How can I solve this problem?

Question

I have $N_{\rm C}=8,$ and $N_{\rm U}=25$

Scenario 1:

$$\frac{l_{c,u}}{\sum\limits_{c=1}^{N_{\rm C}}l_{c,u}}\ge 0.1,\quad\forall u,u=1,2,\cdots,N_{\rm U}$$

and

$$\sum_{u=1}^{N_{\rm U}}l_{c,u}\le 1,\quad\forall c,c=1,2,\cdots,N_{\rm C}$$

For each $u$, a maximum of $4$ out of the $N_{\rm C}$ and $l_{c,u}$s can be non-zero.

Scenario 2:

$$\frac{l_{c,u}}{\sum\limits_{c=1}^{N_{\rm C}}l_{c,u}}\ge 0.5,\quad\forall u,u=1,2,\cdots,N_{\rm U}$$

and

$$\sum_{u=1}^{N_{\rm U}}l_{c,u}\le 1,\quad\forall c,c=1,2,\cdots,N_{\rm C}$$

For each $u$, a maximum of $4$ out of the $N_{\rm C}$ and $l_{c,u}$s can be non-zero.

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Which scenario will give the maximum number of users with at least $2$ non-zero $l_{c,u}$s?

Answer 1

I assume that the $N_C$ is the number of customers (or something similar) and the $N_U$ is the number of utilities (or again something similar) that can be used to satisfy the customers' demand. As I understood you want to maximize the number of customers for whom more than 2 utilities used in the model to satisfy the demand. You have these components in your model then (based on my assumptions):

• $l_{c,u}$: Amount of customer $c$'s demand satisfied by utility $u$.
• constraint 1: $$l_{c,u} \ge (0.1)\sum_{c=1}^{N_C} l_{c,u} \ \ \forall u \in \{1,2,...,N_U\} \ \ \text{and} \ \ \forall c \in \{1,2,..., N_C\}$$ which means amount of satisfied demand for each customer and each utility combination can not exceed $10\%$ of all shipments (or service).
• constraint 2: $$\sum_{u=1}^{N_U}l_{c,u}\le 1,\ \ \forall c \in \{1,2,...,N_C\}$$ that means for all customers the amount of service would be less than or equal to the demand.

• constraint 3: The utilities can not serve more than 4 customers (utilities' capacity constraint). For this constraint, you need to define binary variables for each customer and each utility combination ($\forall c,u$) as follow:

  1. decision variable: $$x_{c,u} = \begin{cases} 1 & \text{ if utility } u \text{ served customer }c \\ 0 & \text{ otherwise} \\ \end{cases}$$
  2. then the constraint can be written as: $$\sum_{c=1}^{N_C} x_{c,u} \le 4 \ \ \forall u \in \{1,2,...,N_U\}$$
  3. and of course the indicator constraints like: $$l_{c,u} \le x_{c,u}\times M \ \ \forall c,u $$

• Now the objective function can be: $$\text{maximize} \ \ z=\sum_{\forall c}\sum_{\forall u} (x_{c,u}-2)$$

With this linear programming, you can compare the two scenarios that you mentioned just by modifying the constraint 2's RHS. I didn't put the formulation in standard LP form just to explain every constraint separately. Even if my assumptions are not correct, I hope this model gives you at list an idea to pursue.