Assume that production of a specific item i costs u_i per unit, but there is an additional fixed charge of w_i if we produce item i at all.

For instance, w_i could be the cost of setting up a production plant, initial cost of equipment etc.

Then the cost of producing x_i units of product i is given by the discontinuous function:

Using AMPL MP-based or Constraint Programing drivers, we can minimize the total production cost of n products with the following objective function:

```
minimize TotalCost:
sum {i in 1..n} (u[i]*x[i] + if x[i]>0 then w[i]);
```

For older MIP drivers, a linearized model is needed. It can be constructed using auxiliary binary variables and connecting constraints as follows:

```
var z{1..n} binary;
minimize TotalCost:
sum {i in 1..n} (w[i]*z[i] + u[u]*x[i]);
s.t. ConnectBinaries {i in 1..n}:
x[i] <= M*z[i]; ## Use a big-M constraint to enforce z[i]=0 ==> x[i]=0
```

Full example of a facility location model with setup costs:

```
# Set up the sets
set I := 1..10; # potential facility locations
set J := 1..50; # customers
# Set up the parameters
param w {i in I} = Normal(60, 20); # fixed costs for each facility
param u {i in I, j in J} = Uniform(10, 30); # transportation costs from each facility to each customer
param d {j in J} = Uniform(5, 10); # demand for each customer
# Set up the decision variables
var x {i in I, j in J} >= 0 <= d[j]; # amount of demand for customer j satisfied by facility i
# Set up the objective function
minimize total_cost:
sum {i in I, j in J} u[i,j]*x[i,j] +
sum {i in I} if sum {j in J} x[i,j] > 0 then w[i];
# Set up the constraints
subject to demand_constraint {j in J}:
sum {i in I} x[i,j] = d[j];
```

Linearized model:

```
# Set up the sets
set I := 1..10; # potential facility locations
set J := 1..50; # customers
# Set up the parameters
param w {i in I} = Normal(60, 20); # fixed costs for each facility
param u {i in I, j in J} = Uniform(10, 30); # transportation costs from each facility to each customer
param d {j in J} = Uniform(5, 10); # demand for each customer
# Set up the decision variables
var z {i in I} binary; # whether or not to build a facility at location i
var x {i in I, j in J} >= 0; # amount of demand for customer j satisfied by facility i
# Set up the objective function
minimize total_cost:
sum {i in I} w[i]*z[i] + sum {i in I, j in J} u[i,j]*x[i,j];
# Set up the constraints
subject to demand_constraint {j in J}:
sum {i in I} x[i,j] = d[j];
subject to capacity_constraint {i in I}:
sum {j in J} x[i,j] <= z[i] * sum {j in J} d[j];
```

Running both models with HiGHS:

```
ampl: option solver highs;
ampl: solve;
HiGHS 1.4.0: optimal solution; objective 4796.760641
0 simplex iterations
1 branching nodes
```

Running both models with Gurobi:

```
ampl: option solver gurobi;
ampl: solve;
Gurobi 10.0.0: optimal solution; objective 4796.760641
44 simplex iterations
1 branching nodes
```

Documentation on the expression-valued `if-then-else`

operator can be found in the MP Modeling Guide.