the ith weight interval-i.e., between w i−1 and Consequently, the expected total number of deliveries by a distributor r to all of its assigned stores that fall in cell <i, j> in period Based on the given data on shipment weight distributions and frequencies, we can compute this parameter a ijrt a priori for all cells, distributors, and periods. Thus, we can express the total expected payment to distributor r in period t as the linear function i j a ijrt P ijt of the fee value decision variables P ijt .
Using the above decision variables and parameters, we can formulate the FTD problem as the following mixed-integer program, which we denote as [FTD]: i j a ijrt P ijt ≥ C rt for all r = 1, 2, . . . , R, t = 1, 2, . . . , T, X k j 1 , j 2 = 0 or 1 for j 1 = 1, 2, . . . , N, The firm's goal is to minimize the excess payments to distributors, over their respective compensation targets, during the planning horizon using the chosen fee table. Equivalently (since the compensation targets are constants), the objective is to minimize the total expected payments (1) to all distributors over all T periods. Constraints (2) and (3) model the distance range selection decisions. Constraints (2) specify that the first and last ranges must, respectively, include intervals 1 and N. Constraints (3) ensure range contiguity by requiring that, if the kth distance range ends at interval j, then the next range (k + 1) must begin at interval (j + 1). Constraints (4) link the range selection decision variables to the fee value variables P ijt . Specifically, if two contiguous distance intervals j and (j + 1) belong to the same distance range (i.e., if k j 1 ≤ j X k j 1 , j = 0), then the constraint requires the fee values in cells (i, j) and (i, j + 1) to be the same. For the weight ranges, Constraints (5) and (6) model the interval coverage and range contiguity requirements, analogous to the distance range selection Constraints (2) and (3). And, like Constraint (4), Constraint (7) specifies that, if two contiguous weight intervals i and (i + 1) belong to the same weight range, then the fee values in the cells (i, j) and (i + 1, j) must be the same. Thus, Constraints (4) and (7) together require that all the cells within a chosen block of delivery distances and shipment sizes have the same fee values. Constraints (8) are the fee adequacy requirements, specifying that, for every distributor r, the expected payment in each period t must be greater than or equal to the distributor's target compensation. Constraints (9) apply the lower and upper bounds on the fee value of each cell <i, j> in each period t. Next, Constraints (10) to (14) incorporate the monotonicity of fee values and restrictions on range choices (increasing widths). Constraints (10), (11), and (12) ensure that the fee values are non-decreasing with distance, weight, and time, respectively. Constraint (13) requires the width of the (k + 1)th range to be greater than or equal to the width of range k. The expressions on the left-hand side and right-hand side of this constraint respectively represent the range widths of the kth and (k + 1)th ranges in terms of the range selection variables. The validity of these expressions stems from the property (implied by Constraints (2) and (3)) that j 1 j 2 ≥ j 1 X k j 1 , j 2 = 1 for every range index k. Constraints (14) similarly impose the increasing width requirement for the weight ranges. Finally, Constraints (15) and (16) are the integrality requirements for the range selection variables.
As is common in integer programming, there are alternative ways to formulate the FTD problem. For instance, in formulation [FTD] we have chosen to define separate range selection variables, X k j 1 , j 2 and Y h i 1 ,i 2 , for each range index (k or h). Instead of using these range-indexed variables, we can alternatively model the range choices using variables that omit the range indices k and h but just indicate which contiguous (distance and weight) intervals to combine into a single range. For illustrative purposes, we discuss this alternate representation for the distance range selection decisions. For every pair of distance intervals j 1 and j 2 , with 1 ≤ j 1 ≤ j 2 ≤ N, suppose we define the binary variable V j 1 , j 2 that takes the value one if a distance range starts at interval j 1 and ends at j 2 and is zero otherwise. We can modify the constraints of the FTD problem formulation in terms of these V -variables, instead of the X-variables, while retaining linearity. Although using the V -variables (and analogous variables for weight range selection) reduces the model size, it also weakens the formulation; i.e., the model's Linear Programming (LP) relaxation is not as tight as that of model [FTD] (see Section 5 for comparative computational results). Furthermore, this model does not permit the modeling enhancements, such as problem reduction and strengthening using valid inequalities, that we discuss in Section 3. Since having a strong LP relaxation is important for solving the problem effectively, we work with (and enhance) our range-indexed formulation [FTD]. Observe that, in practice, the desired number of ranges, H and K, is relatively small (e.g., less than 10), and so the size of the range-indexed model remains well within capabilities of common solvers.
In preliminary computations, a commercial solver (CPLEX) with standard branch-and-bound algorithms was not able to solve formulation [FTD] effectively, due to poor LP lower bounds and ineffective initial upper bounds (from heuristic solutions generated by the solver) that were far from optimal. We therefore pursued a twopronged strategy of (i) strengthening the model formulation by eliminating variables and developing several classes of valid inequalities (Section 3) and (ii) improving the initial upper bound using an optimization-based heuristic procedure (Section 4).

Strengthening the model formulation
By increasing the LP relaxation lower bounds, strong formulations accelerate exact procedures such as branchand-bound, provide more accurate assessment of the quality of heuristic solutions, and yield better starting points for LP-rounding methods to generate integer feasible solutions. Identifying ways to strengthen the model formulation requires first understanding the structure of the problem's LP solutions and the drivers of the integrality gap between the LP value and optimal value of the integer program. For the [FTD] model, the LP relaxation achieves a lower objective value by choosing fractional values for the range selection variables X k j 1 , j 2 and Y h i 1 ,i 2 (effectively selecting multiple fractional starting and ending intervals for each range but satisfying the range selection requirements) so that the solution can set different fee values for different cells even if these cells belong to the same block. To limit the fractional values of range selection variables and raise the LP value, we tighten the range width constraints of formulation [FTD], eliminate some variables in this model, and develop several families of additional valid inequalities. These strategies exploit formulation [FTD]'s use of rangeindexed variables for range selection, the increasing range width property, and the lower and upper bounds on fee values. The following subsections elaborate on these model strengthening strategies. The online supplement contains the formal proofs for all of the Propositions in this section.

Disaggregated increasing range width constraints
To reduce the objective value, the LP solution assigns fractional range-selection values for interval pairs {j , j } spanned by a range k while satisfying the distance range width Constraints (13) in formulation [FTD] on "average" (over all the interval pairs). We can strengthen these constraints by disaggregating them as follows for each ending interval j of every range k: for all j ∈ J\{N}, k = 1, 2, . . . , K − 1. (17) The validity of these disaggregated constraints stems from (and exploits) the contiguity requirement on the range choices, namely, if range k ends in interval j, then range (k + 1) must begin in interval (j + 1). We can show that any fractional solution that satisfies Constraints (17) must also satisfy Constraint (13), but the reverse need not hold; hence, Constraints (17) are tighter. We can similarly replace the increasing range width Constraints (14) for the weight ranges with the following stronger disaggregated version: Although the disaggregate formulation requires more constraints, its benefit in terms of tighter LP bound outweighs the increase in formulation size. Henceforth, we designate the version of the model with Constraints (17) and (18), in place of Constraints (13) and (14), as formulation [FTD].

Cumulative range width forcing inequalities
Even with the disaggregate versions (17) and (18) (17), the LP solution can then choose suitable fractional values for other range selection variables X k j 3 , j and X k+1 j +1, j 4 , for one or more distance intervals j 3 > j 1 and/or j 4 > j 2 .) The following class of inequalities prevents such fractional assignments to consecutive ranges that violate the range width requirement: for all j ∈ J\{N}, j 1 ≤ j, k = 1, 2, . . . , K − 1. (19) The constraint states that, because of the increasing range width requirement, if the solution includes in range k all of the cells from interval j 1 to j (and possibly other intervals before j 1 ), then it must include in range (k + 1) all of the cells from interval (j + 1) to the lowest-indexed interval j 3 (and possibly other intervals after j 3 ) for which We refer to Constraints (19) and (20) as cumulative range width forcing Constraints; the following proposition formally establishes their validity.
Proposition 1. The cumulative range width forcing Constraints (19) and (20) are valid for the FTD problem.
We note parenthetically that we can formulate another set of inequalities, complementary to Equations (19) and (20), that ensure consistency in range widths between one range and its previous range.

Eliminating range selection variables
In formulation [FTD], we define range selection variables X k j 1 , j 2 for all j 2 ≥ j 1 and Y h i 1 ,i 2 for all i 2 ≥ i 1 . We can exploit the increasing range width requirement to omit several of these variables from the formulation. For the distance range selection variables, the following proposition narrows the feasible interval pairs (j 1 , j 2 ) that the kth distance range can span, for k = 1, 2, . . . , K.
Proposition 2. In any feasible integer solution to the FTD problem, the range selection variable X k j 1 , j 2 can be non-zero only if the indices k, j 1 , and j 2 satisfy one of the following three conditions: To understand this result, let us consider condition 1 regarding the feasible spans of the first range (k = 1). Since the fee table contains K distance ranges and the widths of the ranges are non-decreasing with the range index, the width of the first range must not exceed the average range width (d N − d j 2 )/(K − 1) of the remaining (K − 1) ranges. This observation, together with the property that all subsequent ranges (beyond the first range) must span at least one interval, narrows the possible ending intervals j 2 for the first range as shown in condition 1. The other conditions (for middle and last ranges) of the proposition similarly use the increasing range width requirement and range definitions to restrict the range spans.
Proposition 2 permits us to reduce the size of the problem formulation significantly by eliminating the X k j 1 , j 2 variables corresponding to all of the combinations of indices k, j 1 , and j 2 that do not satisfy any of the conditions of the proposition. Moreover, this elimination also strengthens the model by tightening its LP relaxation. For the weight ranges, we can develop a result analogous to Proposition 2 to eliminate many Y h i 1 ,i 2 variables a priori by exploiting the increasing range width requirement and other range selection constraints. For each distance range k, let (k) ⊆ J × J denote the set of distance interval pairs {j 1 , j 2 } satisfying the conditions of Proposition 2 for which the variable X k j 1 , j 2 remains in the model formulation. Similarly, for each weight range h, let (h) ⊆ I × I denote the set of interval pairs {i 1 , i 2 } for which the variable Y h i 1 ,i 2 remains in the model formulation.

Variable lower bounds
Constraints (9) of formulation [FTD] impose the given minimum and maximum fee values in each cell as simple lower and upper bounds on the corresponding fee value variable in Constraint (9). In the model's solution, however, cells are combined into blocks of the proposed fee table, and the fee values for all the cells in a block must be the same. Hence, the fee value for a cell <i, j> must also satisfy the lower and upper bounds for all of the other cells that the solution includes within the same block as cell <i, j>. For instance, if the solution includes intervals j and j 2 (> j) in the same distance range, then the fee value P ijt of cell <i, j> in period t must exceed not only its own lower bound l ijt but also the lower bound l i j 2 t of cell <i, j 2 >. Although all integer solutions of [FTD] satisfy this condition, the solutions to the LP relaxation with fractional range selection values may violate this requirement. To limit such violations, we can impose the following variable lower bounding constraints for fee values: Constraint (21) dominates Constraint (9) because the lower bound values are non-decreasing in distance and because each interval j is covered by exactly one distance range. We can further strengthen Constraint (21) by also incorporating the weight range selection variables as follows.
Suppose the solution includes cell i in a weight range h by setting Y h i 1 ,i 2 equal to one, for some < i 1 , i 2 >∈ (h) with i 1 ≤ i ≤ i 2 . In this case, the fee value P ijt must also satisfy the lower bound l i 2 jt which can exceed l ijt due to monotonicity of bounds. To add the weight range selection decisions to the variable lower bound in Constraint (21), we consider the smallest necessary increment in cell <i, j>'s lower bound for various weight range choices. For this purpose, let λ j be the highest-indexed distance interval that a distance range can span while including interval j; i.e., λ j = max{ j 2 : k, < j 1 , j 2 >∈ (k) for j 1 ≤ j ≤ j 2 }. Then, l i λ j t is the highest coefficient in the right-hand side of Constraint (21). Hence, if the solution includes weight interval i in the hth weight range by setting Y h i 1 ,i 2 equal to one, we can increase the lower bound that the fee value P ijt must satisfy by at least [l i 2 , j,t − l i,λ j ,t ] + . The following constraint incorporates this strengthening: for all i ∈ I, j ∈ J, t = 1, 2, . . . , T.
We can similarly develop another inequality that first accounts for lower bounds implied by the weight range choices and then adds increments based on the distance range choices. To formulate this version of the variable lower bounding constraint, let μ i be the highest-indexed interval that can be part of a weight range that includes interval i; i.e., μ i = max{i 2 : h, < i 1 , i 2 >∈ (h) for i 1 ≤ i ≤ i 2 }. Then, the following variable lower bounding constraint is complementary to Constraint (22): for all i ∈ I, j ∈ J, t = 1, 2, . . . , T.
We refer to Constraints (22) and (23) as the block-based variable lower bound constraints. Constraints (22) and (23) are valid for the FTD problem.

Variable upper bounds
As with the lower bounds, we can also strengthen the upper bounds on the fee values (Constraint (9)) by making them contingent on the range choices. The fee value in a particular distance (and weight range) must not exceed the given upper bounds for all the cells included in that range.
Since the upper bounds are non-decreasing in distance and weight, the fee upper bound corresponding to the starting interval of a range provides the tightest upper bound among all the cells in the range. Building on this observation and using the distance range selection variables, the following constraints, analogous to Constraint (21), establish a variable upper bound on fee values: Since u ijt ≥ u ij't for j ≤ j, Constraint (24) is at least as strong as the simple upper bound Constraint (9). The upper bound values are also non-decreasing with respect to shipment weight, and so Constraint (24) can be further strengthened by incorporating the weight range selection variables. Let ρ j denote the lowest-indexed starting interval for a distance range that includes interval j. Then, u iρ j t is the smallest among the upper bound values in Constraint (24). Using this definition, we can add the weight range selection variables to right-hand side of Constraint (24) (as we did for Constraint (21)) to impose the following tighter block-based variable upper bound on the fee value P ijt : Again, analogous to Constraints (23), we can derive an alternative version of Constraints (25) by first incorporating the weight range selection variables and then accounting for the incremental impact of distance range selection variables. Let σ i denote the lowest-indexed starting interval for a weight range that includes weight interval i. Then, the following variable upper bound constraints can tighten the fee upper bound Constraints (9) in [FTD]: for all i ∈ I, j ∈ J, t = 1, 2, . . . , T. Constraints (25) and (26) are valid for the FTD problem.

Fee difference forcing constraints
In addition to the block-based Constraints (22), (23), (25), and (26), we can impose variable bounds on the maximum permissible fee difference between adjacent cells by simultaneously considering distance and weight range choices. For instance, consider a weight range that begins at interval i 1 ≤ i, spans weight interval i, and ends at interval i 2 ≥ i. In this case, the lower bound l i 2 jt is a valid and tighter bound than l ijt on the fee value P ijt (since i 2 ≥ i and lower bounds are monotonic in weight), and u i 1 , j +1,t is a valid and tighter bound than u i,j+1,t on P i,j+1,t (since i 1 ≤ i 2 and upper bound values are monotonic in weight). Hence, the following constraints are valid and stronger than the original range-to-fee linkage Constraint (4): Analogously, to ensure that fees within a weight range are the same, we can employ the following tighter version of the range-to-fee linkage Constraints (7): Proposition 5. The fee difference forcing Constraints (27) and (28) are valid for FTD problem.

Block selection forcing constraints
Since formulation [FTD] models the weight and distance range choices separately, the LP relaxation solution can fractionally select blocks containing cells whose respective minimum and maximum fees are inconsistent. For instance, suppose cell <i, j> is part of a weight range that begins at i 1 and ends at i 2 (with i 1 ≤ i ≤ i 2 ) and a distance range that spans intervals j 1 through j 2 (j 1 ≤ j ≤ j 2 ). Although these two range choices may individually permit feasible fee values (i.e., l i, j 2 ,t ≤ u i, j 1 ,t and l i 2 , j,t ≤ u i 1 , j,t for all t), selecting both ranges together may not be feasible due to the bounds on the fee values. To address this situation we can impose the condition that, if l i 2 , j 2 ,t ≥ u i 1 , j 1 ,t for any t, the FTD solution must not simultaneously select these two ranges. The following block selection forcing constraints capture this restriction: Analogous constraints, contingent on weight range choices, are as follows: Proposition 6. The block selection forcing Constraints (29) and (30) are valid for the FTD problem.

Summary
Taking advantage of the characteristics and requirements of the FTD problem, in this section we develop a problemreduction approach and five new classes of valid inequalities that do not exclude any feasible integer FTD solutions but eliminate fractional LP solutions. Such inequalities have not been previously proposed in the literature. Our valid inequalities tighten different aspects of the formulation and are complementary. The block-based variable lower bounds ( (22) and (23)), block-based variable upper bounds ( (25) and (26)), and fee difference forcing inequalities ( (27) and (28)) are mixed-integer cuts that tighten the linkage between the binary range selection and the continuous fee setting decision variables. The cumulative range width forcing inequalities ( (19) and (20)) and block selection forcing inequalities ( (29) and (30)) are binary cuts that strengthen the range selection choices to ensure increasing range widths and consistency with fee value bounds. As our computational tests (Section 5) confirm, these constraints are effective in both strengthening the LP relaxation of the FTD problem and providing a good starting point for a LP-based heuristic, thereby accelerating the exact solution procedure.

Solution methodology
Our solution method incorporates problem reduction and the valid inequalities of Section 3 to obtain a strong LP relaxation, and generates a heuristic solution before applying a branch-and-bound approach. Since the number of possible valid inequalities is large, we do not add them all a priori but instead apply a cutting plane procedure (discussed in Section 4.1) to iteratively add violated constraints. If the solution to the strong LP relaxation is fractional, our method (i) generates an LP-based heuristic solution by solving a maximum flow problem on a related layered network (Section 4.2); (ii) refines this initial solution using local improvement rules (Section 4.3); and (iii) initiates a branch-and-bound procedure. We refer to the overall solution procedure with problem reduction, model strengthening, and heuristic as the Composite algorithm.
The following sections discuss the components of this algorithm.

Cutting plane method
We implemented an iterative cutting plane procedure to add select valid inequalities from the five classes of Constraints (19), (20), (22), (23), and (25)-(30). At each iteration, after solving the LP relaxation of the current model, the procedure identifies constraints in each inequality class that the LP solution violates, adds these constraints to the current model, and re-solves the LP. This cutting plane procedure terminates when the current LP solution does not violate any of the valid inequalities. We refer to the augmented formulation that we obtain at the end of the procedure as the strong model. Since each of our inequality classes contains only a polynomial number of constraints, we can identify the constraints that are violated by any given LP solution by evaluating the inequalities for all relevant weight/distance intervals and ranges and time periods. Adding all of the constraints that are violated by the LP solution rapidly increases the model size and hence the computational effort at each iteration; at the other extreme, adding only the most violated inequality necessitates more iterations. We therefore adopted the following intermediate approach to determine the subset of violated constraints to add. Given an LP relaxation solution Q LP = (P LP , X LP , Y LP ) to the incumbent model, we say that a constraint aQ ≤ b is ε-violated by this solution if aQ LP ≥ b + ε; at each iteration, we only add such constraints. In our computations, we initially set ε = 0.1; if no constraints are ε-violated at a particular iteration, we reduce ε by half and add any constraints that are ε-violated with the lower threshold. The cutting plane iterations terminate when the value of ε becomes less than 0.005.

LP-rounding heuristic method
When the solution to the strong model's LP relaxation is fractional, we first apply a rounding method that solves two related network flow problems to determine the weight and distance ranges and then solve a fee setting subproblem (linear program) to determine the corresponding optimal fee values. Our heuristic method determines the table structure in two steps, one each for the weight and distance ranges. We focus here on the procedure to decide distance ranges from the LP solution; a similar procedure determines the weight ranges. Given a solution to the LP relaxation of the FTD problem, rounding up all of the positive fractional X-values in the LP solution will not yield feasible range choices (e.g., the number of ranges may exceed K, the same interval may be included in multiple ranges, and/or the ranges may not satisfy the increasing range width constraints). To judiciously select the LP values to be rounded up, we formulate a network optimization problem defined over a layered network with K layers, one for each distance range. For every pair of distance intervals { j 1 , j 2 } ∈ (k), the network contains a corresponding node {j 1 , j 2 } in layer k. We connect nodes in layer k < K to appropriate nodes in layer (k + 1) that are consistent with the range contiguity and increasing range width requirements. Specifically, a node {j 1 , j 2 } in layer k is connected to a node {j 3 , j 4 } in layer (k + 1) only if j 2 + 1 = j 3 and d j 2 − d j 1 −1 ≤ d j 4 − d j 2 . The network also contains a layer 0 with an origin node O and a layer (K+1) with a destination node D. The origin is connected to all the nodes in layer 1, and every node in layer K is connected to the destination. Figure 4 shows an illustrative network for a problem instance that requires selecting three distance ranges (layers) from 10 equal-width intervals. After eliminating infeasible range selection combinations (using the problem reduction method discussed in Section 3.3), layers 1, 2, and 3 have three, eight, and five nodes, respectively. For intermediate layers, the network connects only those pairs of nodes in successive layers that satisfy increasing range width requirements. With this construction, any path in the network from the origin to destination defines a distance range structure that contains exactly K ranges and satisfies the range selection requirements (contiguity of ranges and increasing range widths). We can therefore use this network to decide which among the range selection values in the LP solution X LP = {(X k j 1, j 2 ) LP } to round up to get a set of distance range choices that satisfies fee range requirements. We favor rounding up X-values that are higher since such values indicate good range choices from the perspective of minimizing the FTD model's objective function value. For this purpose, we assign a length of (X k j 1, j 2 ) LP to every arc entering node {j 1 , j 2 } in layer k, for all k = 1, 2, . . . , K and { j 1 , j 2 } ∈ (k), and a length of zero to all arcs incident to the destination. In Fig. 4, for instance, the arc from the origin to node (1, 2) in layer 1 has a length of (X 1 1,2 ) LP , the arc connecting node (1, 4) in layer 1 to node (4, 7) in layer 2 has a length of (X 2 4,7 ) LP , and so on. Using these arc lengths, the longest origin-to-destination path will likely use arcs with high X-values and also represents a feasible choice of distance ranges, as desired. We can translate this longest-path solution to the distance range solution to our model as follows. For k = 1, 2, . . . , K, suppose the kth arc in longest path is the arc from node {j, j 1 (k)} in layer (k -1) to a node {j 2 (k), j } in layer k. Then, we obtain the corresponding solution to our model by settingX k j 1 (k), j 2 (k) = 1 for all k andX k j 1 , j 2 = 0 for all k, j 1 = j 1 (k), j 2 = j 2 (k). Since the layered network is acyclic, solving the longest path problem requires only modest computational effort (polynomial time). Applying the same procedure to the weight intervals, we solve another longest path problem based on the fractional Y LP values to determine the weight ranges. We denote the integer-valued distance and weight range solutions obtained using this procedure asX andŶ.
Given these distance and weight ranges, we can solve the following fee setting linear program (Balakrishnan et al. 2000), which we denote as [FSP], to determine the optimal fees for every period for the chosen fee ranges.
[FSP] Minimize for all i ∈ I, j ∈ J\{N}, t = 1, 2, . . . , T, for all i ∈ I\{M}, j ∈ J, t = 1, 2, . . . , T, and Constraints (8)-(12). The solutionP of the [FSP] together withX,Ŷ is a heuristic (integer) solution to the FTD problem. Because our procedure selects ranges before deciding fee values, the fee setting problem, for the range choicesX,Ŷ, may not have a solution that satisfies the fee upper bounds (9). In such cases, to ensure feasibility of [FSP], we can make these constraints "soft" by imposing a penalty in the objective function for any fee values that exceed the upper bound. In our computational study, problem [FSP] was feasible (even with fee upper bounds as hard constraints), and our heuristic approach generated near-optimal solutions for all problem instances.

Local improvement procedure
Starting with the LP-based heuristic solution, the local improvement method attempts to reduce the objective value through local changes in the range choices. To systematically identify candidate range structures in the neighborhood of an incumbent solution, the procedure applies the following two rules: 1. For each range 1 < k ≤ K, move the starting interval of range k to the preceding range (k − 1). This move increases the width of the first range, reduces the width of the last range, and possibly alters (depending on the widths of the transferred intervals) the widths of other intermediate ranges. 2. Re-assign the starting interval of every range k > 1 to include the ending interval of the previous range. This rule decreases the width of the first range and increases that of the last range.
Since we can apply either rule to the weight and/or distance ranges, we obtain eight neighbors of the incumbent fee structure. For each neighbor that satisfies the increasing range width requirement, we solve the [FSP] problem to determine the corresponding optimal fee values and select as the new incumbent the neighbor that provides the highest cost reduction. The procedure terminates when none of the neighbors has lower cost than the incumbent. Finally, if the strong LP value is less than the cost of the heuristic solution, our Composite solution method initiates a branch-and-bound procedure, allowing 2 hours for this step. The next section discusses the successful use of this method to generate a practical and effective fee table based on actual data.

Application: developing a fee table for distributing building products
This section elaborates on the building product manufacturer's nationwide distribution context to which we applied our model and solution method, and reports computational results.

Application context and data summary
Before this project, the manufacturer used a manually designed fee table with five weight ranges and four distance ranges to determine delivery fees for its 13 distributors who together serviced over 1500 stores. We refer to this table as the original 5 × 4 table. Due to its limited number of weight and distance ranges that were not optimized, the payments based on this original fee table led to significant excess payments to several distributors. Therefore, the firm wanted to explore the development of a new fee table with more weight and distance ranges. The structure (ranges) of the table would be fixed for 3 years, but the fee values can change each year to reflect operational changes. As part of the fee planning process, the firm first assessed demand at the store level. Based on inputs from the firm's retail customers, the company generated the list of stores that would be operational (taking into account anticipated store closings and new store openings) in each time pe-riod of the planning horizon and the delivery frequency for each store. Each store was assigned to an appropriate RDC (and corresponding distributor) in the manufacturer's distribution system. Next, the firm determined the distribution of shipment weights (in pounds) to each store based on historical data as well as inputs from the retail chains. Given the delivery distance, delivery frequency, and shipment weight distribution to each store in a distributor's coverage area, the firm applied a logistics planning tool to determine the number (and capacities) of trucks needed and their travel mileage to meet the delivery commitments of each RDC. Based on industry expense norms and information provided by distributors, the firm was then able to estimate each distributor's monthly fixed and variable (mileage-and delivery-dependent) costs. These estimates were shared with distributors and minor adjustments were made, if necessary, to obtain each distributor's compensation target for every period in the planning horizon.
To apply our model to the firm's data, we analyzed the delivery distances and shipment weights to determine an appropriate fine-grained grid of weight and distance intervals. Our analysis of store delivery distances revealed that many stores were close to their assigned RDCs. Accordingly, we used non-uniform intervals, starting with narrow 5-mile intervals for the first 50 miles, followed by progressively wider intervals of 10, 25, 50, and 100 miles. Similarly, for the weight axis, based on our analysis of store shipment weights, we chose narrow initial intervals with a width of 100 pounds and increased the interval width progressively to 250, 500, 1000, and 2500 pounds. Altogether, the grid has 43 distance intervals (i.e., N = 43) and 48 weight intervals (M = 48). For each cell in this interval grid, the minimum and maximum fee values were chosen based on commercial LTL rates.
We implemented the FTD model and our Composite solution method using the CPLEX 12.5 callable optimization library (on a Windows server with dual 2.4 GHz Xeon 8 core processors) and applied it to several scenarios, with different range requirements, based on the above data. First, we determined the best possible fee table with five weight and four distance ranges by setting H = 5 and K = 4 in the FTD model. We refer to this table as the optimized 5 × 4 table; it has optimal widths for the five weight and four distance ranges whereas the original 5 × 4 table had pre-specified (manually chosen) widths for the weight and distance ranges. To ensure that our savings estimates are conservative, we determined the optimal fee values (by solving problem [FSP]) even for the original table rather than use the manufacturer's original fee values (that were suboptimal for the original table structure). Comparing the objective values for the original and optimized 5 × 4 tables helps us assess the possible savings by applying our FTD model to optimize range widths for a table with the same size as the original table. Next, to decide whether to increase the number of ranges, we conducted two sets of computational runs. First, as a baseline, we set the desired number of weight and distance ranges at H = 7 and K = 7. Then, to understand the impact of increasing the number of distance (or weight) ranges, we fixed the desired number of weight (or distance) ranges H (or K) at seven and varied the number of distance (or weight) ranges from five to nine. For the baseline case with H = 7 and K = 7, the base model [FTD] (without our valid inequalities) has more than 10 000 variables and nearly 30 000 constraints.

Results
We first discuss our Composite algorithm's effectiveness to generate near-optimal solutions quickly and later address the improvement that the model's solution provides compared to the firm's original fee table. We assess the performance of the Composite method's two main components, namely, the strong formulation incorporating our valid inequalities and the optimization-based heuristic method, by examining (i) the improvement in the LP relaxation value when using the strong formulation and (ii) the ability of the heuristic approach to generate solutions that are close to optimality. We also assess our Composite method's overall improvement in branch-and-bound performance (quality of final upper bound within 2 hours) compared with the results using the base [FTD] formulation without heuristic warm start. We refer to this latter approach as Base B&B. To focus on the effects of our valid inequalities and heuristic procedure and to avoid exaggerating the performance gains, we apply our problem-reduction procedure (Section 3.3) and enable CPLEX's preprocessing and cut generation (at the root node) for both the Base B&B and our Composite method. Table 3 summarizes the results for the 5 × 4 table, the 7 × 7 baseline scenario, and the eight other scenarios obtained by varying the desired number of distance and weight ranges.

Performance of the Composite method
In Table 3, to compare the Composite method's ability to generate good lower and upper bounds compared to the Base B&B, we measure the IP gap (as reported in CPLEX) for each method, based on the incumbent upper and lower bounds, both before the branch-and-bound procedure begins (root node) and at termination. Columns 2 through 5 of the table contain these initial and final integrality gaps for the two methods. The Composite method outperforms the Base B&B approach in all instances, consistently yielding much smaller integrality gaps at both the start and end of the branch-and-bound procedure. At the root node, high gaps can result from both weak LP relaxation values and poor initial heuristic solutions (upper bounds). Since the Composite method uses our valid inequalities to obtain a strong LP relaxation and because it employs the LP-based heuristic solution to obtain a good upper bound, its initial gap is significantly smaller than the gap that CPLEX obtains using the Base B&B method. Remarkably, in seven of the 10 instances in Table 3, the Composite method's initial gap is itself better than the final gap with Base B&B. Consequently, the Composite method, with its strong model and good heuristic method, has superior overall performance (average final gap of 1.3%).
On average, our valid inequalities close nearly 40% of the gap between the base LP value (without our valid inequalities, but after adding CPLEX-generated cuts) and the final lower bound; Column 7 of the table shows this statistic (= (Strong LP value -Base LP value)/(Final LB -Base LP value)) for all problem instances. These results demonstrate that the cuts we developed were very effective in tightening the LP relaxation. Moreover, the cutting plane procedure achieves this improvement by adding just a small fraction (<3%, on average) of the possible inequalities (Column 6).
To measure the quality of the solutions generated by the LP-rounding heuristic procedure, we compute its suboptimality or heuristic gap before local improvement, defined as the difference between the heuristic value (before improvement) and the final upper bound value expressed as a percentage of the final upper bound value. Column 9 shows that the solution from the LP-based heuristic procedure yields solutions that are within 0.9% on average from the final (best) upper bound and requires only modest computational effort (Column 10). The local improvement procedure further improves heuristic quality, generating solutions that were on average only 0.7% away (Column 11) from the final upper bound value. Together, for different settings of the desired number of ranges, the strong LP and the heuristic procedure yield near-optimal solutions quickly even before initiating the branch-and-bound procedure.
We also conducted computational tests to assess the performance of the weaker formulation described in Section 2.3 vis-a-vis the [FTD] model. Recall that this alternative formulation uses aggregate binary variables V j 1 , j 2 and W i 1 ,i 2 instead of the range-indexed variables X k j 1 , j 2 and Y h i 1 ,i 2 in formulation [FTD] and so has fewer variables but a weaker LP relaxation. Omitting the modeling and algorithmic enhancements for both models (that are tailored to the rangeindexed formulation [FTD]), we found that, after 2 hours of computation for each problem instance, the final gap using our base [FTD] model (2.7% on average) is better than that of the alternative formulation (3.3% on average), validating our choice of the range-indexed model as the formulation for the Composite method.
Finally, to validate the robustness of our model and method with respect to parameter changes, we varied the level of fee flexibility, which we define as the range of permissible values between the minimum and maximum permissible fee values of each cell. With lower fee flexibility (i.e., maximum fee value closer to the minimum fee value), we might expect some of our enhancements (e.g., valid inequalities and problem-reduction strategies that rely on these bounds) to be more effective. To study the impact of fee flexibility on the Composite method's performance, we created two new sets of instances by decreasing and increasing the fee flexibility range by nearly 17% relative to the standard case (i.e., the minimum and maximum fee values provided by the manufacturer). The results in Table 4 show that for all three cases-low, standard, and high levels of fee flexibility-the Composite method (Columns 5, 6, and 7) consistently generates good lower bounds and near-optimal solutions that are significantly better than the Base B&B method (Columns 2, 3, and 4) for various distance and weight range combinations. That is, the level of fee flexibility does not adversely affect the performance of our solution method. Other performance measures (not reported in Table 4) also showed consistent results; for instance, even when fee flexibility was high, the strong model was able to bridge a significant portion of the gap (nearly 36.5% on average compared with 39% in the standard case). To examine whether varying the levels of fee flexibility has a significant impact on the total fees paid, we compared the Final UB for the new instances as a fraction of the Final UB for the standard case. As the related statistics in Columns 8 and 9 demonstrate, increasing (decreasing) the fee flexibility reduces (increases) the total fees paid but the deviations from the objective values of the standard case are minimal. With lower flexibility, the total fees increase by around 0.9%, whereas higher flexibility reduces fees by 0.3%.

FTD Savings
To estimate the savings from applying the FTD approach, we first determined the expected compensation using the original 5 × 4 table for the given store demands and distances. In order to avoid overstating the benefits of the FTD approach, rather than use the manufacturer's original fee values (that were not optimized even for the firm's original fee table structure), we obtained the best fee values (to minimize excess payments) for the original 5 × 4 table by solving the fee-setting linear program [FSP] for the corresponding distance and weight ranges. This value, which we denote as F ORIG and refer to as original fees, serves as the benchmark to compare with the best upper bound value of the [FTD] model, denoted as F FTD , that we obtain using our (strengthened) FTD model and Composite solution procedure. We express the savings due to the FTD model over the original fees as (F ORIG -F FTD )/F FTD . Column 12 in Table 3 shows this savings percentage for each scenario. First, by optimizing the range widths of a table