Provide Chain Optimization and Product Mixing Utilizing Python and Mathematical Modeling


Creating environment friendly provide chains is likely one of the best challenges of the 2020s—and never simply due to the disruptions caused by the COVID-19 pandemic. Provide chains had been strained earlier than the pandemic attributable to international bottlenecks and shortages in labor and gear. To maintain up with demand, market gamers should quickly modernize enterprise processes via digitization and clever planning.

My profession as a developer and information science advisor is targeted on heavy business: rail, mining, oil and fuel, transport, and postal logistics. All of those sectors have been vastly impacted by provide chain points over these previous couple of years. On this piece, I discover how mathematical optimization modeling and Python can resolve a core problem within the coal mining business: satisfying custom-made demand and maximizing revenue via product mixing.

An Optimization Strategy for the Trendy Provide Chain

Within the typical provide chain situation, a provider delivers a particular completed product to a buyer. In our instance, to perform this, a provider should:

  1. Collect the mandatory elements from a number of supply areas (e.g., manufacturing websites, warehouses).
  2. Mix the elements, executing a particular process to create a completed good. In a coal provide chain, that is known as product mixing.
  3. Ship the completed product to a single goal location (e.g., the shopper’s web site).

Finished proper, product mixing permits the provider to maximise worth by leveraging trade-offs between buyer wants and the availability chain. Mathematical optimization modeling is the perfect answer for addressing product mixing together with logistical challenges reminiscent of scheduling, planning, packing, and routing.

A graph theoretical strategy, like community movement optimization, works effectively for challenges with a transparent restricted scope (e.g., asking Google Maps the way to get from A to B). However to deal with extra intricate challenges that influence overlapping facets of the availability chain (e.g., product mixing), mixed-integer programming is a robust framework. Quick, well-researched, and established, mixed-integer programming permits customers to handle the overwhelming majority of scheduling, planning, and routing points.

To mannequin and clear up provide chain issues, I like to recommend utilizing Python and its open-source libraries attributable to their robust optimization communities.

Product Mixing in a Coal Provide Chain

For example of product mixing, let’s contemplate a coal provide chain that includes a number of mines and produces a wide range of coal elements. Sometimes, these elements should be routed to seaports. To maintain our instance easy, we are going to connect with only one seaport by way of a rail community that additionally hyperlinks the mines.

We’ll use the next phrases:


A part is a uncooked manufacturing merchandise (e.g., a kind of coal), sourced at a particular location.


A product is a completed good, demanded and outlined by a buyer, usually containing a mix of elements and falling inside a acknowledged high quality vary.


Mixing is the combining of elements to type a product, both on the goal location (usually, the shopper’s vessel) or in some unspecified time in the future within the provide chain.

Spec Assay

A specification, or spec, assay is the measurement of a part property (e.g., moisture content material). Sometimes, engineers carry out about 20 to 100 assays, every of which exams a distinct property of the coal.

Coal elements retrieved from mines are transported by rail to a port, with the shopper vessel as the ultimate vacation spot. Relying on the designated port’s berthing schedule or different circumstances, non permanent storage of the elements at a stockpile could also be crucial. On the port, the prepare will both deposit the load onto a stockpile or unload it instantly onto the shopper vessel (what we name a direct hit).

Elements are saved at mines and seaports. Mines are usually established in distant areas the place space for storing is affordable and plentiful. Ports, alternatively, exist in industrial areas that normally have restricted area, making port stockpiles costly to make use of.

Modeling Particulars

Our hypothetical buyer has demanded product blends that consist of various elements. These blends should conform to the related mineral property requirements, as outlined by the shopper (e.g., CSR worth). For example how this mannequin could be constructed, let’s say that now we have three mines that produce seven elements, as follows:

Mine A

Produces elements A1, A2, A3.

Mine B

Produces elements B1, B2.

Mine C

Produces elements C1, C4.

The letter in a part’s identify signifies the part’s supply mine (e.g., part A3 was sourced at Mine A). Let’s agree that elements that share a quantity are related and as such, we could deal with them equivalently: For instance, A1, B1, and C1 are primarily the identical sort of coal.

On the left side of the screen is Mine A with its stockpiles of A1, A2, and A3 types coal. Below Mine A is Mine B with its stockpiles of B1 and B2 types coal. Below Mine B is Mine C with its stockpiles of C1 and C4 types coal. On the right side of the screen is a ship loaded up with two heaps of coal. Below the ship are four Port Stockpiles, as follows: A stockpile with one heap of coal labeled A3; a stockpile with three heaps of coal labeled A2, B2, and C1, respectively; a stockpile with two heaps of coal labeled B1 and A1, respectively; and a stockpile with two heaps of coal labeled A3 and C4, respectively. At the center of the screen are two trains. The first train is shown as three cars, two of which are filled with coal. Below the train are two green arrows that point to the right. Between the arrows is the language
Instance: Provide Chain Mannequin for Coal Product Mixing

All elements are transported by rail to the port, the place we are able to both carry out a direct hit or deposit every part at an acceptable stockpile. Area limitations could prohibit us from storing elements individually. As such, when mixing a product, we could not have entry to every part individually and should have to extract a number of elements from a single stockpile concurrently.

Now, let’s focus on the mixing guidelines that prospects usually demand for his or her merchandise.

Product Mixing Guidelines

Clients routinely ask for a mix of elements per customer-specific guidelines on each how a mix could also be carried out and which spec assays are crucial. Such guidelines fall into two classes, part mixing guidelines and spec mixing guidelines.



Part Mixing Guidelines

The proportion of every part that composes a product is outlined as a ratio or proportion of the entire.

A product stockpile (aka blended stockpile) with the next breakdown:

Part A2

≥ 50%

50% or extra of the product’s complete tonnage comes from part A2.

Part C1

≤ 20%

20% or much less of the product’s complete tonnage comes from part C1.

Any Mixture of C1, B2

≤ 50%

50% or much less of the product’s complete tonnage comes from the mix of elements C1 and/or B2.

Spec Mixing Guidelines

Worth boundaries for a product are established for every outlined product property.

Product properties are measured by spec assay. Values embrace:

  • Acceptable. The product is accepted by the shopper as a result of its assay values fall inside the buyer’s most well-liked vary.
  • Penalized. The product is accepted by the shopper however with a high quality penalty paid by the provider to the shopper as a result of the product’s assay values fall outdoors the shopper’s most well-liked vary.
  • Unacceptable. The product is just not accepted by the shopper as a result of its assay values deviate too removed from the shopper’s most well-liked vary.

The product in our earlier instance could be accepted with out penalty if:

Moisture Content material

≤ 10%

The whole moisture content material of the product is lower than or equal to 10%.

Ash Content material

≤ 0.1%

The ash content material of the product is lower than or equal to 0.1%.

Coke Energy After Response (CSR)

≤ 35%

The coke energy of the product, after response, is lower than or equal to 35%.

Discover that, usually, the deviation penalty quantity will increase as a linear perform because the boundary violation grows:

A graph depicting three values. The Y axis shows a dollar sign at the bottom of the graph and three dollar signs at the top of the graph. The top of the Y axis is an arrow head pointing upwards. The X axis includes five labels: Unacceptable, Penalized, Acceptable, Penalized, and Unacceptable.
Widespread Modeling Framework for Spec Mixing Rule Worth Boundaries: The penalty for provider deviation from established product property values grows linearly in direct relationship to the extent of deviation.

The optimization of mixing contains trade-offs between accepting penalties for spec mixing and the supply of elements.

When making a product mixing mannequin, we should select between completely different extraction varieties. For mixing, common extraction is the commonest extraction sort. In common extraction, we mannequin based mostly on an assumption that each one elements within the stockpile are totally blended collectively. Layered extraction, the place we mannequin utilizing a last-in, first-out rule, is a substitute for utilizing common extraction:

Common Extraction

  • All elements are blended collectively totally.
  • Extracting a certain quantity from the stockpile wouldn’t have an effect on relative part ratios.

Layered Extraction

  • Elements are layered, not blended.
  • Extraction happens from the highest of the stockpile.
  • After the part on the high of the stockpile is extracted, the part that had been saved instantly beneath it’s revealed and extracted subsequent. This sample repeats itself till we attain the underside of the stockpile.

The thought of layered extraction could also be interesting, because it intently simulates the truth of the storing logistics at most stockpiles. Nonetheless, from a mathematical modeling perspective, common extraction is most well-liked for computational causes. The choice to make use of layered extraction must be rigorously evaluated by enterprise professionals and engineers, in order to keep away from introducing pointless problems right into a modeling strategy.

When utilizing common extraction, the proportions of extracted elements to at least one one other are similar to these of the unextracted elements. For instance, common extraction says that an extraction from stockpile X containing 75% of part A3 and 25% of part C4 accommodates the identical elements and the identical proportions as stockpile X in its entirety.

At left is a port stockpile labeled "Port Stockpile." The first (top) of the stockpile's two layers is labeled A3 and an indicative label states it comprises 75% of the stockpile. The second (bottom) layer of the stockpile's two layers is labeled C4 and an indicative label states it comprises 25% of the stockpile. To the right of the stockpile is an arrow, labeled "Average Extraction" pointing to the right. Further to the right, at the arrow's tip is the ratio "A3/C4.”
Common Extraction Instance: The proportions of extracted elements to at least one one other are similar to these of the unextracted elements within the stockpile.

When layered extraction is used, the proportions of extracted elements to at least one one other are hardly ever, if ever, similar to these of the unextracted elements. Layered extraction says that, for instance, an extraction from stockpile X wouldn’t essentially include the identical elements as stockpile X in its entirety, nor the identical proportions as stockpile X. It is because we’d be extracting no matter part(s) are on the high of the stockpile (last-in, first-out).

A port stockpile made of A3 stock piled on top of C4 stock. There are four arrows from the stockpile. From top to bottom they point to the labels A3, A3 (again), A3/C4, and C4.
Layered Extraction Instance: The proportions of extracted elements to at least one one other are hardly ever similar to these of the unextracted elements within the stockpile.

The inconsistent nature of a layered extraction makes it tough to mannequin loading variables. Due to this fact, common extraction, which avoids advanced interdependencies between the loading variables, is the popular choice when layered extraction is just not a enterprise requirement (see additionally “Coding to Clear up Product Mixing”).

Product Mixing Modeling

Let’s contemplate the case of common extraction. Say we wish to monitor and mannequin elements deposited at a stockpile or buyer vessel. Listed here are three potential extraction and modeling eventualities:

Situation 1: Single Extraction Modeling

We are able to extract any/all elements, no matter sort.

On this instance, we could deal with part A1 (sourced at Mine A) and B1 (sourced at Mine B) as if they’re the identical part as a result of they’re related sufficient.

At top left is a right-pointed arrow labeled

Situation 2: Product Mix Extraction Modeling

We are able to extract a possible product mix.

On this instance, the extracted product mix conforms to the shopper’s product mixing rule necessities:

  • The ratios of elements A2, B2, and C1 at Port Stockpile 2 adhere to the shopper’s part mixing guidelines.
  • As well as, the elements’ spec assays adhere to the shopper’s spec mixing guidelines.

At top left is a right-pointed arrow labeled

Situation 3: Versatile Combine Extraction Modeling

We are able to solely extract an invalid part combine that doesn’t conform to the shopper’s part mixing guidelines and thus doesn’t type a product by itself.

On this instance, since our mix of elements A3 and C4 doesn’t type a legitimate product, we are able to:

  • Await the arrival of and mix the mandatory elements.
  • Mix the mix with different stockpile(s).
  • Load the mix instantly onto a buyer vessel to be mixed with one other mix.

At top left is a right-pointed arrow labeled

From a modeling perspective, I like to recommend creating mixed-integer programming formulations to handle product mixing. We are able to mannequin product mixing through the use of solely real-valued variables and linear constraints, making it comparatively straightforward to calculate and monitor blends.

Issues can get sophisticated when product mixing modeling overlaps with scheduling choices that require binary variables for modeling functions, reminiscent of choices round vessel berthing or prepare schedules.

Coding to Clear up Product Mixing

Python is good for coding and fixing mixed-integer programming formulations. Use the PuLP library to formulate provide chain issues, reminiscent of defining variables, constraints, and goal capabilities. Conveniently, PuLP’s syntax intently resembles a clear mathematical formulation.

You’ll be able to then combine an open-source solver like Cbc or, in case your finances permits, a business solver like Gurobi or CPLEX. The business choices present an incredible efficiency increase in comparison with Cbc.

The next pseudocode examples show how we outline loading variables and constraints. The loading variables are:

load[v=vessel, p=port, c=component, prd=product, t=time]

These variables have 5 indices: vessel, port, part, product, and time. In observe, you’ll outline many extra kinds of loading variables.

Including a product index to the loading variables is helpful for monitoring the particular product for which a part is designated. Since loading variables are actual values (versus integers), they don’t pose an enormous computational problem. Part mixing guidelines can now be modeled as follows:

load[v, p, A2, prd, t] >= 0.5 * sum(load[v, p, c, prd, t] for all c that serve product prd of vessel v)
load[v, p, C1, prd, t] <= 0.2 * sum(load[v, p, c, prd, t] for all c that serve product prd of vessel v)
load[v, p, C1, prd, t] + load[v, p, B2, prd, t] <= 0.5 * sum(load[v, p, c, prd, t] for all c that serve product prd of vessel v)

Spec mixing guidelines might be carried out with the same linear strategy. Nonetheless, these constraints could be a bit extra sophisticated, since spec assays are normally normalized by the loaded quantity. Whereas less complicated to execute, direct modeling would introduce non-linearities and, thus, could be impractical. As an alternative, it might be higher to calculate weighted pseudo-assay values after which reapply the linear equations. Caveat: The constraints could overlap with binary scheduling variables—however that dialogue is past the scope of this text.

I wholeheartedly suggest incorporating mixing guidelines right into a provide chain mannequin. My previous shoppers have had constant successes with optimized customer-specific mixing, which elevated the computational complexity of their scheduling fashions solely by a hair.

Remodeling Your Coal Provide Chain to Incorporate Mixing

Product mixing is strongly linked to rail and port operations, closely impacting day-to-day choices, reminiscent of the place to move which elements or the place to deposit and/or extract elements.

The perfect digital state is a complete scheduling device that gives forward-looking suggestions for rail and port, with a mixing optimization mannequin built-in as a key half. When acceptable (e.g., to reply to altering climate situations), advert hoc problem-solving by licensed rail and port operators can appropriate chosen suggestions.

For every distinctive provide chain, a customized scheduling device is smart. Utilizing an Agile course of, we may establish the influence of product mixing earlier than our full digital device is launched. Shut collaboration with operators and coordinators—always—would go a great distance towards addressing any change administration dangers.

To construction and code the fashions crucial for constructing a scheduling utility, interact the abilities of information scientists, information engineers, and optimization specialists. In at the moment’s difficult and aggressive surroundings, companies that implement product mixing keep forward of the competitors.

The editorial staff of the Toptal Engineering Weblog extends its gratitude to John Lee for reviewing the code samples and different technical content material introduced on this article.


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