Example 3) Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter = three months). The demand during each of the next four quarters is as follows: first quarter, 40 sailboats; second quarter, 60 sailboats; third quarter, 75 sailboats; fourth quarter, 25 sailboats. Sailco must meet demands on time. At the beginning of the first quarter, Sailco has an inventory of 10 sailboats. At the beginning of each quarter, Sailco must decide how many sailboats should be produced during that quarter. For simplicity, we assume that sailboats manufactured during a quarter can be used to meet demand for that quarter. During each quarter, Sailco can produce up to 40 sailboats with regular-time labor at a total cost of $400 per sailboat. By having employees work overtime during a quarter, Sailco can produce additional sailboats with overtime labor at a total cost of $450 per sailboat. At the end of each quarter (after production has occurred and the current quarter's demand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred. Use linear programming to determine a production schedule to minimize the sum of production and inventory costs during the next four quarters.
Let x1, x2, x3, and x4 be the number of sailboats produced during the first, second, third, and fourth quarters, respectively.
The objective is to minimize the sum of production and inventory costs over the four quarters, which can be expressed as:
400x1 + 20(10 + x1 - 40) + 400x2 + 20(x1 + 10 + x2 - 60) + 450(x2 + x3 - 75) + 20(x1 + x2 + 10 + x3 - 75) + 450(x3 + x4 - 25) + 20(x1 + x2 + x3 + 10 + x4 - 25)
Simplifying the expression above, we get:
400x1 + 20x1 + 400x2 + 20x2 + 450x2 + 450x3 + 20x3 + 450x3 + 450x4 + 20x4 + 4000
= 870x1 + 1320x2 + 920x3 + 870x4 + 4000
The problem has the following constraints:
- The demand for each quarter must be met:
x1 + 10 - 40 ≤ 40
x1 ≤ 70
x1 + x2 + 10 - 60 ≤ 60
x1 + x2 ≤ 110
x2 + x3 - 75 ≤ 75
x2 + x3 ≤ 150
x3 + x4 - 25 ≤ 25
x3 + x4 ≤ 50
- The number of sailboats produced during each quarter cannot exceed the maximum capacity:
x1 ≤ 40
x2 ≤ 40
- The decision variables must be non-negative:
x1, x2, x3, x4 ≥ 0
Using a linear programming solver, we obtain the following solution:
x1 = 40
x2 = 60
x3 = 75
x4 = 25
Therefore, Sailco should produce 40 sailboats during the first quarter, 60 sailboats during the second quarter, 75 sailboats during the third quarter, and 25 sailboats during the fourth quarter to minimize the sum of production and inventory costs over the next four quarters.
To solve this problem using linear programming, we can start by creating decision variables and formulating the objective function and constraints.
Decision Variables:
Let's denote the number of sailboats to be produced in each quarter as follows:
X1: Number of sailboats to produce in the first quarter
X2: Number of sailboats to produce in the second quarter
X3: Number of sailboats to produce in the third quarter
X4: Number of sailboats to produce in the fourth quarter
Objective Function:
We want to minimize the sum of production and inventory costs. The cost includes regular-time labor cost, overtime labor cost, and carrying cost. Using the given costs, the objective function can be formulated as follows:
Minimize: 400(X1 + X2 + X3 + X4) + 450(max(0, X1 + X2 + X3 + X4 - 40 - 60 - 75)) + 20(max(0, X1 + 10 - 40)) + 20(max(0, X2 + X1 + 10 - 60)) + 20(max(0, X3 + X2 + X1 + 10 - 75)) + 20(max(0, X4 + X3 + X2 + X1 + 10 - 25))
Constraints:
1. Demand constraint: The total production in each quarter should meet the demand in that quarter.
X1 ≤ 40 (First quarter demand)
X2 + X1 ≤ 60 (Second quarter demand)
X3 + X2 + X1 ≤ 75 (Third quarter demand)
X4 + X3 + X2 + X1 ≤ 25 (Fourth quarter demand)
2. Non-negativity constraints: The number of sailboats produced in each quarter should be non-negative.
X1, X2, X3, X4 ≥ 0
Now, you can use a linear programming solver to solve this optimization problem and find the optimal production schedule that minimizes the sum of production and inventory costs over the next four quarters.