A company is manufacturing 3 different types of products A, B and C. each product has to be processed on 3 machines M1, M2, M3. Product A requires 3 hours on M1, 2 hours on M2 and 3 hours on M3. B requires 2 hours on M1, 1 hour on M2 and 2 hours on M3. Product C requires 3 hours each on machines M1, M2 and M3 respectively. Available capacity on machine M1 is 100 hours, M2 is 60 hours and M3 is 72 hours. Profit per unit is Rs. 10, Rs. 8, and Rs. 6 for products A, B and C respectively. - 1. Formulate the LPP - 2. Determine optimal solution by simplex method - 3. Write status of each resource and explain shadow prices -

1. Formulation of LPP:

   Let X, Y and Z be the number of units of A, B and C produced.

   Maximize profit: Z = 10X + 8Y + 6Z

   Constraints on the resource (machine):

   M1 max. 100 hours: 3X + 2Y + 3Z ≤ 100

   M2 max. 60 hours: 2X + Y + 3Z ≤ 60

   M3 max. 72 hours: 3X + 2Y + 3Z ≤ 72

2. Determination of optimal solution:

   Introducing slack variables into the constraints:

   $3X + 2Y + 3Z ≤ 100 → 3X + 2Y + 3Z + S_1 = 100 \\ 2X + Y + 3Z ≤ 60 → 2X + Y + 3Z + S_2 = 60 \\ 3X + 2Y + 3Z ≤ 72 → 3X + 2Y + 3Z + S_3 = 72$

   Maximization function becomes: $Z = 10X + 8Y + 6Z+ 0S_1+ 0S_2+ 0S_3$

   

Maximum profit: Z = Rs. 288

Number of units of A & C produced = 0

Number of units of B produced = 36

1. Status of each machine (resource):

   Slack variable $S_3$ is not present in the base; this indicates that machine 3 is fully utilized.

   Slack variables $S_1 \ \& S_2$ are present in the base, and have values of 28 and 24 respectively. This means that 28 hours of machine $M_1$ and 24 hours of machine $M_2$ remain unused.

   Shadow prices:

   The Cz-Zj row values under the slack variables show the shadow prices of all the machines.

   ‘-4’ under $S_3$ shows that if machine $M_3$ is used an hour less, then the profit will reduce by Rs. 4, and if used an hour more, then the profit will increase by Rs. 4. So the company is advised to increase the capacity of machine $M_3$.

   Rs. 0, Rs.0 &Rs. 4 are the shadow prices of machines $M_1, M_2 \ and \ M_3$ respectively.