An appliance company has a warehouse and two terminals. To minimize shipping costs, the manager must decide how many appliances should be shipped to each terminal. There is a total supply of 1200 units in the warehouse and a demand for 400 units in terminal A and 500 units in terminal B. It costs $12 to ship each unit to terminal A and $16 to ship to terminal B. How many units should be shipped to each terminal in order to minimize cost?
Can someone help me come up with a function and the constraints from this question?
The equation I would use is C = 12a + 16b. Does that seem alright? And how would I form the contraints?
yes
Sure, I can help you with that! Let's start by setting up a variable for the number of units to be shipped to terminal A, and another variable for the number of units to be shipped to terminal B. Let's call these variables "x" (for terminal A) and "y" (for terminal B).
Now, let's set up the objective function, which represents the cost. The cost is calculated by multiplying the number of units shipped to each terminal by the respective shipping costs. So, the objective function is:
Cost = 12x + 16y
Next, let's set up the constraints based on the given information. We know that the supply (total number of units in the warehouse) is limited to 1200 units, so the constraint for the supply is:
x + y ≤ 1200
We also have demand constraints for each terminal. Terminal A has a demand of 400 units, so the demand constraint for terminal A is:
x ≥ 400
Similarly, terminal B has a demand of 500 units, so the demand constraint for terminal B is:
y ≥ 500
Finally, both x and y should be non-negative, since we can't ship negative units:
x ≥ 0
y ≥ 0
So putting it all together, we have the following function and constraints for this problem:
Objective function:
Cost = 12x + 16y
Constraints:
x + y ≤ 1200
x ≥ 400
y ≥ 500
x ≥ 0
y ≥ 0
I hope this helps! Let me know if you have any other questions.
To minimize shipping costs, we can set up a linear programming problem.
Let x be the number of appliances shipped to terminal A and y be the number of appliances shipped to terminal B.
The objective is to minimize the total shipping cost Z, which can be expressed as:
Z = 12x + 16y
Subject to the following constraints:
1. The total supply of appliances: x + y ≤ 1200
2. The demand for appliances in terminal A: x ≥ 400
3. The demand for appliances in terminal B: y ≥ 500
The first constraint ensures that the total number of appliances shipped does not exceed the total supply. The second constraint ensures that terminal A's demand is met, while the third constraint ensures that terminal B's demand is met.
These constraints can be written as a system of linear inequalities:
x + y ≤ 1200
x ≥ 400
y ≥ 500
To come up with a function and constraints for this problem, we need to define some variables:
Let x be the number of appliances shipped to terminal A.
Let y be the number of appliances shipped to terminal B.
We know that the total supply of appliances is 1200 units, so we have the constraint:
x + y ≤ 1200
We also have the constraints for the demand at each terminal:
x ≥ 400
y ≥ 500
Additionally, we need to consider the objective function, which represents the cost of shipping the appliances:
Cost = cost of shipping to terminal A + cost of shipping to terminal B
The cost of shipping to terminal A can be calculated as:
Cost of shipping to A = $12 * x
Similarly, the cost of shipping to terminal B can be calculated as:
Cost of shipping to B = $16 * y
Now we can define the objective function:
Cost = $12 * x + $16 * y
To summarize, we have the function and constraints for this problem:
Objective Function:
Cost = $12 * x + $16 * y
Constraints:
x + y ≤ 1200
x ≥ 400
y ≥ 500
These equations and inequalities can be used to solve the problem and determine the optimal number of appliances to be shipped to each terminal to minimize cost.