This toy problem is presented only to illustrate how GAMS lets you model in a natural way. GAMS can handle much larger and highly complex problems. Only a few of the basic features of GAMS can be highlighted here.

Here is a standard algebraic description of the problem, which is to minimize the cost of shipping goods from 2 plants to 3 markets, subject to supply and demand constraints.

$i = $plants

$j = $markets

$a_{i} = $supply of commodity of plant $i$ (cases)

$b_{j} = $demand for commodity at market $j$ (cases)

$d_{ij} = $distance between plant $i$ and market $j$ (thousand miles)

$c_{ij} = F \times d_{ij}$ shipping cost per unit shipment between plant $i$ and market $j$ (dollars per case per thousand miles)

Distances | Markets | |||
---|---|---|---|---|

Plants | New York | Chicago | Topeka | Supply |

Seattle | 2.5 | 1.7 | 1.8 | 350 |

San Diego | 2.5 | 1.8 | 1.4 | 600 |

Demand | 325 | 300 | 275 |

$F=$ $ per thousand miles

$x_{ij}=$ amount of commodity to ship from plant $i$ to market $j$ (cases), where $x_{ij} > 0$, for all $i,j$.

- Observe supply limit at plant $i: \sum_{j}{x_{ij}} \le a_{i}$ for all $i$ cases
- Satisfy demand at market $j: \sum_{i}{x_{ij}} \ge b_{j}$ for all $j$ cases

The same model modeled in GAMS. The use of concise algebraic descriptions makes the model highly compact, with a logical structure. Internal documentation, such as explanation of parameters and units of measurement makes the model easy to read.

` ````
Sets
i canning plants / Seattle, San-Diego /
j markets / New-York, Chicago, Topeka / ;
Parameters
a(i) capacity of plant i in cases
/ Seattle 350
San-Diego 600 /
b(j) demand at market j in cases
/ New-York 325
Chicago 300
Topeka 275 / ;
Table d(i,j) distance in thousands of miles
New-York Chicago Topeka
Seattle 2.5 1.7 1.8
San-Diego 2.5 1.8 1.4 ;
Scalar f freight in dollars per case per thousand miles /90/ ;
Parameter
c(i,j) transport cost in thousands of dollars per case ;
c(i,j) = f * d(i,j) / 1000 ;
Variables
x(i,j) shipment quantities in cases
z total transportation costs in thousands of dollars ;
Positive variables x ;
Equations
cost define objective function
supply(i) observe supply limit at plant i
demand(j) satisfy demand at market j ;
cost .. z =e= sum((i,j), c(i,j)*x(i,j)) ;
supply(i) .. sum(j, x(i,j)) =l= a(i) ;
demand(j) .. sum(i, x(i,j)) =g= b(j) ;
Model transport /all/ ;
Solve transport using LP minimizing z ;
```