The question is about GAMS programming language which is used in linear programming methods. So that, the question should be coded in GAMS language. Asking for writing this problem in proper code.

## Expert Answer

_{1}units of Product A be sold in Sales Area 1

Let a_{2} units of Product A be sold in Sales Area 2

Let a_{3} units of Product A be sold in Sales Area 3

Let a_{4} units of Product A be sold in Sales Area 4

Let b_{1} units of Product B be sold in Sales Area 1

Let b_{2} units of Product B be sold in Sales Area 2

Let b_{3} units of Product B be sold in Sales Area 3

Let b_{4} units of Product B be sold in Sales Area 4

Let c_{1} units of Product C be sold in Sales Area 1

Let c_{2} units of Product C be sold in Sales Area 2

Let c_{3} units of Product C be sold in Sales Area 3

Let c_{4} units of Product C be sold in Sales Area 4

Let d_{1} units of Product D be sold in Sales Area 1

Let d_{2} units of Product D be sold in Sales Area 2

Let d_{3} units of Product D be sold in Sales Area 3

Let d_{4} units of Product D be sold in Sales Area 4

Let e_{1} units of Product E be sold in Sales Area 1

Let e_{2} units of Product E be sold in Sales Area 2

Let e_{3} units of Product E be sold in Sales Area 3

Let e_{4} units of Product E be sold in Sales Area 4

Our objective function is to maximize profit i.e.

Maximize Z = 4.8a_{1} + 3.75a_{2} + 4a_{3} + 4.75b_{1} + 5b_{3} + 4.5b_{4} + 3.75c_{1} + 3.3c_{2} +3.5c_{4} + 5.25d_{1} + 5d_{2} +5.15d_{3} + 5.05d_{4} + 5.04e_{2} + 5.25e_{3} +5.15e_{4}

Now applying the time constraints:

1(a_{1} + a_{2} + a_{3}) + 1.25(b_{1} + b_{3} + b_{4}) + 0.75(c_{1} + c_{2} + c_{4}) +1.25(d_{1} +d_{2} + d_{3} + d_{4}) + 1.2(e_{2} + e_{3} + e_{4}) <= 50000

Demand Constraints

Production of each product should be less than or equal to the sum of demand in all Sales Area.

a_{1} + a_{2} + a_{3 < =12000}

b_{1} + b_{3} + b_{4<= 6000}

c_{1} + c_{2} + c_{4 <= 17000}

d_{1} +d_{2} + d_{3} + d_{4<= 15000}

e_{2} + e_{3} + e_{4 <=14000}

_{Supply for the product in each area should be less than or equal to the demand for that product in that area}

a_{1}_{< =6000 ,} a_{2}_{< =2000,} a_{3} < = 4000

b_{1} <= 3000, b_{3} <= 1000 , b_{4} <= 2000

c_{1} <=6000 c_{2} < =6000, c_{4} < =5000

d1<=5000, d_{2}<=5000, d_{3}<=3000, d_{4}<=2000

e_{2} <= 9000, e_{3}< =2000, e_{4} <= 3000

Finally the time constraint,

The time required for production should be less than or equal to 50000

i.e. (Time required per unit production of product * Number of Units) for all products <= 50000

1(a1+a2+a3+a4) + 1.25(b1+b2+b3+b4) + 0.75(c1+c2+c3+c4) + 1.25(d1+d2+d3+d4) + 1.2(e1+e2+e3+e4) < =50000

Solved this problem using excel solver. Please check the image with answer. Let me know if you require the spreadsheet.

Optimal Solution is:

Product | Number of Units |

A | 6000 |

B | 0 |

C | 17000 |

D | 11560 |

E | 14000 |