(Solved Homework): The question is about GAMS programming language which is used in linear programming methods. So that, the

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.Products A, B, C, D and E require the same processing facility for their manufacture. At any one time, therefore, this facility can produce only one of the five product types. The time required per unit of product is Table 1 Product Time( Minutes) A 1 B 1,25 C 075 DI 1.25 E 12 The estimated potential demand in the next planning period in the four sales areas supplied by the plan is as follows: Table 2 Demand(no. of units) Product Type A B C D E Sales Area 16.000 3.000 6.000 5.000 6.000 5.000 9.000 Sales Area 22.000 Sales Area 34.000 1.000 3.000 2.000 Sales Area 4 2.000 5.000 2.000 3.000 Total 12.000 6.000 17.000 15.000 14.000 Se ng prices of the same products are identical in all areas; differences in transportation costs, however, mean that the profit per unit sold will be different in the areas, as indicated below. Table 3 Profit(mu/unit) A B C D E Sales Area 14,804,75 3,755,25 Sales Area 23,75 3,305,005,04 Sales Area 34,005,00 5,155,25 Sales Area 4- 4,50 3,50 5,05 5,15 The factory operates an around the clock shifts system 7 days per week. The available capacity of production facility during the planning period is 50.000 minutes. Every time there is a changeover from the production of one product to another, set-up time is required. This set-up time is negligible a) Formulate an LP model to achieve profit maximization within the planning period. Determine the quantity of each product to be produced, and allocation to be made to each sales area Clearly define all decision variables Write explanations for each constraint b) Solve the proposed LP model using GAMS In the output, display the quantity of each product to be produced, allocation to be made to each Sales area c) Form Excel file/files that include Tables 1 to 3. Modify your GAMS model that you have developed in part b so that it imports Tables 1to 3 from Excel files. Export the results you have found to an Excel file d) If you could satisfy all of the demand in four areas, how many extra minutes would you like to have? e) Determine the products which all demands are satisfied for all sales areas. For unsatisfied demand products, conduct a sensitivity analysis by changing the time required per unit of productto ensure that all demands would be satisfied for that product

Expert Answer

 Let a1 units of Product A be sold in Sales Area 1

Let a2 units of Product A be sold in Sales Area 2

Let a3 units of Product A be sold in Sales Area 3

Let a4 units of Product A be sold in Sales Area 4

Let b1 units of Product B be sold in Sales Area 1

Let b2 units of Product B be sold in Sales Area 2

Let b3 units of Product B be sold in Sales Area 3

Let b4 units of Product B be sold in Sales Area 4

Let c1 units of Product C be sold in Sales Area 1

Let c2 units of Product C be sold in Sales Area 2

Let c3 units of Product C be sold in Sales Area 3

Let c4 units of Product C be sold in Sales Area 4

Let d1 units of Product D be sold in Sales Area 1

Let d2 units of Product D be sold in Sales Area 2

Let d3 units of Product D be sold in Sales Area 3

Let d4 units of Product D be sold in Sales Area 4

Let e1 units of Product E be sold in Sales Area 1

Let e2 units of Product E be sold in Sales Area 2

Let e3 units of Product E be sold in Sales Area 3

Let e4 units of Product E be sold in Sales Area 4

Our objective function is to maximize profit i.e.

Maximize Z = 4.8a1 + 3.75a2 + 4a3 + 4.75b1 + 5b3 + 4.5b4 + 3.75c1 + 3.3c2 +3.5c4 + 5.25d1 + 5d2 +5.15d3 + 5.05d4 + 5.04e2 + 5.25e3 +5.15e4

Now applying the time constraints:

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

Demand Constraints

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

a1 + a2 + a3 < =12000

b1 + b3 + b4<= 6000

c1 + c2 + c4 <= 17000

d1 +d2 + d3 + d4<= 15000

e2 + e3 + e4 <=14000

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

a1< =6000 , a2< =2000, a3 < = 4000

b1 <= 3000,  b3 <= 1000 ,  b4 <= 2000

c1 <=6000 c2 < =6000, c4 < =5000

d1<=5000,   d2<=5000,   d3<=3000,   d4<=2000

e2 <= 9000,  e3< =2000,  e4 <= 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
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