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Food insecurity affects more than 41 million people annually in the United States. Within the Feeding America network, approximately 200 food banks are working throughout the US to serve people in need with donated food. Satisfying hunger need of food insecure people with limited supply is a challenge for these food banks. A numerical study is performed on data from Food Bank of Central and Eastern North Carolina (FBCENC) to capture the major attributes controlling its food distribution system. FBCENC seeks to distribute donated food equitably so that each service area (county) receives food proportional to its demand while minimizing the undistributed food donations. In addition to seeking equitable and effective food distribution policies, FBCENC wants to identify distribution branches to maximize the accessibility of the counties to donated food. An assignment and distribution model is developed to minimize the cost of maintaining a user-specified cap on the maximum inequity in food distribution. A sensitivity analysis between the user-specified maximum inequity cap and effectiveness shows the effectiveness of donated food distribution can be improved significantly by sacrificing equitable distribution slightly.