alright.. this is the question i got..
JJ Manufacturing has three factories (1,2,3) and three distribution centers (A,B,C). shipping costs from the factories to the distrubution centers is as follows
from factory 1 to center A: $6
from factory 2 to center A: $8
from factory 3 to center A: $11
from factory 1 to center B: $5
from factory 2 to center B: $10
from factory 3 to center B: $14
from factory 1 to center c: $3
from factory 2 to center A: $8
from factory 3 to center A: $18
production capability of fact 1 = 6000
production capability of fact 2 = 8000
production capability of fact 3 = 10000
demand at distributtion center A = 7000
demand at distributtion center B = 12000
demand at distributtion center C = 5000
the product is is produced and deliverd in a lot size of 100
im at a total blank at how to write the objective function and contraints. can u please help me with this ASAP :)
Sure! I can help you with that. To write the objective function and constraints for this problem, we need to define the decision variables, the objective, and the constraints.
Decision Variables:
Let's define the decision variables as the number of lots produced and shipped from each factory to each distribution center. We can represent these variables as follows:
Xij represents the number of lots produced at factory i and shipped to distribution center j.
Objective:
The objective is to minimize the total shipping cost. In this case, the shipping cost is the cost per lot multiplied by the number of lots shipped. So, the objective function can be written as:
Minimize: 6*X11 + 8*X21 + 11*X31 + 5*X12 + 10*X22 + 14*X32 + 3*X13 + 8*X23 + 18*X33
Constraints:
Now let's define the constraints based on the given information:
1. Production capacity constraint:
The total number of lots produced at each factory should not exceed its production capability. This can be represented as:
X11 + X12 + X13 <= 6000 (for factory 1)
X21 + X22 + X23 <= 8000 (for factory 2)
X31 + X32 + X33 <= 10000 (for factory 3)
2. Demand constraint:
The total demand at each distribution center should be met. This can be represented as:
X11 + X21 + X31 >= 7000 (for distribution center A)
X12 + X22 + X32 >= 12000 (for distribution center B)
X13 + X23 + X33 >= 5000 (for distribution center C)
3. Non-negative constraint:
The number of lots produced and shipped cannot be negative. This can be represented as:
Xij >= 0 (for all i and j)
These are the objective function and constraints for your problem.
Sure! Let's break it down step-by-step:
Step 1: Define the decision variables:
Let's define the decision variables as follows:
xij = Quantity of product shipped from factory i to distribution center j.
Step 2: Define the objective function:
The objective of this problem is to minimize the total shipping cost. So, the objective function can be written as:
Minimize Z = 6x11 + 8x12 + 11x13 + 5x21 + 10x22 + 14x23 + 3x31 + 8x32 + 18x33
Step 3: Define the constraints:
3.1 Production capacity constraints:
The total quantity produced at each factory cannot exceed its production capability. So, we have the following constraints:
x11 + x12 + x13 <= 6000 (for factory 1)
x21 + x22 + x23 <= 8000 (for factory 2)
x31 + x32 + x33 <= 10000 (for factory 3)
3.2 Demand constraints:
The total quantity received at each distribution center must meet the demand. So, we have the following constraints:
x11 + x21 + x31 >= 7000 (for distribution center A)
x12 + x22 + x32 >= 12000 (for distribution center B)
x13 + x23 + x33 >= 5000 (for distribution center C)
3.3 Non-negativity constraint:
The quantity shipped cannot be negative. So, we have the following constraint:
xij >= 0 for all i and j
Step 4: Write the complete mathematical model:
Minimize Z = 6x11 + 8x12 + 11x13 + 5x21 + 10x22 + 14x23 + 3x31 + 8x32 + 18x33
Subject to:
x11 + x12 + x13 <= 6000 (for factory 1)
x21 + x22 + x23 <= 8000 (for factory 2)
x31 + x32 + x33 <= 10000 (for factory 3)
x11 + x21 + x31 >= 7000 (for distribution center A)
x12 + x22 + x32 >= 12000 (for distribution center B)
x13 + x23 + x33 >= 5000 (for distribution center C)
xij >= 0 for all i and j
Note that this formulation assumes that the quantities shipped can be fractional. If that's not the case, you may need to include additional constraints to enforce integer values for xij.