the production manager of koulder refrigerators

Question

The production manager of Koulder Refrigerators must decide how many refrigerators to produce in each of the next four months to meet demand at the lowest overall cost. There is a limited capacity in each month although this will increase in month 3.  Due to a new contract, costs are expected to increase.  The relevant information is provided in the table below.

Month	Capacity	Demand	Cost of production
1	140	110	$80 per unit
2	140	150	$85 per unit
3	160	130	$90 per unit
4	160	140	$95 per unit

Each item that is left at the end of the month and carried over to the next month incurs a carrying cost equal to 10% of the unit cost in that month (e.g. anything left in inventory at the end of month one incurs an $8 cost). Management wants to have at least 30 units left at the end of month four to meet any unexpected demand at that time. A linear program has been developed to help with this. However, this may or may not be totally correct. You should verify that it is the correct formulation before solving the problem. If it is not correct, make any necessary changes to the linear program before solving it on the computer.

X₁ = number of units produced in month 1;  X₂ = number of units produced in month 2; 

X₃ = number of units produced in month 3; X₄ = number of units produced in month 4;

N₁ = number of units left at end of month 1; N₂ = number of units left at end of month 2;

N₃ = number of units left at end of month 3; N₄ = number of units left at end of month 4

Minimize cost = 80X₁ + 85X₂ + 90X₃ + 95X₄+ 8N₁ + 8.5N₂ + 9N₃+ 9.5N₃₄

X₁< 140

X₂< 140

X₃< 160

X₄< 160

X₁ = 110 + N₁

X₂ + N₁ = 150 + N₂

X₃ + N₂ = 130 + N₃

X₄ + N₃ = 140 + N₄

N₄>30

All variables > 0