Old Japan Pottery Co. produces glazed bowls and jugs using two furnaces, one to

Question

Old Japan Pottery Co. produces glazed bowls and jugs using two furnaces, one to fire the clay and another to set the glaze. A bowl takes 2 hours to fire in the first furnace, and 1.5 hours for the glaze to set in the second furnace. A jug takes 3 hours to fire in the first furnace, and does not need to glaze in the second furnace. The first furnace has 42 hours of available time, while the second furnace has 22.5 hours of available time. A bowl sells for $30 and a jug sells for $42. How many bowls and jugs should they create to maximize their gross profit? Formulate a linear programming model to solve this problem. List the extreme points and determine the solution graphically. You do not need to submit your graph

Explanation / Answer

Let us consider the no. bowls that we are producing are x and no. of jugs that we are producing is y.

Now our Target Function is f(x,y) = 30x +22.5y

The above functions shows the amount gained by seling x no. of bowls and y no. of jugs.

The constraint functions are as follows:

1. Hours of first furnace : 2x+3y <= 42

2. hours of second furnace: 1.5x<=22.5

Plotting the graph with no. of jugs on y-axis and no. of bowls on x-axis, we get a trapezoidal figure.

The extreme points of the figure would be (x,y)={(0,0), (15,0), (0,14), (15,4)}

The point (15,4) maximises the target function. Hence the now.of bowls to be produced are 15 and no. of jugs to be produced are 4.

Note: Here Target function is considered to maximise the amount gained by selling the bowls and jugs. But what we should actually do is to maximise the profit gained by selling i.e S.P - C.P . Since here we dont have the data of C.P available with us, the only resort remaining is to maximise the amount received by selling the commodities.

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