Chapter 4, Problem 26RP

Explanation of Solution

LP formulation:

• Suppose that,
  • a_t= Number of robots available during quarter t.
  • b_t= Number of robots bought during quarter t.
  • s_t= Number of robots sold during quarter t.
  • i_t= Number of cars available during quarter t.
  • c_t= Number of cars produced during quarter t.
  • $i_t{}^{\text{'}}$= Blockage demand for cars at the end of quarter t.
  • z= Cost of the company.
• The cost of purchasing a robot is $5,000 and maintenance cost of a robot is $500
• The selling cost of a robot at the beginning of each quarter is $3,000.
• Holding cost of a car at the end of each quarter is $200.
• For the backlog demand, the holding cost of a car at the end of each quarter is $300.
• Therefore, it is necessary to minimize the cost of company
  • z= $500{\displaystyle \sum _{t=1}^4{a}_t+200{\displaystyle \sum _{t=1}^4{i}_t}}+5000{\displaystyle \sum _{t=1}^4{b}_t}-3000{\displaystyle \sum _{t=1}^4{s}_t+300{\displaystyle \sum _{t=1}^4{i}_t{}^{\text{'}}}}$
• The demand to be met for quarter 1 is 600. Therefore,
  • $i_1=i_1{}^{\text{'}}+c_1-600$
• The demand to be met for quarter 2 is 800. Therefore,
  • $i_2=i_2{}^{\text{'}}+i_1-i_1{}^{\text{'}}+c_2-800$
• The demand to be met for quarter 3 is 500