Chapter 3.10, Problem 8P

Explanation of Solution

Formulation of Linear Programming (LP):

• Assume the following,
  • Let “${\text{R}}_{\text{m}}$” be the quarter “m” with regular time production.
  • “${\text{O}}_{\text{m}}$” be the quarter “m” with overtime production.
  • “${\text{I}}_{\text{m}}$” be the quarter “m” ending inventory after spoilage.
• The Linear Programming (LP) is formulated to minimize the total cost of meeting the next three quarters’ demands as follows,

${\text{min z=40R}}_{\text{1}}{\text{+40R}}_2{\text{+40R}}_3{\text{+60(O}}_{\text{1}}{\text{)+60(O}}_2{\text{)+60(O}}_3{\text{)+15I}}_1{\text{+15I}}_2{\text{+15I}}_3$

subject to the following conditions,

• All variables >= “0”.
• “${\text{I}}_{\text{1}}\text{ = 0}\text{.9(20+0}{\text{.8R}}_{\text{1}}+0.8{\text{O}}_{\text{1}}-3\text{0 )}$” that denotes the demands for the pair of shoes in inventory at the end of each year