a small soft drink company produces only two types of drink, awefull A small sof

Question

a small soft drink company produces only two types of drink, awefull A small soft drinks company produces only two types of drink, Awefull (A) and Burpit (B). The production manager is concerned to get the best use of the bottling machinery at her disposal, and has asked you to investigate the problem. Essentially, the bottling process involves washing, filling and capping. One bottle of Awefull takes an average of 0.8 seconds to wash, 0.9 seconds to fill and 1 second to cap. A bottle of Burpit requires 1.2 seconds to wash, 1.5 seconds to fill and 1 second to cap. The company has one machine of each type (washing, filling and capping), and each machine is available for 10 hours each day. The company has long standing daily orders for at least 10000 bottles of Awefull, and at least 5000 bottles of Burpit, which must still be honoured. An investigation of the profit margins for the two drinks reveals that the contribution for Awefull is 204 per bottle, whilst Burpit generates 304 per bottle. The time to switch between productions of the two drinks is negligible, and the company is sure that they can sell all they produce. Formulate a linear programming model for this problem and solve it using the araphical method

Explanation / Answer

Let x be the number of bottles produced of the drink type Awefull and y be the number of bottles produced of the drink type Burpit.

Time constraint on Washing Maching
0.8*x + 1.2y <= 36000 (seconds)
8x + 12y <= 360000
2x + 3y <= 90000

Time constraint on Filling machine
0.9x + 1.5y <= 36000 (seconds)
9x + 15y <= 360000
3x + 5y <= 120000

Time constrint on Capping machine
x + y <= 36000 (seconds)

Demand constraint
x >= 10000
y >= 5000

Here the profit needs to be maximized
Zmax = 0.20x + 0.30y

Hence formulation is
maximize Z = 0.20x + 0.30y
subject to,
2x + 3y <= 90000
3x + 5y <= 120000
x + y <= 36000
x >= 10000
y >= 5000

Solving above set of equations simultaneously we get the below set of vertices for the feasible region

Vertex Lines Through Vertex Value of Objective
(30000,6000) 3x+5y = 120000; x+y = 36000 7800 Maximum
(10000,18000) 3x+5y = 120000; x = 10000 7400
(31000,5000) x+y = 36000; y = 5000 7700
(10000,5000) x = 10000; y = 5000 3500

hence the optimal production of Awefull and Burpit is 30000 bottles and 6000 bottles respectively

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