Cost Optimization across Different Suppliers for a Product - math

Question

I've this following optimization problem. A company produces a product, say Big A. To produce this product, it requires 5 processes. (Please find the detail table below). For each process, there are number of supplier that supply raw material for that particular process. E.g. For process 1, there are 3 supplier 1,2 & 3.
The constrain for the CEO of this company,say C, is that for each process the CEO has to purchase supplies from Supplier 1 first, then for additional supplies from 2nd Supplier and so on.
The optimization problem is C wants 700 units for total material to produce for 1 unit of Big A then how will he do it at minimum cost. How the optimization will change if the amount of units require increases to 1500 units.
I'll be grateful if I get the solution of this answer. But if somebody can suggest me some reference regarding this problem it will be a great help too. I'm mainly using R software here.
Process Supplier Cost Units Cumm_Cost Cumm_Unit
1 1 10 100 10 100
1 2 20 110 30 210
1 3 10 200 40 410
2 1 20 100 20 100
2 2 30 150 50 250
2 3 10 150 60 400
3 1 40 130 40 130
3 2 30 140 70 270
3 3 50 120 120 390
4 1 20 120 20 120
4 2 40 120 60 240
4 3 20 180 80 420
5 1 30 180 30 180
5 2 10 160 40 320
5 3 30 140 70 460
Regards,

Answer 1

I will start by solving the specific problem that you have posted and then will demonsrate how to formulate the problem more abstractively. For simplicity, I will use Excel's Solver add-in to solve the problem, but any configuration of a modeling language (such as AIMMS, AMPL, LINGO, OPL, MOSEL and numerous others) with a solver (CPLEX, GUROBI, GLPK, CBC and numerous others) can be used. If you would like to use R, there exists an lpSolve package that calls the lpSolve solver (which is not the best one in the word to be honest, but it is free of charge).
Note that for "real" (large scale) integer problems, the commercial solvers CPLEX, GUROBI and XPRESS perform a lot better than others. The first completely free solver that performs decently in most tests (including Hans Mittelman's page) is CBC. CBC can be hooked up in excel and solve the built-in solver model without restrictions in the number of constraints or variables, by using this add-in. Therefore, assuming that most CPU is going to be spent by the optimization algorithm, using CBC/OpenSolver seems like an efficient choice.
SPREADSHEET SETUP
I follow some conventions for convenience:
Decision variable cells are marked Green.
Constraints are marked red.
Data are marked grey.
Objective function is marked blue.
First, lets augment the table you presented as follows:
The added columns explained briefly:
Selected?: equals 1 if the (Process, Supplier) combo is allowed to produced a positive quantity, zero otherwise.
Quantity: the quantity produced, defined for each (Process, Supplier) combo.
Max Quantity?: Equals 1 if the Suppliers produces the maximum amount of units for that particular Process.
Quantity UB: equals Units * Selected?. This makes the upper bound either equal to Units, when the Supplier is allowed to produce this Process, or zero otherwise.
Quantity LB: equals Units * Max Quantity?. This is to ensure that whenever the Max Quantity? column is 1, the produced quantity will be equal to Units.
Selection: For the 1st supplier, it equals 0. For the 2nd and 3rd suppliers, it equals the Max Quantity? of the previous supplier (row) minus the Selected? of the current supplier (row).
A screenshot with formulas:
There exist two more constraints:
There must be at least one item produced from each process and
The total number of items should be 700 (or later 1,500).
Here is their setup:
and here are the formulas:
In brief, we use SUMIF to sum the quantities that are specific to each supplier, which we are going to constrain to be more than 1 item for each process.
To finish the spreadsheet setup, we need to calculate the objective function, namely the cost of the allocation. This is easily done by taking the SUMPRODUCT of columns Quantity and Cost. Note that the cumulative quantities are derived data and not very useful in the current context.
After the above steps, the spreadsheet looks like below:
SOLVER MODEL
For the solver model we need to declare
The Objective
The Decisions
The Constraints
The Solver (and tweak some parameters if necessary).
For ease of exposition, I have given each range the name of its header. The solver model looks as follows:
It should all be explanatory, except possibly the Selected >= 0 part. The column selected equals the difference between the binary max Quantity of the previous supplier minus the Selected of the current supplier. Selected >= 0 => max Quantity of previous supplier >= Selected of current supplier. Therefore, if the previous supplier does not produce at max quantity (binary = 0), the current supplier cannot produce.
Then we need to make sure that the solver setting are OK:
and solve the problem.
Solution for req = 700 :
As we see, the model tries to avoid procedures 3 and 5 as much as possible, and satisfies the constraint "at least 1 item per process" by picking up exactly 1 item for processes 3 and 5. The objective function value is 11,710.
Solution for req = 1,500 :
Here we need more capacity, but yet process 3 seems expensive and the model tries to avoid it by allocating whatever is necessary (just 1 unit to supplier 1).
I hope this helps. The spreadsheet can be downloaded here. I include the definition of the mathematical model below, in case you would like to transfer it to another language.
MATHEMATICAL FORMULATION
A formal definition of your problem is as follows.
SETS:
PARAMETERS:
Decisions:
Objective:
Constraints:
Constraint explanation:
C1: A supplier cannot produce anything from a process if he has not been allocated to that process.
C2: If a supplier's maximum indicator is set to 1, then the production variable should be the maximum possible.
C3: We cannot select supplier s for process p if we have not produced the max quantity available from the previous supplier s_[-1].
C4: We need to produce at least 1 item from each process.
C5: the total production from all processes and suppliers should equal the required amount.

Answer 2

Looks like you should look at the simplex algorithm (or some existing implementation of it).
Wikipedia has a fairly nice description of the algorithm, http://en.wikipedia.org/wiki/Simplex_algorithm