Chapter 4.17, Problem 5P

Explanation of Solution

Finding optimal solution:

The demand for sail boats during each of the next four quarters is;

First quarter-40 sailboats

Second quarter-60 sailboats

Third quarter- 75 sailboats

Fourth quarter- 25 sailboats

An inventory of 10 sailboats is at the beginning of the first quarter. Also, it is assumed that sailboats manufactured during a quarter can be used to meet demand for that quarter.

Also, during each quarter, Sailco can produce up to 40 sailboats with total cost of $400 per sailboat.

Additional sailboats with overtime labor can be produced at a total cost of $450 per sailboat.

A carrying cost of $20 per sailboat is incurred at the end of each quarter. A production schedule to minimize the sum of production and inventory costs during the next four quarters using linear programming is to be formulated and using the solver add in excel, the number of sailboats during each quarter is to be determined.

Let,

${\text{x}}_{\text{t}}$= Number of sailboats produced by regular time labour (at $400 pre boat) during quarter $\text{t}$ where $\text{t=1,2,3,4}$.

${\text{y}}_{\text{t}}$= Number of sailboats produced by overtime time labour (at $450 per boat) during quarter $\text{t}$ where, $\text{t=1,2,3,4}$.

${\text{i}}_{\text{t}}$= Number of sailboats on hand at the end of quarter $\text{t}$ where, $\text{t=1,2,3,4}$.

Sailco’s total cost can be determined as follows.

$\begin{array}{c}\text{ Cost=Cost of producing regular time boats }\\ \text{+ Cost of producing overtime boats }\\ \text{+ inventory costs}\\ {\text{=400(x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}{\text{)+450(y}}_{\text{1}}{\text{+y}}_{\text{2}}{\text{+y}}_{\text{3}}{\text{+y}}_{\text{4}}{\text{)+20(i}}_{\text{1}}{\text{+i}}_{\text{2}}{\text{+i}}_{\text{3}}{\text{+i}}_{\text{4}}\text{)}\end{array}$

Since the total cost to be minimized, the objective function of Sailco is given below,

Minimize, ${\text{z=400(x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}{\text{)+450(y}}_{\text{1}}{\text{+y}}_{\text{2}}{\text{+y}}_{\text{3}}{\text{+y}}_{\text{4}}{\text{)+20(i}}_{\text{1}}{\text{+i}}_{\text{2}}{\text{+i}}_{\text{3}}{\text{+i}}_{\text{4}}\text{)}$

The constraints are given below,

For each period’s regular time production will not exceed 40. So the below given constraints are obtained.

$\begin{array}{l}{\text{x}}_{\text{1}}\le 40\\ {\text{x}}_{\text{2}}\le 40\\ {\text{x}}_{\text{3}}\le 40\\ {\text{x}}_{\text{4}}\le 40\end{array}$

Also suppose that,

$\begin{array}{l}\text{Inventory at the end of quarter t= Inventory at the end of quarter}\left(\text{t-1}\right)\hfill \\ \text{ +quarter t production – quarter t demand}\hfill \end{array}$

This means,

${\text{i}}_{\text{t}}{\text{=i}}_{\text{t-1}}{\text{+(x}}_{\text{t}}{\text{+y}}_{\text{t}}{\text{)-d}}_{\text{i}}$ where, $\text{(t=1,2,3,4)}$

When submitting the values, the following constraints are obtained.

$\begin{array}{l}{\text{i}}_1{\text{=10+x}}_1{\text{+y}}_1\text{-40}\\ {\text{i}}_2{\text{=i}}_{\text{1}}{\text{+x}}_2{\text{+y}}_2\text{-60}\\ {\text{i}}_3{\text{=i}}_2{\text{+x}}_2{\text{+y}}_3\text{-75}\\ {\text{i}}_4{\text{=i}}_3{\text{+x}}_4{\text{+y}}_4\text{-25}\end{array}$

Also, ${\text{i}}_{\text{t}}\ge \text{(t=1,2,3,4)}$ implies that each quarter’s demand is met on time and $x_{\text{t}}\ge 0$ implies that the production levels are positive at all time