I really really need help with this question, i've tried it different ways and end up confusing myself more. It's a multi schedule question and my biggest block is coming up with the objective function. HELP!!!
The Homespun Company produces an air conditioning unit for use in vehicles. The company has firm orders for six months in the future. The company can schedule its production over the next six months to meet these orders in either a regular (<=500 units per month) or an overtime (<=300 units per month) Customer orders and associated production costs for the next six months are as follows:
Cost ($per Unit)
MONTH ORDERS REGULAR OVERTIME
Jan 590Units $50 $62
Feb 610Units $52 $58
Mar 650Units $51 $63
April 700Units $55 $60
May 500Units $47 $55
June 700Units $50 $52
At the beginning of Jan, the company has 75 units in inventory. At the end of June it wishes to have at least 100 units in inventory. The inventory carrying cost for the air conditioners is $10 per unit per month. Formulate as linear programming model.
To formulate a linear programming model for this problem, we need to determine the decision variables, the objective function, and the constraints.
Decision Variables:
Let x_i represent the number of units produced in the regular schedule in month i (where i=1,2,...,6).
Let y_i represent the number of units produced in the overtime schedule in month i (where i=1,2,...,6).
Objective Function:
The objective is to minimize the total production cost and inventory carrying cost over the six months.
The total production cost is calculated by multiplying the number of units produced in each month with the respective cost per unit:
Total production cost = 50x_1 + 52x_2 + 51x_3 + 55x_4 + 47x_5 + 50x_6 + 62y_1 + 58y_2 + 63y_3 + 60y_4 + 55y_5 + 52y_6
The inventory carrying cost is calculated by multiplying the number of units in inventory at the beginning of the month with the carrying cost per unit:
Inventory carrying cost = 10(75 + x_1 + y_1 + x_2 + y_2 + x_3 + y_3 + x_4 + y_4 + x_5 + y_5 + x_6 + y_6)
Therefore, the objective function is:
Minimize: 50x_1 + 52x_2 + 51x_3 + 55x_4 + 47x_5 + 50x_6 + 62y_1 + 58y_2 + 63y_3 + 60y_4 + 55y_5 + 52y_6 + 10(75 + x_1 + y_1 + x_2 + y_2 + x_3 + y_3 + x_4 + y_4 + x_5 + y_5 + x_6 + y_6)
Constraints:
1) The production in regular schedule must be less than or equal to 500 units per month: x_1 <= 500, x_2 <= 500, x_3 <= 500, x_4 <= 500, x_5 <= 500, x_6 <= 500
2) The production in overtime schedule must be less than or equal to 300 units per month: y_1 <= 300, y_2 <= 300, y_3 <= 300, y_4 <= 300, y_5 <= 300, y_6 <= 300
3) The total production (including regular and overtime) in each month must be able to meet the customer orders: x_1 + y_1 >= 590, x_2 + y_2 >= 610, x_3 + y_3 >= 650, x_4 + y_4 >= 700, x_5 + y_5 >= 500, x_6 + y_6 >= 700
4) The inventory at the end of June must be greater than or equal to 100 units: 75 + x_1 + y_1 + x_2 + y_2 + x_3 + y_3 + x_4 + y_4 + x_5 + y_5 + x_6 + y_6 >= 100
With these decision variables, objective function, and constraints, you can set up and solve a linear programming model to determine the optimal production schedule that minimizes the total production cost and inventory carrying cost, while meeting the customer orders and inventory requirements.