Lakeside Boatworks is planning to manufacture three types of molded fiberglass recreational boats-fishing (bass boat), a ski boat, and a small speedboat. The estimated selling price and variable cost for each type of boat are summarized in the following table:

boat | Variable cost | selling price |
---|---|---|

Bass | \$12,500 | \$23,000 |

Ski | \$8,500 | \$18,000 |

Speed | \$13,700 | \$26,000 |

The company has incurred fixed cost of \$2,800,000 to set up its manufacturing operation and begin production. Lakeside has entered into agreements with several boat dealers in the region to provide a minimum of 70 bass boats, 50 ski boats, and 50 speedboats. Alternatively, the company is unsure of what actual demand will be, so it has decided to limit production to no more than 120 of any one boat. The company wants to determine the number of boats that is must sell to break even while minimizing its total variable cost.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer.

#### Solution

Let $X_1$ = number of bass boats, $X_2$ = number of ski boats , $X_3$ = number of speed boats.

Number to manufacture

$X_1\geq 70$ and $X_1\leq 120$

$X_2\geq 50$ and $X_2\leq 120$

$X3\geq 50$ and $X_3\leq 120$

In order to break even, the total revenue must equal total cost:

$23,000X_1 + 18,000X_2 + 26,000X_3 = 12,500X_1 + 8,500X_2 + 13,700X_3 + 2,800,000$.

That is

$10,500 X_1 + 9,500X_2 + 12,300X_3 = 2,800,000$

That is

$105X_1 + 95X_2 + 123X_3 = 28000$

The objective function is

Minimize $Z = 12500 X_1 + 8500 X_2 + 13700 X_3$

So the formulated LPP is

` $$ \begin{aligned} \text{Minimize } Z &= 12500 X_1 + 8500 X_2 + 13700 X_3\\ \text{Subject to} & \\ & 105X1 + 95X2 + 123X3 = 28000\\ & X_1\geq 70 \\ & X_1\leq 120 \\ & X_2\geq 50 \\ & X_2\leq 120 \\ & X_3\geq 50 \\ & X_3\leq 120 \end{aligned} $$ `

See the attached Excel file for complete solution using Solver.

$X_1$ = number of bass boats = 70

$X_2$ = number of ski boats = 120

$X_3$ = number of speed boats = 75.203 $\approx$ 75

The minimum value of the objective function is

Z = \$2925285

Excel Solver file : Boat LPP