Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below. Fliptop Model Tiptop Model Available Plastic 3 4 36 Ink Assembly 5 4 40 Molding Time 5 2 30 The profit for either model is $1000 per lot. 1. What is the linear programming model for this problem? 2. Find the optimal solution. 3. Will there be excess capacity in any resource?

Step 1

Let us assume that x1 amount of Fliptop and x2 amount of Tiptop models are produced, then the objective function is to maximize profitability with the constraints on the production limited by the available plastic, ink and time. Hence the LP model is given by the objective function and the three constraints as shown below:

Objective function ($) (OF):                maximize z = 1000x1 + 1000x2

Plastic material constraint ( Eqn. 1): 3x1 + 4x2 <= 36

Ink material constraint (Eqn. 2):         5x1 + 4x2 <= 40

Time constraint (Eqn. 3):                    5x1 + 2x2 <= 30

Non negativity constraints:                 x1, x2 >= 0

Step 2

Since it is a 2 variable problem it can be solved graphically or using a model solver such as MS-Excel ®. The feasible region is defined by the corner points (boundary points) A, B, C, D, E and the boundary lines of the constraint equations 1,2,3 and the objective function OF as shown in the diagram.

The coordinates and the OF values at the corner points are given below:

A (0,0); OF = 0 (intersection of non-negativity constraints)

B (6,0); OF = 6000 (intersection of x2=0 and eqn 3)

C (4,5); OF = 9000 (intersection of eqn 2 and 3)

D (2, 7.5); OF = 9500 (intersection of eqn 1 & 2)

E (0,9); OF= 9000 (intersection of x1=0 and eqn 1)

Step 3

Hence the optimal solution is given by the point D where the OF equation touches the feasible region with the maximum value. There is an excess of 5 units of molding time available.