Charge Mix Optimization using Linear Programming

The example of popular taxi services “Ola” and “Uber” has taught us that if we look at the issues innovatively, we can come up with solutions which are meeting the requirements of all the stake-holders (except for the old taxi-driving community but slowly they are also moving to Ola / Uber) and are cost effective as well. Another example in the same league could be Tesla cars developed by Mr. Elon Musk and his team. Considering the issues with fossil fuels and the cars running on them, Mr. Musk developed these new generation cars. These cars are so advanced that they can update their software over internet and various new features (such as enabling auto-pilot option) to the car can be added after the car is connected to the internet (so no need to go to the service centre) and at the same time the Tesla cars can also ‘know’ the faults in them and can communicate with the car manufacturer over internet and get the solution for those issues as well.

Introduction

With the world advancing so fast and coming up with innovations at never before speed in various aspects of our life, why can’t we think innovative about our foundry operations and make them world class by adopting technology which can pay for itself and make our operations cost-efficient. This can lead to reduction in our costs, time, etc. and thereby giving us competitive advantages either by reducing the sales prices or increasing our profit margins.

While adopting these new self-financing technologies we also have to see that we are moving away from the concept of producing castings as an ‘art’ to producing casting as a ‘science’. We, the foundry fraternity have been adopting many new concepts such as laser based 3 D imaging giving direct inputs to make solid modelling software, 3D printing of patterns, simulating of liquid flow in casting and thereby fixing the positions of runners, risers, partition lines, etc. Now we also have to think new avenues in the same league to have cutting edge technology to reduce the foundry operation costs.

Cost Break-up in foundry operations

In order to understand the scope of cost optimizations in foundry operations, let’s first break up the costs of typical foundries in India, which is given below:

Costs Break-up Of Foundry Operations

Figure 1 – Cost Break up of Foundry Operations (excluding profit / loss)

Notes: 1. Source¹

2. ‘Other expenses’ include ‘energy costs’ as well.

What costs can be optimized

As per our popular management principals let us first concentrate on the high cost areas as a moderate saving of say 5 % in 47 % (raw material costs) would be say lead to overall cost savings of 2.35%. Compared to this, say 5% cost savings in 3% (finance costs) would be giving a meagre overall cost savings of 0.15%. So considering this management principal let us concentrate on raw materials for cost savings as it would give us significant results.

Having said this, let us see where we can reduce the costs of raw materials. Few of the probable areas of cost reduction are listed below:

1. Purchasing the materials in bulk by clubbing the requirements smartly and thereby getting economics of scale (or Economic Order Quantity / EOQ). I suppose most of us are already carrying out this activity. You may get advantage of say upto 1% to 5% of the material costs depending upon your negotiation skills in today’s tight market. There could also be some small savings in clubbing the quantities in terms of transport cost savings instead of transporting smaller quantities (less than truck-load). Generally most of the ERP packages have an option of calculating the EOQ based on the ordering costs, inventory carrying costs, etc. and most of us are using some or the other ERP package. So this option is already available with most of us.
2. Hedging of Raw Material costs e. future buying of raw materials when the costs are low and then use it when the costs go up. Some savings depending upon the volatility of the market and risk prediction and risk taking capacity of the organization governs this. However in steady market not much savings can be accrued. This type of savings is achieved by commercial persons having very deep knowledge of trading and it depends more on individuals, even though many new software are coming in market to predict the future prices based on regression analysis, the costs of the software is still quite high and they also need very high end computers for number crunching. Considering these issues, this option is ruled out for most of the small and medium foundries.
3. Using of returns to the maximum extent in the charge mix optimization. If used properly, the re-use of returns and rejects can reduce the costs to a great extent, as the costs of the returns / rejects is already recovered in the costing of good castings. So practically these returns / rejects are free costs and can reduce the manufacturing costs to that extent. However if the returns or rejects have chemistry different than what is required in the final chemistry, sometimes the returns turns out to be a costly affair to reach to the desired chemistry. A very basic example of this could be say I want to use Nickel based returns for making castings which does not need Nickel in their chemical composition. So I have to add other elements to take out all the Nickel for the molten metal. Considering this aspect before using the returns, we need to be doubly sure of the final output chemistry and the chemistry of the returns / rejects.
4. Judiciously mixing the raw materials to get the exact output chemistry in first go or charge mix optimization. This strategy has 2 advantages first advantage is savings in costs due to judicious mixing of raw materials and second advantage is saving in energy costs as we get the correct chemistry in the first go and do not have to heat the furnace till the time we get correct chemistry. This energy savings to the tune of 1% to 5% of the energy costs could be an additional bonus.

How to optimize the charge costs

Being technical minded and rational thinkers, let us concentrate on the third and fourth of the above options for cost reduction. In these options, we know the output chemistry, input chemistry, recovery of each alloying element in the raw materials, melting losses, etc. We input all these details and then by chemical balance find out the best raw material combination to achieve the desired chemistry by trial and error. We may use following techniques:

1. Calculator, pen and paper (old school of thought). This may take some time and the accuracy of the calculations depend upon person to person. This method may lead to lots of dependency on a particular person and hence the charge mix optimization technique becomes an “art” of that person. Also since lots of calculations are involved it takes good amount of time. So in order to reduce the time, generally this person uses a template and make small changes here and there to reduce the calculation time. Sometimes other distractions on the shop-floor can lead to wrong charge mix optimization calculations. So this method is not dependable for mass production as it has too many IFs and BUTs and since it is based on the trial and error method of the calculating individual, we cannot be sure that this is the optimal charge mix optimization giving us the least cost. Moreover when the person calculating this charge mix optimization is not present, the operations may hamper or other persons use their ‘skills’ to calculate the charge based on previous experience. In such cases the charge mix optimization may still move away from optimal charge mix costs.
2. Spread-sheets calculation method. This is one of the most popular methods of charge mix optimization calculations. In this method all the details stated above are fed in various worksheets of a spreadsheet and then the actual calculations are performed by the spreadsheet program. Since the calculations are made by the computer program, the speed of calculations and accuracy is high. In this method couple of persons are aware of the inputs and working of the “customized” spread-sheets and this can again lead to the charge mix optimization as an “art” controlled by those individuals. Sometimes errors in entering the chemistry of raw materials / output chemistry can lead to errors in charge mix optimization calculations. However since most of the persons are aware of working of spreadsheets, many companies / organizations have developed their “customized” spreadsheet programs for internal use as it takes very less effort to train the new persons to use the ‘customized’ spreadsheet program.
3. Customized Software method. A company / organization develops a customized charge mixing software based on their requirements from internal / external software vendor after taking inputs from their in-house charge mix calculating experts. These in-hose experts give inputs about chemical balance, calculation methods, etc. to the software professionals. So this kind of software is advanced version of the above option 2, wherein the graphical user interface (GUI) or front end is designed in better way for the users, however the back-end works on the similar fashion as stated in the above option 2. These type of systems are not developed and used by many companies due to additional development cost involved and not significant cost saving compared to the above option 2.
4. Linear Programming based software method. These types of software are specifically designed for charge mixing taking best practices from various foundries all over the globe. These types of software can also be integrated with other system such as Spectrometers to reduce the errors in entry of chemistry / duel data entry. ERPs such as SAP can also be integrated to these software to get the inventory and purchase related data. The best practices in these kind of software when combined with linear programming solver can lead to most optimal solutions which cannot be optimized further and thus giving good amount of consistent cost savings over the above 3 options.

Figure 2 – Difference between Linear Programming based optimal solution and trial & error based solution for 2 dimensional problems (in foundry there always are more than 2 dimensions).

Please refer to the above figure 2, the first 3 of the above options would lead us to the trial and error solution. When arriving at the solution, we would not know whether the solution is the optimal solution as we can control only few known variables and then come to a conclusion that this the possible best solution (ignorance is bliss). As against this, using Linear programming which is a very systematic approach, we arrive at a solution which is for sure the most optimal solution and considering the properties of linear programming method, it may not be possible to further optimize this solution.

Most of the readers would be aware of the first 2 options. Few of you would be aware of the third option as well. For the benefit of all the users, let us understand the 4^th option in detail.

Linear Programming

When one has to find out the value of only one variable, then one equation is sufficient. Few examples of this type of equations are:

Difference between Linear Programming based optimal solution and trial & error based solution for 2 dimensional problems

1. x + 5 = 0, which gives us the value of x = -5.
2. 3x = 18, giving us a value of x = 6.

When one has to find out value of say two variables, then we need at least two equations to find the correct value of those variables. With one equation, we cannot get correct value of two variable as mathematically there could be infinite solutions to this 2 variable one equation problem. Example of this is popular line slope equation y = mx + c in which we have to find the value of variable y based on the value of x. So by inputting various values of x coordinates, we get value of y coordinate and thereby path of a continuous line and not a single point. So from this one equation, we can see that for every value of x there is corresponding value of y and there is no exact solution as in the above point.

So to get the value of two variables, we solve these equations by simultaneous equation method. Few of the examples of this type of equations are:

1. 3x + 2y = 12 & 10x + 5y = 35. After solving these two equations, we get the values of x = 2 and y =3.
2. 3x . 4y = 38 & 10x / 50y = 1, giving us values of x = 10 and y = 2.

Now think that we have 3 unknown variables, so to get their exact values, we would be ideally needing 3 different equations. Hence we can generalize that to get the values of ‘n’ unknown variables, we would be needing ‘n’ equations.

With the above mathematical background in mind, now let us come to our issue of output chemistry of castings. Wherein we generally control percentages of at least 5 alloying elements in a casting (say Fe, Si, Mn, Carbon & Sulphur), which is fixed by external agencies such as clients and can be treated as imposed constraints.

Other constraints could be:

• Availability / stock of each raw material.
• Internal conditions (know – how, quality, previous experiences in mechanical tests).
• Any other customer requirements.

The list of the constraints would be specific to each foundry and this list can vary depending upon the working style at each foundry and the client combination.

To achieve the above output chemistry, we would be using various raw materials which could include say:

• Pig Iron
• Casting returns,
• HMS of various types say
  • Turning and boring scrap,
  • Sheet metal scrap (low carbon steel)
  • Rods / TMT bar scrap, etc.
• Ferro-x (manganese, silicon, chromium, etc.) alloys, etc.
• Pure elements such as copper, graphite, etc.

Now each of the above raw material has its own chemistry. Let us consider only 5 alloying elements of each of the above raw materials (except for pure elements), so if we are using 5 of the above raw materials, then we are talking about 5 x 5 = 25 variables.

Also for each of the five alloying elements in the above raw materials, there would be different recovery factor (as we would not get 100% recovery for each of the alloying elements as per the chemistry of the raw material, furnace temperature, flux chemistry, etc.). So that would again 25 variables for the recovery factor of the above 5 raw materials.

Since the prices of the raw materials are not constant and vary depending upon the demand supply curve, we need to consider them while optimizing the costs of the raw materials used. This input would go towards our objective of “to reduce the overall cost of the charge”.

If we just add the above inputs, then we have 5 imposed constraints (of chemistry) for finished goods. Then 25 variables for raw materials and their chemistry + 25 variables for recovery percentage of each of the alloying element and the objective of overall cost reduction. When we club all this it becomes too complicated. With all said and done about human brain and its capacity, it becomes almost impossible for any average person to practically calculate the accurate charge mix just before the start of the next batch of furnace with all these different inputs using either “Paper & pencil method” / “Spread-sheet method”. So the best solution to solve these kind of problems is Linear Programming.

The linear programming (LP) method is first developed in 1827 by French mathematician Mr. Joseph Fourier. This method is generally used when you have many variables compared to the number of equations required to solve them. In linear programming we always get the most optimal solution for a given set of constraints and variables. For the given set of constraint, it may be possible that there could be two solutions of same value, however mathematically / technically there would not exist any better solution than the one arrived using Linear Programming method. For more details of Linear programming method, the interested readers can read on Wikipedia² or similar websites.

So with linear programming, we can arrive at the best / optimal charge mix for given set of conditions. If any of the condition changes, i. e. say price of one raw material is reduced / increased. Then again the earlier solution arrived using Linear Programming might not be optimal and we have to calculate the new optimal solution.

Currently what we in the foundry industry do is, we arrive at the charge mix based on the expertise of our foundry supervisor / foundry manager. This person based on his / her own judgement, calculates the charge mix. Most of them are not aware of the current raw material prices, so their solution may be good based on the chemical balance of few alloying elements and guidelines from top management about using a particular raw material more because it is cheaper than other raw materials (however how much cheap is not known to them). So this charge mix calculated by the foundry persons may not be optimal price-wise or in overall manner.

Let us take couple of examples to elaborate this point. For one company ‘x’, they are creating almost 95% to 99% pure Fe by removing all the alloying elements in it. Rest of the amount is carbon depending upon the casting requirements. Then they add suitable alloying elements as per their requirements in the ladle.

If we calculate the cost of raw materials as per conventional practices (i. e. chemical balancing) then the per ton cost of the charge mix is 26,233. Following is their raw material input this company uses per ton of finished goods produced. If we put in the constraints in terms of output chemistry and chemistry, prices & recovery of each alloying element in the raw materials in the leaning programming method, we would get following solution:

Raw Materials used		Conventional Method in Kg per ton		Linear Programming Method in Kg per ton
Pig Iron		370		277
HMS		493		576
DRI		123		89
Turning Scrap		31		78
Misc		62		55
TMT Bars		31		33
Total Cost		26,233		25,807

So with linear programming method there is a savings of 1.63% for getting a plain metal. This 1.63% of 47% of raw material costs would give us overall savings of 0.76 in the costs. I am sure none of us would have thought of getting a cost saving in producing pure Fe. But in case of linear programming it is possible to get the cost saving in this seemingly impossible case as well. If we consider proper cast iron with all the alloying elements, we can get a cost savings of approx. of upto 4% (instead of 1.63% shown above). This can help us a lot in making our foundry operations profitable to that extent.

Now let us take another example of company ‘y’ producing complex chemistry of GX- 40 CrNi 25-12 type steel³. This company wanted to use a fixed quantity of 450 Kg of GX-40 returns per ton of steel. Hence you may see that the quantity of this raw material in conventional method (spreadsheet method) and linear programming method is same. As per the desired chemistry, the spread sheet method calculated the charge as shown below.

Materials Used		Conventional Method Quantity (Kg)		Linear Programming Quantity (Kg)
Steel Scrap		250		0
FeCr		210		163
FeMn		13		0
FeSi		14		13.6
Graphite		1		0
Nikel		62		38.4
Return GX-40 (Stock = 450 kg)		450		450
Return CF8M (Stock = 450 kg)		0		234
Return GS-52 (Stock = 250 kg)		0		16
Ingot 3.8C-0.7Mn-3.15Si		0		85
Charge Cost per Ton		1,04,655.84		90,072.95

First of all in this example, the linear programming software ‘searched’ for other low cost returns / raw materials before consuming high value steel scrap, FeMn alloy, FeCr alloy, etc. This change in approach lead to the raw material savings to the tune of 17.24% of 47% giving us the final results of 8.1% overall cost savings!

Even though, we have taken above examples of charge mix optimization for ferrous foundries. Since the mathematical principals are same, we can effectively use linear programming for optimizing the costs of non-ferrous foundries as well. Based on our understanding following could be typical values of cost savings achieved through the use of Linear Programming based software.

• Cast Iron: 4% (About 15,000 Us$ /(1000 Tons))
• Aluminium: 8% (About 120,000 Us$ / (1000 Tons))
• High Alloyed Steel: 15% (About 250,000 Us$ / (1000 Tons))

I am sure most of the foundries do not have such a huge margins of 8%. So all this cost savings can lead to additional profit or reduction in costs to get the competitive advantage. It is upto the individual foundry to keep 100% additional cost savings with them, or pass on some percent of this cost savings to the client by reducing the costs and thereby beating the competition and gaining additional business.

Conclusions

Linear programming even though about 190 years old technique, for variety of reasons nobody had thought of using it in calculating charge mix optimization calculations earlier. If we put in ‘innovative’ constraints in the linear programming to let the system calculate the charge mix optimization, we can get very good and encouraging results in reducing the costs and thereby improving the sales volumes. Another advantage of this method is since the mundane work of charge  mix optimziation calculations is performed by the computers, the foundry supervisor / managers get more time in doing some creative and productive work for improving quality, 5S, TPM, TQM, etc. activities and thereby improving the overall performance of the foundry.

References

1. From the Annual reports (audited balance sheet) published on the websites of 5 publicly traded foundries / companies on various stock exchanges in India. These details are for Financial Year ended on 31^st Mar 2016. Under each head shown in the chart, we have added costs for these 5 companies and taken simple average of those figures and then converted into the percentages.
2. Wikipedia – https://en.wikipedia.org/wiki/Linear_programming.
3. Article “Software for the Optimization of Raw Materials in Casting: A Key to Quality and Competitiveness” by Mr. Alberto Montenegro Correa published in WFC 2014.

Written By

C. S. Joshi – BE (Mech.), PGDST, Chartered Engineer, PMP

Managing Consultant, CompuBee Technology Pvt. Ltd.

Can be reached @ info@compubee.in