Question

Let P1 = number of Product 1 to be produced

          P2 = number of Product 2 to be produced

          P3 = number of Product 3 to be produced

          P4 = number of Product 4 to be produced

Maximize 15P1 + 20P2 + 24P3 + 15P4 Total profit

Subject to

        8P1 + 12P2 + 10P3 + 8P4 ≤ 3000 Material requirement constraint

            4P1 + 3P2 + 2P3 + 3P4 ≤ 1000 Labor hours constraint

                                             P2 > 120 Minimum quantity needed for Product 2 constraint

                   And P1, P2, P3, P4 ≥ 0 Non-negativity constraints

The Excel Solver output for this problem is given below.

(a) Determine the optimal solution and the optimal value and interpret their meanings.

(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

Answer 1

ANSWER::

We solve the given LPP in Excel using Excel Solver as shown below:

Variables P1 P2 0 120 156 0 P3 P4 Objective Max P $ 6,144 Constraints Condition 1 2 3000 672 120 <= 3000 1000 120 Material requirement Labor hours Minimum quantity needed for Product 2 3 >=

Hence,

(a)

The optimal solution =

P1 = 0

P2 = 120

P3 = 156

P4 = 0

The optimal value = $6144

As seen from the optimal solution, product P1 and P4 is not produced, 120 units of P2 are produced, and 156 units of P3 are produced.

The maximum profit which is achieved from this combination of production = $6144

b)

As seen from the solution, the Right Hand Side of constraints > Left hand side of constraint for labor hours constraints. Hence, slack of RHS - LHS = 1000 - 672 = 328 labor hours is available.

The above solution in the form of formulas along with Excel Solver extract is shown below for better understanding and reference:

B C D E F G Н Solver Parameters Set Objective: $B$8 1 To: O Max O Min O Value of: 0 By Changing Variable Cells: $B$2:$B$5 1 Condition Subject to the Constraints: $B$11:$B$12 <= $D$11:$D$12 $B$13 >= $D$13 <= Add <= 3000 Material requirement 1000 Labor hours 120 Minimum quantity needed for Product 2 Change >= Delete 1 Variables 2 P1 0 3 P2 120 4 P3 156 5 P4 0 6 7 Objective 8 Max P =15*B2+20*B3+24*34+15*B5 9 10 Constraints 11 1 =8*B2+12*B3+10*B4+8*B5 12 2 =4*B2+3*B3+2*B4+3*B5 13 3 =B3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Reset All Load/Save Make Unconstrained variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help Solve Close

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