# Economics in Engineering

Basic principles of economics in engineering include some of the classic concepts such as Economic Order Quantity, Break Even Point, Interest and Loan Repayment and Diminishing Return. Basic management tools include Gantt and PERT charts. Most economic decisions in engineering can be traced down to basic (and often overlooked) concepts like these.

**From MEM30008A Delivery Plan**

*3. Economic Considerations*

*Production quantities (mass/batch)**Cost of manufacture**Cost of maintenance**Quality of product consideration**Practicability of manufacture**Standardised components.*

#### Contents

- Break Even
- Economic Order Quantity
- Compound Interest and Loan Repayment
- Depreciation
- Diminishing Returns
- Economies of Scale
- False Economy
- Quality
- Project Charts

## 1. Break Even

How long will it take to pay off a business investment?

The "break even" point is the number of items (or hours) needed to pay off an investment.

**Examples**

A more expensive mould makes cheaper parts. Break even is the number of parts that pays for tooling difference.

A superior tool speeds up work. The payback is measured in number of hours of increased productivity that pays for the tool.

A redesigned product is more profitable. Break even is the number of parts that pays for product development.

The **break even point(BEP)** is the volume of sales required so that the expenses of the business are equal to the income received. There is neither a profit nor a loss. After that point you are making money (in the black).

In short, at the BEP all associated costs are paid off but the profit is zero.

To find the break-even point, we need to know fixed and variable costs. The fixed cost includes things like tooling, capital expenditure, rent. Variable costs are associated with the quantity of product - like labour, material costs, electricity and resources.

These costs are then compared to the profit per product, and the BEP is calculated.

**Worked Example**

Work out the break-even point for the production of a plastic doova-lacky (looks like a spoon-shaped half side of a thinga-me-jig). Maybe it's a fashion accessory.

**Data**

- Tooling costs = $20000 (This is about right for this mould - I kid you not)
- Production cost per part = $5 (includes material $0.20 + machine time $1.00 + labour $1.00 + tool amortisation $2.80)
- Profit per part = $25 (Yes, I know it's a complete rip-off, but this is a fashion accessory)

**Solving Graphically**

So costs start at $20000 and increase by $5 per product.

Income starts at $0 and increase by $25 per part.

The break even point is 1000 units. This is when the tool has been paid off and we are finally starting to make some money ($20 per part to be precise).

**Solving Mathematically**

**BEP = TFC ****/ (P - V)**

**TFC**is Total Fixed Costs = $20000**P**is Unit Sale Price = $25**V**is Unit Variable Cost = $5

BEP = 20000 / (25-5) = 20000 / 20 = 1000 units.

##### More resources

http://en.wikipedia.org/wiki/Break-even_(economics)

## 2. Economic Order Quantity

Just-in-Time manufacturing methods are championed by high flying mass-producers, but there are traps for the unwary. Attempting to achieve zero inventory can make production more expensive due to increased setup (ordering costs). For every manufacturing situation, there is an *ideal* inventory size which is determined by the EOQ (Economic Order Quantity).

The economic order quantity is the optimum quantity to order that has the lowest cost taking into account the carrying costs (inventory), ordering costs (setup), and stockout costs (lost sales/back orders).

When large quantities are ordered, ordering costs and stockout costs go down, but carrying costs go up.

### EOQ Model 1: basic model

The basic model for determining the optimal economic order quantity (EOQ) was first proposed by Harris (1913). This basic model assumes the following parameters are constant and known with certainty:- Demand rate per unit time : consumption is constant
- Replenishment lead time: no delays
- Per-unit purchase cost: no quantity discounts
- Per-unit inventory: holding cost is constant
- Per-unit ordering cost: Setup is constant

ln addition, the model assumes that all demand must be met and the complete order is delivered instantaneously. Hence, the basic EOQ model does not consider purchase quantity discounts, back orders, or inventory shortage costs.

Moreover, there is no safety stock considered in this model.

* *

##### Terms

**D**is the annual demand by the client for inventory in units/year. However, the time period could be weeks, months, or some other period.**Q**is the quantity of material ordered each time a requisition is made and is measured in units/ purchase order.**C**is the cost of carrying one unit in inventory for one year, $/unit/year when one year is the time period in consideration. This value of C may be a given financial amount such as $10/unit/year or may be expressed as a percentage of the unit purchase price of the material.**S**is the average cost of placing a purchase order, $/order.**TSC**is the total annual inventory cost, $/year. The assumptions of the model are that the values of D, C, and S can be determined precisely, and remain**constant**.

Since the assumption is that the entire inventory is used up before the next order quantity arrives then the minimum value of inventory is 0, and the maximum is Q.

##### Therefore:

**C**= purchase price, unit production cost**Q**= order quantity**Q***= optimal order quantity**D**= annual demand quantity**K**= fixed cost per order, setup cost (*not*per unit, typically cost of ordering and shipping and handling. This is not the cost of goods)= annual holding cost per unit, also known as carrying cost or storage cost (capital cost, warehouse space, refrigeration, insurance, etc. usually not related to the unit production cost)*h*

Classic EOQ model: trade-off between ordering cost (blue) and holding cost (red). Total cost (green) admits a global optimum. Purchase cost is not a relevant cost for determining the optimal order quantity.

The single-item EOQ formula finds the minimum point of the following cost function:

Total Cost = purchase cost or production cost + ordering cost + holding cost

- Purchase cost: This is the variable cost of goods: purchase unit price × annual demand quantity. This is c × D

- Ordering cost: This is the cost of placing orders: each order has a fixed cost K, and we need to order D/Q times per year. This is K × D/Q

- Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2, so this cost is h × Q/2

Total Cost:

Reorganizing and making Q the subject leads to the** EOQ:**

This is the economic order quantity (the ideal quantity to purchase that will keep inventory-related costs to a minimum)

The expected order cycle length, or the time between orders (TBO), is **TBO = EOQ/D **in years.

An order is placed when inventory position reaches the reorder point **R = d*L**, where L is the length of the replenishment lead time and d is the expected demand per unit of time.

##### Points to Note

- Demand increase: If demand in the marketplace doubles, the optimal order quantity, and hence the average inventory level, increases only by the square root of 2 or by 41% over the original amount.
- Warehouse consolidation: The total average inventory required to maintain identical regional warehouses is n(Q*/2). However, consolidating the regional warehouses into a central facility results in system order cycle inventory of only square root(n)* Q/2. Hence, a regional system required square root(n) times as much inventory as a central system.
- Reduction in ordering costs: Suppose an investment in an automated electronic ordering system (EDI…) reduces ordering costs by 75% of their former level. The optimal order cycle inventory reduces by 50% reduction in inventory over the previous level of ordering cost.
- Just in Time: It is never economical to reduce inventory below the EOQ. The inventory can only be lowered by reducing the ordering cost (
**K**). So JIT is not about getting rid of inventory, but lowering order cost (i.e. setup costs).

More complicated and realistic stock replenishment models are outlined in the original article:

##### More resources

http://en.wikipedia.org/wiki/Economic_order_quantity

## 3. Compound Interest and Loan Repayment

**Compound interest** is interest added to the principal of a deposit or loan so that the added interest also earns interest from then on. This addition of interest to the principal is called *compounding*.

But this is normal.

A bank account, for example, may have its interest compounded every year: in this case, an account with $1000 initial principal and 20% interest per year would have a balance of $1200 at the end of the first year, $1440 at the end of the second year, $1728 at the end of the third year, and so on.

To define an interest rate fully, allowing comparisons with other interest rates, both the interest rate and the compounding frequency must be disclosed. For instance, the yearly rate for a loan with 1% interest per month is approximately 12.68% per annum (1.0112 − 1). This equivalent yearly rate may be referred to as *annual percentage rate* (APR), *annual equivalent rate* (AER), *effective interest rate*, *effective annual rate*, and other terms. When a fee is charged up front to obtain a loan, APR usually counts that cost as well as the compound interest in converting to the equivalent rate.

For any given interest rate and compounding frequency, an equivalent rate for any different compounding frequency exists.

Compound interest may be contrasted with simple interest, where interest is not added to the principal (there is no compounding). Compound interest is standard in finance and economics, and simple interest is used infrequently (although certain financial products may contain elements of simple interest).

**Formula for calculating annual compound interest:**

where

*S*= value after*t*periods*P*= principal amount (initial investment)*j*= annual nominal interest rate (not reflecting the compounding)*m*= number of times the interest is compounded per year*t*= number of years the money is borrowed for

**Example**

Suppose an amount of $1500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Then the balance after 6 years is found by using the formula above, with *P* = 1500, *j* = 0.043 (4.3%), *m* = 4, and *t* = 6:

So, the balance after 6 years is approximately 1938.84. The amount of interest received can be calculated by subtracting the principal from this amount ($1938.84 - $1500 = $438.84).

http://en.wikipedia.org/wiki/Compound_interest

#### Amortisation

Amortisation is paying off a loan by regular instalments over a period of time. A home mortgage is an example of an amortised loan.

An exact formula for monthly payment is;

where

*P*= Payment per month*L*= Initial Loan amount*i*= interest rate (per payment)- n = number of months (or payments)

**Example: **

At an interest rate of 6% per annum over 25 years on $150000. What is the monthly payment?

L = $150000

i = 0.06/12 = 0.005 per month (use a fraction, not %)

n = 25 years * 12 = 300 payments

P = (150000 * 0.005) / ( 1 - ( 1 / (1 + 0.005)^300)) = 966.45 per month

So the monthly payment will be $966.45. This assumes the bank adds interest monthly.

By changing to a weekly payment rather than monthly;

L = $150000

i = 0.06/52 = 0.005 per month (use a fraction, not %)

n = 25 years * 52 = 1300 payments

P = (150000 * 0.001154) / ( 1 - ( 1 / (1 + 0.001154)^1300)) = 222.84 per week

Which is a monthly payment of $965.65. This seems like a trivial difference - which it is. However, if the bank accrues interest at short intervals and you pay at longer intervals the difference can be significant.

**Frequency Effects **

Increasing the frequency of repayment or interest charge has the effect of speeding up the compounding effect.

## 4. Depreciation

Expensive assets cannot be written off in one tax year. For example, a $200 tool can be fully deducted form income, but a $10000 machine will need to be depreciated over the effective life of the asset.

There are several methods for calculating depreciation, generally based on either the passage of time or the level of activity (or use) of the asset.

### Straight-line depreciation - US (Prime Cost Method - Aus)

Straight-line depreciation is the simplest and most often used method. In this method, the company estimates the salvage value (scrap value) of the asset at the end of the period during which it will be used to generate revenues (useful life). (The salvage value is an estimate of the value of the asset at the time it will be sold or disposed of; it may be zero or even negative. Salvage value is also known as scrap value or residual value.) The company will then charge the same amount to depreciation each year over that period, until the value shown for the asset has reduced from the original cost to the salvage value.

**Straight-line method:**

Book value at the beginning of the first year of depreciation is the original cost of the asset. At any time book value equals original cost minus accumulated depreciation.

**book value = original cost − accumulated depreciation** Book value at the end of year becomes book value at the beginning of next year. The asset is depreciated until the book value equals scrap value.

For example, a vehicle that depreciates over 5 years is purchased at a cost of $17,000, and will have a salvage value of $2000. Then this vehicle will depreciate at $3,000 per year, i.e. (17-2)/5 = 3. This table illustrates the straight-line method of depreciation.

Depreciation expense |
Accumulated depreciation at year-end |
Book value at year-end |
---|---|---|

- | - | (original cost) $17,000 |

$3,000 | $3,000 | $14,000 |

3,000 | 6,000 | 11,000 |

3,000 | 9,000 | 8,000 |

3,000 | 12,000 | 5,000 |

3,000 | 15,000 | (scrap value) 2,000 |

If the vehicle were to be sold and the sales price exceeded the depreciated value (net book value) then the excess would be considered a gain and subject to depreciation recapture. In addition, this gain above the depreciated value would be recognized as ordinary income by the tax office. If the sales price is ever less than the book value, the resulting capital loss is tax deductible. If the sale price were ever more than the original book value, then the gain above the original book value is recognized as a capital gain.

If a company chooses to depreciate an asset at a different rate from that used by the tax office then this generates a timing difference in the income statement due to the difference (at a point in time) between the taxation department's and company's view of the profit.

### Declining Balance Method - US (Diminishing Value Method - Aus)

Suppose a business has an asset with $1,000 original cost, $100 salvage value, and 5 years of useful life. First, the straight-line depreciation rate would be 1/5, i.e. 20% per year. The table below illustrates this:

Depreciation rate |
Depreciation expense |
Accumulated depreciation |
Book value at end of year |
---|---|---|---|

- | - | - | original cost $1,000.00 |

40% | 400.00 | 400.00 | 600.00 |

40% | 240.00 | 640.00 | 360.00 |

40% | 144.00 | 784.00 | 216.00 |

40% | 86.40 | 870.40 | 129.60 |

129.60 - 100.00 | 29.60 | 900.00 | scrap value 100.00 |

When using the diminishing value method, the salvage value is not considered in determining the annual depreciation, but the book value of the asset being depreciated is never brought below its salvage value, regardless of the method used. Depreciation ceases when either the salvage value or the end of the asset's useful life is reached.

Since diminishing value depreciation does not always depreciate an asset fully by its end of life, some methods also compute a straight-line depreciation each year, and apply the greater of the two. This has the effect of converting from declining-balance depreciation to straight-line depreciation at a midpoint in the asset's life.

Calculating the book value of an asset after*t* years is exactly the same as the compounding interest formula - except the interest is negative of course.

where

*NPV*= Net Present Value, the book value of the asset after*t*years*PC*= Prime Cost or principal amount. The initial investment.*j*= annual depreciation rate (typically twice the prime cost depreciation rate, which is 1/life)*t*= number of years of depreciating the asset

**And, to be really tricky...**

With the declining balance method, one can find the depreciation rate that would allow exactly for full depreciation by the end of the period, using the formula: Residual value = scrap value.

,

where N is the estimated life of the asset (for example, in years).

**Comparision of annual depreciation amounts deducted each year for an $80000 capital purchase.**

Prime Cost Method: If the asset cost $80,000 and has an effective life of five years, you can claim 20% of its cost, or $16,000, in each of the five years.

Diminishing Value method: The rate is calculated on 40% - twice the prime value amount.

A comparision of Prime Cost (red) depreciation vs Diminishing Value (blue) method shows that DV gives a higher tax advantage at first, but this reduces each year.

Note that the Diminishing Value method never quite reaches zero. There are a few ways to deal with this. The logical solution is to write off the asset after the effective life has been reached (which must be known in order to obtain the depreciation rate). The asset might also be written off when it dips below the cut-off (typically $300 for personal tax - based on ATO rules. QC 2610). For very high value assets, this many take too long, so the DV method can be swapped to the PC method after a certain point (or when the PC method catches up - as in year 6 in the example chart). This is, or course, equivalent to writing off the asset after the effective life has been reached. Another option is to write the asset off when it reaches the scrap value.

The decision to use PC or DV, and all the variations, depends on the tax planning for the business at the time. E.g. Making too much profit and need to reduce tax? Use DV for high initial deductions. Want to store up some deductions for future years? Use PC.

##### Notes

**Date of purchase**. The first year might only be 6 months from June 30, so the depreciation would be half the (10% instead of 20%, for example).

**Sale of Asset.** If the asset is sold, income will need to be adjusted if the sale price does not match the book value (pretty likely).

**Depreciation Rates.** The ATO has rules on the maximum depreciation rates on most common assets - which is supposed to represent the useful life of the item. For example, a laptop has shorter life than a car.

From the ATO website:

**Computers:**ATO

**Computers****Effective Life**

**(Years)****Diminishing**

**Value****Prime Cost**Desktops 4 50% 25% Laptops 3 66.67% 33.33% -
**Vehicles:**ATO**Vehicle designed to carry**

a load**of less than 1 tonne and****fewer than 9 passengers****Effective Life**

**(Years)****Diminishing**

**Value****Prime Cost**Generally 8 25% 12.5% Hire & travellers’ cars 5 40% 20% Taxis 4 50% 25%

##### Resources

- http://en.wikipedia.org/wiki/Depreciation
- ATO Depreciation Guide 2013-14: download here

## 5. Diminishing Returns

In economics, diminishing returns (also called law of diminishing returns, law of variable proportions, principle of diminishing marginal productivity, or diminishing marginal returns) is the decrease in the marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, while the amounts of all other factors of production stay constant.

For example, the use of fertilizer improves crop production on farms and in gardens; but at some point, adding increasingly more fertilizer improves the yield by less per unit of fertilizer. Excessive quantities can even reduce the yield - which is NEGATIVE gain which should never happen in engineering - but it can!

A common sort of example is adding more workers to a job, such as assembling a car on a factory floor. At some point, adding more workers causes problems such as workers getting in each other's way or frequently finding themselves waiting for access to a part. In all of these processes, producing one more unit of output per unit of time will eventually cost increasingly more, due to inputs being used less and less effectively.

As labour usage increases from L1 to L2, total output (measured vertically in the top graph) increases by the amount shown. But at a higher labour level, when the labour usage is increased by the same amount again, output goes up by less, implying diminishing marginal returns to the use of labour as an input.
The marginal product of labour is diminishing everywhere to the right of point A.
From a mathematical point of view, "return" or "marginal" is simply the gradient of the total product curve. So the marginal product curve is the gradient of the total product curve. |

In engineering design, diminishing return can apply in various ways:

- Using components that are twice as expensive does not necessarily mean you can charge twice as much.
- Increasing the scale/speed of a machine cannot go on forever, eventually any increase in size or performance will be prohibitively expensive (square/cube law).
- Saturation of the local market can mean that increased production will result in the need to ship greater distances.
- Physical laws that are power functions. (e.g. the power required against air resistance increases to the cube of velocity. To double the maximum speed requires an 8-fold increase in power)

## 6. Economies of Scale

Economy of scale is like "buying in bulk". As an engineering or manufacturing operation increases in size (scale) multiple factors work together to reduce costs.

- Mass production is cheaper than custom manufacture.
- Purchase discounts are common for large quantities (bulk buying). (e.g. shipping and manufactured goods discounts)
- A larger workforce allows specialisation, where specialists are more efficient (e.g. CAD specialist, financial staff)
- Expensive promotions can be used - e.g. TV, billboard campaigns, sales people. (e.g. McDonalds franchise TV advertising)
- Specialised equipment can be purchased as there are more opportunities to spread the work over multiple jobs (e.g. welding robot)
- The administrative load of certain types of accreditation can be met, giving access to new markets (e.g. defence force work, quality accreditation)
- A larger company can be more tolerant to unexpected changes (e.g. law changes, riding out of economic downturns)
- Drag loss of vehicles like aircraft or ships generally increases less than proportional with increasing cargo volume. So, making them larger usually results in less fuel consumption per ton of cargo at a given speed.
- Heat losses from industrial processes vary per unit of volume for pipes, tanks and other vessels in a relationship somewhat similar to the square-cube law. So larger pipes are more effectivley insulated.
- Startup and setup costs can be spread over larger batches. (e.g. setting up rolling mill to different profile)

##### The Square-Cube Law

(See also Square-Cube Law in page Simple Machines)

Economy of scale is also directly related to area and volume (The square-cube law). For example, the COST of a vessel is proportional to the surface area, which increases by the square of the scale, while the volume increases by the cube of the scale.

Therefore a tank that cost twice as much to build will hold more than twice the volume. Hence larger tanks cost less per litre of fluid.

In the above example, polyethylene water storage tanks are obviously more expensive as they get larger. However, economy of scale allows the material cost to reduce with increasing size ($/kg of PE drops from $19.80 to $5.58, a factor of 3.5 times). However, for the volume of water, this economy of material scale also combines with the square/cube law to bring a dramatic reduction in water storage cost. ($/litre drops from $1.46 to $0.10, a factor of 14.6 times!)

Interestingly, there appears to be some slight effects of diminishing return for the largest tanks. This may be due to lower sales, increased handling costs (cranes), higher manufacturing risks of scrap or production difficulties when working near the size limit.

The square cube law also works the other way round - causing an engineering limit to the scale of equipment. According to the square/cube law, stresses increase with scale. This defines a maximum size of a structure built from a certain material. Beyond a certain size, exotic high strength materials would be required, making further increases in scale uneconomical (diminishing return).

##### Synergy

**Synergy** is the creation of a whole that is greater than the simple sum of its parts. Synergy in business management refers to combining the different operations to gain a mutual advantage.

For example, a tree lopper (arborist) might want to start a business selling firewood. This would give several advantages;

- When arborist work is quiet (e.g. bad weather) he can cut and split his logs into firewood.
- Logs that he would have to dump or chip can produce more cash as firewood.

This is often a motivation behind business mergers, offering economy of scale with additional profits due to synergistic partnerships.

http://en.wikipedia.org/wiki/Economies_of_scale

## 7. False Economy

False economy generally refers to doing something to save money, but it ends up costing more in the long run (or overall).

Examples of false economy:

- Buying a cheap new whipper-snipper every summer, instead of buying a good one that lasts 10 years.
- Driving an old vehicle, where the added fuel cost outweighs the saving in capital (purchase cost).
- Taking shortcuts to save time, resulting in expensive accidents or mistakes
- Eating cheap food and getting tired or sick

**Engineering Change Procedures**

In an engineering environment, false economy can occur when making decisions that do not take the full picture into account. A typical example might be like this:

A manufactured product is chugging along nicely. Some "bean-counter" (derogatory term for finance-related managers) decides to change some of the components to a cheaper supplier. All goes well at first, but after a few months products begin to fail and return under warranty. This ends up costing more than the original saving: False economy.

This is an example of why an engineering change system is used in manufacturing - to avoid falling for some bright idea that turns out to be very expensive.

An engineering change procedure may need to consider the costs involved in;

- New tooling
- Wasted product inventory
- Re-training staff, sales
- Documentation
- Issues with certification, accreditation
- Unknowns - unforeseen consequences

A typical engineering change will need to be assessed by relevant managers - (design, manaufacturing, sales etc), and only accepted if all parties agree. This puts pressure on product designers to avoids excessive numbers of trivial changes that can reduce the profitablity of production.

## 8. Quality

Although quality accreditation is often masked by increased paperwork requirements, the core principles of TQM are supposed to improve the bottom line. This is primarly due to:

- Less mistakes, waste and scrap
- Continuous improvement of methods and processes

## 9. Project Charts

##### Gantt Charts

A Gantt Chart is a project plan laid out on a timeline. In a Gantt chart, the rows are the individual steps or processes in the project breakdown. The horizontal axis is the time - in hours, days, weeks or whatever.

The example Gantt chart (above) is for building a house. Each row represents a division into (major) construction steps. Certain processes can only begin after a previous step is completed. This is called a dependency.

These charts are best done by computer where it is easy to edit. During the project (such as a maintenance shutdown, or major installation etc), variations and unforeseen problems may arise. A computerised Gantt chart can be quickly edited and the new schedule and dates sent (by email or online access) as an update to all contractors.

An extension to the basic Gantt chart is to include resources. For example, a limited number of workers are defined, and linked to each activity. This may effect the leadtimes of certain activities, or highlight the need for extra staff/sub-contractors in certain steps in the project. Other resources could include machine time, warehouse space, stock levels, financial restraints etc.

**A free online program for building Gantt charts;**

https://www.smartsheet.com/gantt-chart-software

##### Pert Charts

A PERT chart is graphical way of showing the interconnections in a project. It can be used to highlight the critical path (the limiting path that determines the minimum duration of the project).

Most charting programs can automatically generate a PERT chart from a Gantt chart.

PERT stands for the Project Evaluation and Review Technique. It is a statistical tool used in project management, which was designed to analyze and represent the tasks involved in completing a given project. First developed by the United States Navy in the 1950s, it is commonly used in conjunction with the critical path method (CPM).

The example above shows a simple PERT network chart for a seven-month project with five milestones (10 through 50) and six activities (A through F). The milestone numbers are arbitrary. The duration of each process (activity) is labeled as "t=4 mo" which means "time = t months".

The critical path is BC, with a duration of 4+3 = 7 months. There is also another 7 month path ADF, with a duration of 3+1+3 = 7 months, making it another critical path. The non-critical path is AE, or duration 3+3 = 6 months. So the project cannot be sped up by trying to shorten activities A or E.

## Glossary