Chapter 3, Problem 48RP

Explanation of Solution

LP to maximize cash on hand:

• Assume the following,
  • Let “${\text{C}}_{\text{m}}$” be the cash on hand at the end of month “m”.
  • Let “${\text{X}}_{\text{m}}$” be the bushels of corn bought at the starting of month “m”.
  • Let “${\text{Y}}_{\text{m}}$” be the bushels of corn sold at the end of month “m”.
  • Let “${\text{I}}_{\text{m}}$” be the corn on hand after purchase of the month “m” but before the sale of month “m”.
• An appropriate Linear Programming (LP) is given as follows,

${\text{Max z=250Y}}_{\text{1}}+400{\text{Y}}_2+350{\text{Y}}_3+550{\text{Y}}_4-300{\text{X}}_{\text{1}}-350{\text{X}}_2-400{\text{X}}_3-500{\text{X}}_4$

• The above function subject to the following conditions,
  • “${\text{I}}_1={\text{X}}_1+50$” denotes the corn on hand after purchase of month “1”.
  • “${\text{I}}_2={\text{I}}_1-{\text{Y}}_1+{\text{X}}_2$” denotes the corn on hand after purchase of month “2”.
  • “${\text{I}}_3={\text{I}}_2-{\text{Y}}_2+{\text{X}}_3$” denotes the corn on hand after purchase of month “3”.
  • “${\text{I}}_4={\text{I}}_3-{\text{Y}}_3+{\text{X}}_4$” denotes the corn on hand after purchase of month “4”.
  • “${\text{I}}_1\le 100$” denotes that the corn on hand after purchase of month “1”should be less than or equal to “100”.
  • “${\text{I}}_2\le 100$” denotes that the corn on hand after purchase of month “2”should be less than or equal to “100”.
  • “${\text{I}}_3\le 100$” denotes that the corn on hand after purchase of month “3”should be less than or equal to “100”.
  • “${\text{I}}_4\le 100$” denotes that the corn on hand after purchase of month “4”should be less than or equal to “100”