By Jenny Morber
When a fledgling company writes its business plan, the first worry is often demand — namely whether enough people will buy the product. Nonprofit supply chains face different motivations and uncertainties, and those that address food scarcity may instead ask how to best distribute a limited supply. But what does “best” mean? Food banks aim to fairly provide the most food to those with the greatest need while also minimizing waste. Sometimes the former and latter goals are at odds.
In a recent paper, Irem Sengul Orgut (Lenovo), Julie Ivy and Reha Uzsoy (North Carolina State University), and Charlie Hale (Food Bank of Central & Eastern North Carolina (FBCENC)) describe a model to help food banks balance equity and effectiveness when food supply is uncertain [1]. The researchers draw on an eight-year partnership with the FBCENC to optimize food distribution at the county level.
The FBCENC, a Feeding America affiliate based in Raleigh, N.C., distributes food to 34 counties through six branch locations. Unfortunately, high need and low supply make it impossible to satisfy demand. Hence, the FBCENC instead seeks to dispense food across counties in proportion to the population experiencing poverty, so that each food-insecure person ideally receives the same amount of donated food. In short, it strives for equity. However, data suggests that distribution is inequitable, with some counties remaining underserved and others reporting food waste.
More effective food distribution strategies would minimize food waste by directing the majority of donations to high-capacity food banks. But this solution means that people in some areas would be consistently underserved. “If you only care about equity, one very trivial solution is just to not ship anything,” Sengul Orgut said. “It’s perfectly equitable but it’s not effective. On the other hand, if you only want to ship out maximum food, you would look at each distribution location and stack its capacity to the limit. But then you are not looking at the population size that the locations serve, and their distribution ends up becoming inequitable. So we have these two contradicting objectives.”
Adding to this complexity is the fact that the capacity—amount of food each agency can receive in a given week or month—fluctuates in the FBCENC distribution area. Demand remains mostly stable, but budget, transportation, workforce, and food storage capability change over time. Some food banks must temporarily close if they fail to meet required equity targets, and volunteer numbers wax and wane with the changing seasons.
Fairness in the Face of Uncertainty
The robust optimization model for capacity uncertainty (C-RM) helps food banks distribute goods in a way that can absorb random capacity changes across the region’s multiple counties. The team reasoned that while capacity may be uncertain, it is likely uncertain within a range. The C-RM allows users to specify a particular capacity robustness within this range while attempting to maximize the total amount of distributed food. One can then balance equity and effectiveness according to each decision-maker’s specific needs.
The researchers formulated the C-RM as follows, where \(n\) is the number of counties in the service region:
\[\textrm{max} \sum_{l=1}^n \:\:X_l, \tag1 \]
subject to the following constraints:
\[\frac{X_j}{\Sigma^n_{l=1} X_l} - \frac{D_j}{\Sigma^n_{l=1}\:\:D_l} = 0\forall j \in J, \tag2\]
where \(j\) is the county index, \(X_j\) is the amount of food shipped to county \(j\), \(J\) is the set of counties in the food bank service area, \(D_j\) is the demand of county \(j\), and \(\widehat{C_j}\) is the nominal capacity value for county \(j\).
\[ X_j \le \widehat{C_j} - \varepsilon_j \varphi_j \forall j \in J, \tag3 \]
where \(\varepsilon_j\) is the maximum negative deviation of county \(j\)'s capacity from its nominal value \(\widehat{C_j}\), and \(\varphi_j\) is the fraction of deviation in county \(j\)'s capacity from its nominal value.
\[\sum^n_{l=1} \:\:\:\varphi_l \ge \rho, \tag4 \]
where \(\rho\) is the robustness control parameter (the uncertainty budget).
\[\sum^n_{l=1} \:\:\:X_l \le S, \tag5 \]
where \(S\) is total supply.
\[\varphi_j \le 1 \forall j \in J. \tag6\]
The objective function \((1)\) seeks to maximize the total amount of food shipped from the food bank while maintaining equity, as expressed in \((2)\). With constraint \((3)\), the model prevents capacity overestimation by stipulating that the quantity of food sent to a county must be less than or equal to its nominal capacity minus the allowed deviation. Since county agencies are not as well equipped for long-term food storage as their suppliers, overestimation of agency capacity leads to greater food waste.
To avoid an overly-conservative solution, \(\rho\) limits the number of parameters that can take their worst-case values. In this model, the minimum capacity-to-demand ratio \(R\) is given by
\[R = \min_{j \in J}\{\frac{C_j}{D_j}\}, \]
where \(C_j\) is a random variable that fluctuates over the interval \([\widehat{C_j} - \varepsilon_j, \widehat{C_j}]\). \(R\) reaches its minimum and maximum when capacities of all counties respectively achieve their lowest and highest values. Unlike in previous models, it must be treated as a random variable. Because county capacities are uncertain, the county with the minimum \(CD\) ratio may fluctuate. The same is true of the value of that ratio.
The C-RM’s optimal solution finds deviations from an agency’s nominal capacity that satisfy constraint \((4)\) with minimal effect on total food distribution. Sengul Orgut et al. again discern that a bottleneck county’s capacity-to-demand ratio sets the overall food distribution when capacity is the constraining factor. In this case, one can express the excess capacity fraction as
\[\varphi^{\min}_j = \min\{1, \frac{\widehat{C_j}D_B - \widehat{C_B} D_j}{D_B \varepsilon_j}\}.\]
No bottleneck exists if the nominal instance is supply-constrained instead. Here, one defines the proportion of excess capacity fraction by
\[\varphi^{\min}_j = \min\{1, \frac{\widehat{C_j}\triangle - SD_j}{\varepsilon_j \triangle}\},\]
where \(\triangle\) is the capacity deviation coefficient and \(S = \triangle \min_{j \in J} \frac{\widehat{C_j}-\varepsilon_j \varphi_j}{D_j}.\) If the calculated excess capacity fraction is greater than or equal to the uncertainty budget \(\rho\), total distribution is equal to the nominal problem, such that \(\Sigma^n_{l=1}\: X_l^* = \min \{S, R \triangle\}\). But if the excess capacity fraction is less than \(\rho\), a set of counties can achieve maximum deviation from their nominal values without affecting the optimal solution. That set is expressed as
\[J_E = \{j \in J| \frac{\widehat{C_j} - \varepsilon_j}{D_j} > \min \{\frac {S}{\triangle}, \frac {\widehat{C_B}}{D_B}\}\}.\]
One can determine the optimal solution by listing the counties in decreasing order of their \(\frac{\widehat{C_j}-\varepsilon_j}{D_j}\) ratios. The optimal distribution is \(X_i^* = D_i \min_{j \in J}\{\frac{\widehat{C_j}-\varepsilon_j \varphi_j^*}{D_j}\}\).
To help food banks implement this model, the team translated it into an algorithm that first detects whether the problem is supply- or capacity-constrained, and then finds optimized distributions for each county according to the model. This is the robust optimization algorithm for capacity uncertainty (see Figure 1).
Figure 1. The robust optimization algorithm for capacity uncertainty provides a practical roadmap for food distributors who seek to optimize the sometimes-competing goals of effectiveness and equity via the robust optimization model for capacity uncertainty (C-RM). Figure courtesy of [1].
But does it work in real life? To illustrate the model’s applicability to real-life scenarios, the researchers input historical data from the FBCENC and ran an experiment based on the assumption that the food bank has received 2,600,255 pounds of food — the average monthly dry goods donation in 2014. They set \(\widehat{C_j}\) equal to the 90th percentile of the empirical distribution of the amount of food shipped to each county from July 2012 to June 2015, and performed experiments for capacity deviation coefficient \(\theta = \{0.1, 0.5, 0.9\}\) to examine capacity uncertainty’s effect on total food distribution (see Figure 2).
Figure 2. The plot of total optimal food distribution versus uncertainty budget for changing capacity range indicates that uncertainty and greater capacity fluctuations necessitate decreases in total food distribution to prevent food waste. Figure courtesy of [1].
As the robustness control parameter increases, one must reduce food shipments to protect against larger deviations from nominal capacity. To optimize distribution at a real food bank, employees can set capacity ranges using historical data and their own experiences/goals. A less risk-averse food bank—or one with well-known capacity ranges—would be expected to set \(\theta\) closer to one, while a more risk-averse bank—or one with increased capacity fluctuation—may set \(\theta\) at a lower value.
Bypassing the Bottlenecks
The results indicate that perfect equity means some food banks consistently receive less food than their capacity allows. What if distributors were more equity-flexible? In their current and previous work, Sengul Orgut et al. show that small deviations from perfect equity yield large increases in the total quantity of distributed food [2]. Using the robust optimization algorithm for equity deviation (E-RA), the team explores how distribution is affected when fewer counties are allowed to deviate from perfect equity, but can do so by a larger margin. The E-RA assumes that capacity is deterministic and known, but deviation from equity is a random variable. This nontraditional use of the robust optimization method permits large deviations at the more granular county level but maintains overall equity within the larger system.
The researchers introduce parameter \(a_j\) to define the proportion of total food shipped to any single county, equal to \(\frac{X_j}{\Sigma^n_{l=1}\:\:X_l}\). The value of \(a_j\) can vary over the range of equity requirements, so that
\[a_j \in [\frac{D_j}{\triangle} - K, \frac{D_j}{\triangle} + K]{j \in J}.\]
Counties with low \(CD\) ratios are at high risk of becoming bottlenecks. The model allows underserving of such counties, allocating the remaining food to those with greater capacity. This solution works in actuality because the FBCENC enables its distributors to ship excess food to whomever can take it, sacrificing equity in lieu of food waste. The team’s model again provides a practical algorithm for real-world use (see Figure 3).
Figure 3. The robust optimization algorithm for equity deviation (E-RA) allows food banks to relax equity requirements for some bottleneck counties in order to achieve greater food distribution for the overall service area. Real-world scenarios that utilize this algorithm show that even very small deviations from perfect equity generate large increases in total food distribution for these counties. Figure courtesy of [1].
Sengul Orgut et al. use historical data to show that the model allows considerable increase in total distribution in other counties, and that distribution deviates from perfect equity only by about one percent in bottleneck counties. Unlike many academic models, this work provides food banks with a practical tool for real decision-making. “In some of the other work we do, you must estimate probability distributions and things like that to solve a model,” Sengul Orgut said. “But in this case, it’s very simple to explain. You’re just looking at ranges.”
This work and other related papers by Sengul Orgut and her colleagues [1-2] highlight the often-overlooked fact that capacity is just as important as supply when demand is high and decision-makers seek equity. In the future, they plan to include stochastic supply to address the problem of supply scarcity.
The aforementioned algorithms are useful for any system with high demand and uncertain capacity. In addition to implementation in food banks across the U.S., potential applications include natural disaster response, wartime supplies, and allocation of funds for growth and supply storage infrastructure within similar organizations.
References
[1] Sengul Orgut, I., Ivy, J.S., Uzsoy, R., & Hale, C. (2018). Robust optimization approaches for the equitable and effective distribution of donated food. Euro. J. Operation. Res., 269(2), 516-531.
[2] Sengul Orgut, I., Ivy, J.S., Uzsoy, R., & Wilson, J.R. (2015). Modeling for the equitable and effective distribution of donated food under capacity constraints. IIE Trans., 48(3), 252-266.
Jenny Morber holds a B.S. and Ph.D. in materials science and engineering from the Georgia Institute of Technology. She is a professional freelance science writer and journalist based out of the Pacific Northwest.