Provide Chain Optimization and Product Mixing Utilizing Python and Mathematical Modeling


Creating environment friendly provide chains is likely one of the biggest 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 on account of world bottlenecks and shortages in labor and tools. To maintain up with demand, market gamers should quickly modernize enterprise processes by means of digitization and clever planning.

My profession as a developer and information science marketing consultant is targeted on heavy trade: rail, mining, oil and fuel, transport, and postal logistics. All of those sectors have been tremendously 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 mining trade: satisfying custom-made demand and maximizing revenue by means of product mixing.

An Optimization Strategy for the Fashionable Provide Chain

Within the typical provide chain state of affairs, a provider delivers a selected completed product to a buyer. In our instance, to perform this, a provider should:

  1. Collect the required parts from a number of supply places (e.g., manufacturing websites, warehouses).
  2. Mix the parts, executing a selected process to create a completed good. In a mining provide chain, that is known as product mixing.
  3. Ship the completed product to a single goal location (e.g., the client’s website).

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 resolution for addressing product mixing together with logistical challenges reminiscent of scheduling, planning, packing, and routing.

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

To mannequin and remedy provide chain issues, I like to recommend utilizing Python and its open-source libraries on account of their sturdy optimization communities.

Product Mixing in a Mining Provide Chain

For example of product mixing, let’s take into account a mining provide chain that contains a number of mines and produces a wide range of uncooked materials parts. Usually, these parts must be routed to seaports. To maintain our instance easy, we’ll connect with only one seaport through 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 copper or iron ore), sourced at a selected location.


A product is a completed good, demanded and outlined by a buyer, sometimes containing a mix of parts and falling inside a said high quality vary.


Mixing is the combining of parts to kind a product, both on the goal location (sometimes, the client’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). Usually, engineers carry out about 20 to 100 assays, every of which checks a distinct property of the uncooked materials.

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

Parts are saved at mines and seaports. Mines are usually established in distant places the place cupboard space is affordable and plentiful. Ports, however, 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 parts. These blends should conform to the related mineral property requirements, as outlined by the client (e.g., CSR worth). For instance how this mannequin could be constructed, let’s say that now we have three mines that produce seven parts, as follows:

Mine A

Produces parts A1, A2, A3.

Mine B

Produces parts B1, B2.

Mine C

Produces parts 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 parts that share a quantity are related and as such, we might deal with them equivalently: For instance, A1, B1, and C1 are primarily the identical sort of uncooked materials.

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

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

Now, let’s talk about the mixing guidelines that prospects sometimes demand for his or her merchandise.

Product Mixing Guidelines

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



Element 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:

Element A2

≥ 50%

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

Element C1

≤ 20%

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

Any Mixture of C1, B2

≤ 50%

50% or much less of the product’s whole tonnage comes from the mixture of parts 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 embody:

  • Acceptable. The product is accepted by the client as a result of its assay values fall inside the buyer’s most popular vary.
  • Penalized. The product is accepted by the client however with a high quality penalty paid by the provider to the client as a result of the product’s assay values fall outdoors the client’s most popular vary.
  • Unacceptable. The product isn’t accepted by the client as a result of its assay values deviate too removed from the client’s most popular vary.

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

Moisture Content material

≤ 10%

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

Property 𝝰 (e.g., residue)

≤ 0.1%

The proportion of 𝝰 content material within the product is lower than or equal to 0.1%.

Property 𝛃 (e.g., a chemical response relationship)

≤ 35%

The proportion of 𝛃 content material within the product is lower than or equal to 35%.

Discover that, sometimes, the deviation penalty quantity will increase as a linear operate 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.
Frequent 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 parts.

When making a product mixing mannequin, we should select between completely different extraction sorts. For mixing, common extraction is the most typical extraction sort. In common extraction, we mannequin based mostly on an assumption that each one parts within the stockpile are totally blended collectively. Layered extraction, the place we mannequin utilizing a last-in, first-out rule, is an alternative choice to utilizing common extraction:

Common Extraction

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

Layered Extraction

  • Parts are layered, not blended.
  • Extraction happens from the highest of the stockpile.
  • After the part on the prime 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 concept 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 popular for computational causes. The choice to make use of layered extraction ought to be rigorously evaluated by enterprise professionals and engineers to keep away from introducing pointless problems right into a modeling strategy.

When utilizing common extraction, the proportions of extracted parts to 1 one other are an identical to these of the unextracted parts. For instance, common extraction says that an extraction from stockpile X containing 75% of part A3 and 25% of part C4 comprises the identical parts 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 pointing to the ratio "A3/C4.”
Common Extraction Instance: The proportions of extracted parts to 1 one other are an identical to these of the unextracted parts within the stockpile.

When layered extraction is used, the proportions of extracted parts to 1 one other are not often, if ever, an identical to these of the unextracted parts. Layered extraction says that, for instance, an extraction from stockpile X wouldn’t essentially include the identical parts as stockpile X in its entirety, nor the identical proportions as stockpile X. It is because we might be extracting no matter part(s) are on the prime 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 parts to 1 one other are not often an identical to these of the unextracted parts 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 isn’t a enterprise requirement (see additionally “Coding to Remedy Product Mixing”).

Product Mixing Modeling

Let’s take into account the case of common extraction. Say we want to monitor and mannequin parts deposited at a stockpile or buyer vessel. Listed here are three attainable extraction and modeling eventualities:

State of affairs 1: Single Extraction Modeling

We will extract any/all parts, no matter sort.

On this instance, we might 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

State of affairs 2: Product Mix Extraction Modeling

We will extract a possible product mix.

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

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

At top left is a right-pointed arrow labeled

State of affairs 3: Versatile Combine Extraction Modeling

We will solely extract an invalid part combine that doesn’t conform to the client’s part mixing guidelines and thus doesn’t kind a product by itself.

On this instance, since our mix of parts A3 and C4 doesn’t kind a sound product, we will:

  • Await the arrival of and mix the required parts.
  • 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 deal with product mixing. We will 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 Remedy Product Mixing

Python is right 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 may then combine an open-source solver like Cbc or, in case your finances permits, a industrial solver like Gurobi or CPLEX. The industrial choices present an amazing efficiency increase in comparison with Cbc.

The next pseudocode examples display 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 apply, you’ll outline many extra forms 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. Element 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 will be carried out with an analogous linear strategy. Nonetheless, these constraints could be a bit extra sophisticated, since spec assays are normally normalized by the loaded quantity. Whereas easier 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 might overlap with binary scheduling variables—however that dialogue is past the scope of this text.

I wholeheartedly advocate 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 by solely a hair.

Reworking Your Mining Provide Chain to Incorporate Mixing

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

The best digital state is a complete scheduling software 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 approved rail and port operators can appropriate chosen suggestions.

For every distinctive provide chain, a customized scheduling software is smart. Utilizing an Agile course of, we may establish the affect of product mixing earlier than our full digital software 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 obligatory for constructing a scheduling utility, have interaction the abilities of information scientists, information engineers, and optimization consultants. In right now’s difficult and aggressive atmosphere, 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 offered on this article.


Please enter your comment!
Please enter your name here