$20$:$07$

$.$II LTE

Sample Final Questions

$12/22$

Mathematical model of the p$-$median problem

Decision variables and parameters that will be used in the mathematical model.

$x_{i,j}=$ amount $of$ customer $is$ demand satisfied $by$ facility $j$

$Y_{i,j}=1,if$ customer $iis$ assigned $to$ facility $j$; $0,o.w.$

$Z_j=1,if$ there exists a facility $at$ location $j$; $0,o.w.$

$C_{i,j}=$ the distance between i and $j$

$D_i=$ demand $of$ customer $i$

Write down the mathematical model of p$-$median problem when each customer has a demand of $D_i.$ In the p $-$median problem all customers are served, we want to locate $"$ p $"$ facilities, and each customer is served by one facility only. The goal is to minimize the total transportation cost. It is assumed that all facilities have the same fixed cost of opening and this is why it is ignored in the model.

A$)$Propose the mathematical optimization model of p$-$median problem without using $Y_{i,j}$ decision variable. $(15$ points$)$

B$)$ Now we relax the assumption that each customer may be served by one facility and add the following restrictions $(15$ points$)$:

a$.$ each facility can only serve at most $3$ customers.

b$.$ the capacity of located facilities cannot exceed $Q.$

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