Problem Definition:

The Northern Food Bank serves over $500$ low$-$income and homeless families in $5$ counties struggling with food insecurity. The food bank distributes customized $$meal kits$$ containing recipes and pantry ingredients tailored by family size several times per week.

There are four meal kit offerings: Individual $(1$ person$),$ Small Family $(2-3$ people$),$ Medium Family $(4-5$ people$)$ and Large Family $(6+$ people$).$ Each requires different ingredients and quantities to meet nutritional guidelines.

However, fluctuating ingredient donations, limited storage, and tight volunteer packing capacity constraints total weekly meal kits. But total demand, especially from big families, often exceeds what's available. Each week, the cost of the meal kits Northern Food Bank can provide is $$400.$ Due to storage limits$,$ there is a weekly cap of $150$ individual kits and $100$ for pre$-$made small kits. They have enough freezer capacity to stock $60$ medium kits. Since large kits require a significant amount of storage space, the weekly capacity for them is at $50$ kits.

Therefore, similar to airline ticket revenue management, the food bank needs an optimization model to determine the right mix of meal kits to prepare during each distribution $$bank$$ to maximize vulnerable community members supported under constrained supplies while prioritizing larger families.

The problem is to optimize the preparation of meal kit types to maximize the total family members served under changing ingredient availability and logistical constraints.

Problem Formulation: We define the sets, parameters, and variables used to formulate the problem.

Sets:

K : the type of kits $($individual$,$ small, medium, large$)$

F : the family sizes $(1$ member, $2$ members, $3$ members, $4$ members, etc.$)$

Parameters:

Dkf : demand for meal kits of type k prepared for families of size f

Skf : supply of ingredients available for meal kits of type k prepared for families of size f

Ckf : capacity constraint for meal kits of type k for families of size f

Pkf : cost constraint weekly for meal kits of type k for families of size f

Decision variables:

Xkf$$: The number of meal kits of type k prepared for families of size f

Objective function:

Maximize total family members served $=$ Xkf $*$ f

Constraints:

Demand:

Xkf $<=$ Dkf

Supply:

Xkf $<=$ Skf

Cost:

Xkf $*$ Pkf $<=400$

Capacity:

XI $<=150$

Xs $<=100$

Xm $<=60$

XL $<=50$

Sign restriction:

Xkf$>=0$

Solve this on Excel