The Effect of Scale Economies

Economies of scale play a fundamental role in network design and operations decisions, and thus affects network performance (Camargo et al. Reference Canning2009). Hubs operating on a large scale enjoy lower operating costs compared with smaller-scale hubs (King Reference Martinez, Hand and Pra1973, Jacobsen Reference King2013). Large-scale operations experience more efficient aggregation, storage, distribution, and marketing of products from local producers to consumers (Matson et al. 2013). As a result, smaller hubs face challenges, such as inefficiency in operating those hub agribusiness services for local producers which allow for an increase in their operating costs and thus limit their ability to supply larger markets (Pressman and Lent 2013). If economies of scale are ignored in empirical estimates of the benefits from food system infrastructure investment and food system operations, the empirically generated optimal hub network solution could be misleading (Ippolito Reference Jablonski, Schmit and Kay1975).

In this study, the characteristics and functionalities of modeled aggregation hubs are more similar to those mainstream and conventional aggregation and distribution facilities such as merchant wholesalers. The operation pattern of those counterparts in reality provides valuable information to build a model that mirrors the actual system. To measure scale effects inherent in fresh produce aggregation hub operations, we first compiled 2012 Economic Census data on annual sales and operating costs of fresh fruit and vegetable wholesale establishments (U.S. Census Bureau 2012). In many states, a subset of the counties are not reported for reasons of disclosure, so in these cases, we compile multicounty data by taking the difference between statewide totals and published county totals.Footnote ³ The resulting dataset is comprised of 114 county observations and 38 multicounty observations for (i) total annual operating costs, and (ii) total annual product sales.

Our hypothesis is that for a county (s) with a specific annual capacity (c) (the quantity handled at a county in a given year), annual operating costs (TC_s) include a fixed component $\lpar h_c^0 \rpar$ that is independent of product volume handled, and a variable component $\lpar h_c^1 \rpar$ that varies with volume handled (Q _s). We consider sizes of regional aggregation hub operations C  =  1 to c, where 1 is the smallest and c is the largest. Then the relationship between annual operating costs and county throughput of fresh produce is:

(1)$${\rm T}{\rm C}_s\, = \, \, \; h_c^0 + \, h_c^1. \, Q_s $$

From the 152 county-wide observations of annual operating costs and throughput or hub sales, we convert the latter from dollars to pounds by dividing by the national weighted average 2012 wholesale price of 44.01 cents per pound for all fresh produce marketed in the United States (USDA National Agricultural Statistics Service 2018).

To represent the actual cost of hub operations in the model, we measure the effect of economies of scale inherent in hub operations based on our observations. Due to the limited number of observations, we disaggregate them into four groupings. After conducting sensitivity analyses with numerous groupings of the 152 observations, we identified and defined the four hierarchy groupings that provide the best empirical fit for the regression. The aggregation allows for the appropriate scale level for each grouping, which is based on the range of handled product quantity for observations in each grouping,

• C ₁: 130,000 units and below (36 lowest counties or equivalents by total annual throughput)

• C ₂: 130,001–450,000 units (next 39 counties by total annual throughput)

• C ₃: 450,001–1,500,000 units (next 39 counties by total annual throughput)

• C ₄: 1,500,001 units and above (highest 38 counties by total annual throughput)

To assess whether the mean values for the dependent variable (operating costs) differ between any two of the four groupings, we conduct two-tail t-tests between every two groupings. The absolute value of t-statistic ranges from 6.37 to 11.21. There is a statistically significant difference in the means of operating costs between groupings. We reject the null hypothesis that any two of the four groupings are equal.

For each grouping, the relationship between total operating cost (TC) and sales volume (Q) is obtained using OLS and shown in Table 1. The cost estimations will be used later to build the base scenario of the model.

Table 1.

Regression Results for Economies of Scale Effect Based on 2012 Economic Census Data

Note: A single asterisk indicates significance at a 5 percent level, and a double asterisk indicates significance at a 10 percent level. These are county-level, not establishment-level hubs.Footnote ⁴

Source: Compiled by Authors based on U.S. Census Bureau (2012)

The regression results demonstrate a noticeable scale effect in hub operations. While fixed costs (intercept) increase with hub size, the marginal costs decrease with hub size. Specifically, the larger the hub size, the lower the marginal cost per unit (one thousand pounds) handled. In this manner, handling costs of an aggregation hub are based on the amount of product flow carried by the hub and endogenously respond to flow by rewarding the shipper for greater volumes shipped. Under cost minimization principles, the numbers and scale of facilities to be established typically are endogenously determined.

Each of the four marginal cost parameter estimates exhibits a highly significant t-statistic. While the fixed cost parameters are consistent with expectations and contribute to the overall goodness of fit in all four hub size equation estimates, they are also statistically significant at certain levels. Building on these results, the cost estimates in Table 1 are incorporated into the hub location model to demonstrate how they affect spatial network structures.

Data

USDA's National Agricultural Statistics Service (NASS) reports state-level annual production statistics for 21 different fresh market vegetable crops across 37 states (USDA National Agricultural Statistics Service 2018). The 2012 statistics are allocated to counties in these states based on harvested acreage by county, using the 2012 Census of Agriculture for total fresh market vegetables (USDA National Agricultural Statistics Service 2018). For states that are not covered in the annual NASS reports but are covered in the Census, harvested acreage data from the Census are multiplied by yield per acre data in nearby states to impute production.

In the Census data, harvested acreage of some counties is suppressed in order to protect the confidentiality of individual responders. To overcome data suppressions in the Census data, a constrained maximum likelihood mathematical programming model is used. The model minimizes adjustments to the variance-weighted initial estimates of suppressed county harvested acreage statistics to align these estimates with the adding up requirements of the hierarchical data in the 2012 Census. This hierarchy includes requirements that the sum of harvested acreage across all commodities equals the published county-wide total harvested acreage across all covered crops, and that the sum of harvested acreage for specific crops across all counties in a State equals the published statewide total for each crop. By simultaneously solving a state/county model and a national/state model for the same commodities, the model produces maximum likelihood estimates of all suppressed county harvested acreage statistics using the method described in Canning (Reference Cook2011). The same approach was used for estimates of both fresh market fruit production and fresh market vegetable production in US counties. Combined production for the subset of these 55 crops produced in each county is converted to a common unit (one thousand pounds) and summed to a single production statistic per county. Production estimated from suppressed county harvested acreage only accounts for a very small proportion of annual domestic production (1 percent) so the estimation error should have very limited influence on the aggregation hub location solution.

County-to-county shipping distances are based on the Oak Ridge National Laboratory multimodal impedance network data product, and average shipping cost statistics are reported by USDA Agricultural Marketing Service (Reference MacDonald, Korb and Hoppe2010).

According to 2012 USDA National Agricultural Statistics Service data, 2,467 counties, or 75 percent of all counties in the contiguous US, grew vegetables or fruits. Annual production statistics are disaggregated into seasonal marketing segments to more accurately account for the highly variable geographic disposition of annual fresh produce production. This is achieved by identifying the beginning and ending dates for the marketing seasons of each fresh produce crop in each state (USDA National Agricultural Statistics Service 2006, 2007). Many states have multiple growing regions with different marketing seasons for the same crop, and in some cases, a single crop may have two marketing seasons. Annual production data by county is allocated to each day of the identified marketing season(s) according to a normal distribution under the assumption that 100 percent of available product is marketed within 4 standard deviations of the marketing season midpoint. This imputed daily marketing data is then aggregated into four marketing seasons: January to March; April to June; July to September; October to December.

Monthly 2012 fresh produce import data by county of unladingFootnote ⁵ are compiled from U.S. Census Bureau (2012) sources. 81 counties in 30 states import fresh produce. Statistics reported in metric tons are converted to 1,000-pound units and aggregated into the same four marketing seasons as domestic production.

Figure 1 shows continental US fresh produce production plus imports mapped across four seasons. Production and imports are unevenly distributed across seasons, and the main production/import areas are on the West Coast and Florida. Among them, California has the highest production and import levels, followed by Florida and Washington. The Northeast and Upper Midwest also have high production and import levels, while production and imports in some Rocky Mountain States and Plains States are low.

Figure 1.

Distribution of fresh produce production in the U.S. across seasons. (a). Production in Season 1 (January–March). (b). Production in Season 2 (April–June). (c). Production in Season 3 (July–September). (d). Production in Season 4 (October–December).

Source: USDA/NASS (2012) and author calculations.

Market Setting and Optimization Model

Produce assembly at aggregation hub s involves shipment from surrounding production and import regions (or nodes) via a domestic freight network that connects all nodes and aggregation hubs. Volpe, Roeger, and Leibtag (Reference Volpe, Roeger and Leibtag2013) show that domestic shipments for produce commodities are almost exclusively by truck. It is assumed in this study all shipments are transported to regional hubs by truck. Transportation costs between node-hub pairs are often defined to be proportional to the distance between them. However, transportation costs are also subject to constraints, such as road condition, speed reduced on evening commute, and traffic congestion which influences travel time and speed (Novaco et al. Reference Pressman and Lent1990). To model transportation costs between production nodes and aggregation hubs, each link in the network is assigned an impedance value other than actual mileage. Impedance represents a measure of the amount of resistance, or cost, required to traverse a path in a network or to move from one element in the network to another. High impedance values indicate more resistance to movement. For this study, we use a national average shipping rate per impedance mile (the average cost of transporting 1,000 pounds of products for one impedance mile). If the shipping rate per impedance mile is converted to the shipping rate per highway mile, the latter is higher in more congested areas, and vice versa. We use county-to-county highway impedance measures from version 3 of the national multimodal impedance network database developed by Oak Ridge National Laboratory (Reference Teo and Shu2011), which is based on population weighted centroids.

Establishment home regions (business addresses) of fresh produce first-handlersFootnote ⁶ are increasingly based in key production areas, but they source from shifting production regions that follow climate-determined seasonal patterns throughout the year (Cook Reference Dimiero and Mayfield2011). To stay profitable, an establishment seeking to source from outside its home region must assemble and handle the produce at least as efficiently as establishments based in the originating region. Our approach is to design a national “home field” hub system where aggregation hubs built to one of the four scales reported in Table 1 are located throughout select growing/import regions in order to minimize total assembly and first-handler costs across all seasons and growing/import regions.

Whether the same hub locations and sizes would result from a model that allows for individual establishments to operate several “virtual hubs” in multiple locations is beyond the scope of this study. Such a model would require operating cost and throughput data that distinguishes between “home field” and “virtual” hub operations to calibrate the cost functions. Such data was not available for this study. We recognize that both types of operations exist in practice.

Our approach is realistic provided the cost structure of operating either hub type for a given annual throughput is similar. We are not aware of data or compelling arguments that suggest, a priori, a systematic bias is introduced by our assumption of similar cost structures. This issue does merit further investigation in future research. A practical limitation of our approach is the disconnect between the model results which report the physical location (county), cost, and throughput of all aggregation hubs, and the published Economic Census data which reports total costs and throughput reported by all establishments with business mailing addresses in each county. It is important that we be mindful of this difference when comparing the model results with the published data.

While the classic facility location model considers only a trade-off between facility fixed cost and transportation cost, this hub network design problem considers trade-offs between facility fixed cost, transportation cost, and operation cost (Teo and Shu 2004). We formulate this national hub system optimization problem as a mixed integer linear programming (MILP hereafter) model with the objective of minimizing total costs associated with product assembly and aggregation hub setups and operations. The optimization problem requires a definition of restrictions to ensure that total production and imports by county and average per unit supplier and shipping costs meet observed statistics. The following notation is introduced for the model:

Sets:

• I = {1,…,i} denotes marketing seasons in a year;

• F = {1,…,f} denotes a set of production locations;

• S = {1,…,s} denotes a set of aggregation hub candidate locations;

• C = {1,…,c} denotes capacity level of aggregation hubs; each capacity level has an interval span;

Parameters:

• $p_{i\comma f}^{}$ denotes production plus imports at location f in marketing season i;

• $d_{f\comma s}^{}$ denotes distance between node location f and aggregation hub location s (impedance miles);

• t denotes fixed transportation cost ($ per thousand-pound impedance mile);

• $h_c^0 \;$ denotes fixed annual handling cost of a c level aggregation hub;

• $h_c^1 \;$ denotes marginal handling cost per unit of annual throughput of a c level aggregation hub;

• $V_c^1 \;$ denotes maximum annual throughput of a c level aggregation hub;

• $V_c^2 \;$ denotes minimum annual throughput of a c level aggregation hub;

Variables:

• $x_{i\comma f\comma s}^{}$ denotes quantity shipped from node f to aggregation hub s in marketing season i;

• $Z_{c\comma s}^{}$ denotes an integer variable  =  1 if location s has an aggregation hub with capacity c, and 0 otherwise;

• Q _s denotes annually assembled throughput for aggregation hub s;

Functions:

• TC denotes system-wide total assembly plus first-handler costs.

The objective function and system constraints of a model to solve this problem are given in equations 2–9.

Minimize:

(2)$${\rm TC} = \; \lpar{\mathop \sum \limits_{c\in C} \mathop \sum \limits_{s\in S} \lcub {h_c^0 \cdot Z_{c\comma s} + h_c^1 \cdot Q_s} \rcub \; } \rpar+ \lpar{\mathop \sum \limits_{i\in I} \mathop \sum \limits_{\,f\in F} \mathop \sum \limits_{s\in S} \lcub {x_{i\comma f\comma s}\cdot d_{\,f\comma s\; } \cdot t} \rcub } $$

Subject to:

(3)$$\mathop \sum \limits_{s\in S} x_{i\comma f\comma s} = p_{i\comma f}^{} \quad \forall \; i\comma \; f $$

(4)$$\mathop \sum \limits_{i\in I} \mathop \sum \limits_{\,f\in F} x_{i\comma f\comma s} = Q_s\quad \forall \; s $$

(5)$$Z_{c\comma s}\in \lcub {1\comma \; 0} \rcub \quad \forall \; c\comma \; s$$

(6)$$\mathop \sum \limits_{s\in S} Z_{c\comma s} \le 1\quad \forall \; s$$

(7)$$Q_s \le Z_{c\comma s}V_{c\comma s}^1 \quad \forall \; c\comma \; s$$

(8)$$Q_s \ge Z_{c\comma s}V_{c\comma s}^2 \quad \forall \; c\comma \; s$$

(9)$$x_{i\comma f\comma s} \ge 0\quad \forall \; i\comma \; f\comma \; \; s$$

The objective function (Equation 2) minimizes the total cost, which includes fresh produce transportation (expression in second outer brackets) plus first-handler services (expression in first outer brackets). Equation 3 ensures that in each marketing season the total quantity transported from county f to all aggregation hubs S are equal to total quantity produced and imported in county f in the season. That is, all products must be assembled into aggregation hubs across marketing seasons. Equation 4 is an accounting identity, equation 5 provides the binary condition-build/not-build, and equation 6 ensures that 0 or 1 aggregation hub of scale c is built in county s. Equations 7 and 8 define the quantity-handling thresholds and limits for an aggregation hub of each capacity level. Together, equations 7 and 8 require that aggregation hubs of size c, for c  =  1 to 4, will only be constructed in county s if it attracts enough shipments to meet its handling threshold $\lpar V_{c\comma s}^2 \rpar$ and does not attract shipments in excess of its handling limit $\lpar V_{c\comma s}^1 \rpar$. Equation 9 ensures that shipments only flow from nodes to aggregation hubs and not vice versa.

The model defined in equations 2–9 describes a national system of regional fresh produce aggregation hubs that seek to aggregate all available fresh produce sourced from US farms and US custom ports spanning vast geography. Any candidate hub location can offer lower per-unit operating costs as it reaches each of four handling thresholds with per-unit operating costs decreasing as each higher threshold is reached. But to reach each of the increasing threshold targets, a candidate hub location must aggregate produce from a larger geographic area. Achieving each higher level of aggregation will result in higher transportation costs. Candidate hub locations that can achieve these thresholds with lower transportation costs will be favored. For a solution to the model defined in equations 2–9 to be optimal, fresh produce originating across all locations cannot decrease its combined aggregation (transportation) and handling costs by redirecting any product to a different aggregation hub location in any marketing season.

Model Setting

In this study, the handling capacity of an aggregation hub is the quantity that can be managed at the aggregation hub during a given year. To incorporate scale effects inherent in aggregation hub operations into the model, we use the operating cost estimates reported in Table 1. The handling capacity of hubs defined for our regression analysis also defines the four different annual thresholds for the quantity of products handled at aggregation hubs over the four marketing seasons. Following the regression pattern of operating costs reported in Table 1, when the scale effect is present, an aggregation hub handling a higher quantity of products enjoys lower marginal costs. An aggregation hub must handle a minimal level of product quantity to achieve a certain level of scale effect, and this motivates the construction of large capacity aggregation hubs.

In reality, factors influencing hub operations and maintenance are complex. The food system has experienced significant changes over the past decades and today's food system has been shaped historically by a range of internal and external social and economic drivers that have evolved with time as well. In this study, the regression results for fixed costs and marginal costs quantitatively reflect the effects of different influential factors at an aggregation level. Fresh produce facility operating data from the 2012 Economic Census indicate the increased dominance of large-scale facilities in fresh produce assembly and distribution across the US when compared with data from the 2007 Economic Census. The marginal cost margin in regressions of 2012 data increases when compared with the regression results using 2007 annual sales and operating costs data (see Table A1 in Appendix I). Here the marginal cost margin is defined as the marginal cost difference between two adjacent aggregation hub levels. The difference between sets of results indicates a possible trend of a more pronounced scale effect stemming from scaling up hubs in the future. In order to identify the effect of scale economies on the optimal solutions of the problem, we design four scenarios in which different values of the marginal handling costs are assumed.

We set Scenario 1 as a scenario in which the marginal cost is not related to economies of scale. The cost remains constant across aggregation hub levels, that is, $55.74, which is the marginal cost of Level 1 aggregation shown in Table 1. Scenario 1 captures the incremental costs of building to a larger scale while omitting the incremental efficiencies realized by scaling up. We will demonstrate how the modeling results in Scenario 1 will differ from those of the base scenario (Scenario 2) as the existing scale effect is ignored.

Scenario 2 is designed as the base scenario in which the marginal costs to handle 1,000 pounds of product for four levels of aggregation hubs follow the regression results shown in Table 1, that is, $59.74, $49.71, $38.67 and $36.89 for aggregation hub Levels 1 to 4, respectively. The marginal cost margin between Level 1 and Level 2 aggregation hubs is $10.03. This value is $11.04 between Level 2 and Level 3 aggregation hubs, and is $1.78 between Level 3 and Level 4 aggregation hubs. Empirical estimation of operating costs based on more recent observations are more likely to appropriately reflect the current operational environment for hubs, so the solution of Scenario 2 can provide empirical implication for the optimal structure of a hub network in the current situation.

To allow for the possible trend of more pronounced scale effect in the future, two additional scenarios are designed. Based on the base scenario, we increase the marginal cost margins between aggregation hub sizes by 20 percent to generate new marginal costs for the four aggregation hub sizes in Scenario 3, and increase the margins by 40 percent for setting up Scenario 4. Eventually, the fixed and marginal costs for each aggregation hub size in each scenario are listed in Table 2. The magnitude of scale effect increases from Scenario 1 to Scenario 4.

Table 2.

Fixed Costs and Marginal Costs for Hubs

^a Note: C ₁: 130,000 units and below; C ₂: 130,001–450,000 units; C ₃: 450,001–1,500,000 units; C ₄: 1,500,001 units and above (unit: one thousand pounds).

^b Note: Scenario 1- no scale effect on marginal cost; Scenario 2- base scenario; Scenario 3–20 percent increase in marginal cost margin; Scenario 4–40 percent increase in marginal cost margin.