A. Determine the equations for each of the three constraints that are plotted on the attached “Graph 1.”
1. Identify each constraint as a minimum or a maximum constraint.
Nutrient C constraint: 4X + 4Y <= 30 (Maximum constraint) Flavor constraint: 12X + 6Y <= 72 (Maximum constraint ) Color additive constraint: 6X + 15Y <= 90 (Maximum constraint ) B. Determine the total contribution to profit that lies on the objective function (profit line) as it is plotted on the graph if the company produces a combination of cases of Brand X and Brand Y.
Shaded area on the graph shows the feasible region.
P = 40X + 30Y
Based on the graph it is clear that the two corner points are (0, 6) and (6,0).
In order to find more points, we will have to solve the below mentioned equations
(4X + 4Y = 30, 6X + 15Y = 90) and (4X + 4Y = 30, 12X + 6Y = 72)
On Solving 4X + 4Y = 30, 6X + 15Y = 90, we get
X = 2.5, Y = 5
On Solving 4X + 4Y = 30, 12X + 6Y = 72, we get
X = 4.5, Y = 3
The corner points as identified are
(6, 0), (4.5, 3), (2.5, 5) and (0, 6).
Putting (4.5, 3) in P = 40X + 30Y
We get = P = 40*4.5 + 30*3 = 270
We get = Putting (2.5, 5) in We get = P = 40X + 30Y
We get = P = 40*2.5 + 30*5 = 250
We get = Putting (6, 0) in We get = P = 40X + 30Y
We get = P = 40* + 0 = 240
We get = Putting (0, 6) in We get = P = 40X + 30Y
We get = P = 0 + 30*6 = 180
C. Determine how many cases each of Brand X and of Brand Y you recommend should be produced during each production period for optimum production if Company A wants to generate the greatest amount of profit.
We get = P = 40*4.5 + 30*3 = 270
Maximum of We get = P is $270
at X = 4.5 and Y = 3
D. Determine the total contribution to We get = Profit that would be generated by the We get = Production level you recommend in We get = Part C.
We get = Profit at (4.5, 3)
We get = P = 40*4.5 + 30*3 = $270
References
Beckman, M. (2005). An activity analysis approach to location theory. Kyklos, vol. 8
Bonini, C. (2007). Quantitative Analysis for Management. Columbus: McGraw Hill
Cooper, W. (2004). Linear programming. Scientific American, vol. 2, 22-3
Dantzig, B. (2009). Programming interdependent activities, II, mathematical model. Econometrica, vol. 17, pp. 200-11
Goldratt, E. (2006)., The Goal: A Process of Ongoing Improvement. Great Barrington: New River Press.
