Chapter 3.10, Problem 9P

Explanation of Solution

Formulation of Linear Programming (LP):

• Assume the following,
  • Let “${\text{X}}_{\text{m}}$” be the number of workers who get quarter “m” off.
  • Let “${\text{I}}_{\text{m}}$” be the inventory of mixers at the end of quarter “m”.
  • Let “${\text{M}}_{\text{m}}$” be the number of mixers made during quarter “m”.
• The Linear Programming (LP) is formulated to minimize the total cost of meeting the next year’s demand as follows,

$\begin{array}{l}\text{z=(worker's salary)(Number of workers)+(Inventory of mixers)(Inventory of mixers at the end of quarter)}\\ {\text{z=30(I}}_{\text{1}}{\text{+I}}_2{\text{+I}}_3{\text{+I}}_4{\text{)+30000(X}}_{\text{1}}{\text{+X}}_2{\text{+X}}_3{\text{+X}}_4\text{)}\end{array}$

• The above function is subject to the following conditions,
  • All variables >= “0”.
  • “${\text{I}}_{\text{1}}{\text{ = 600+M}}_{\text{1}}-400\text{0 }$” that denotes the inventory of mixers during the quarter “1”.
  • “${\text{I}}_2{\text{ = I}}_{\text{1}}{\text{+M}}_2-200\text{0 }$” that denotes the inventory of mixers during the quarter “2”.
  • “${\text{I}}_3{\text{ = I}}_2{\text{+M}}_3-300\text{0 }$” that denotes the inventory of mixers during the quarter “3”