engineering economy ( ned this in 2 hours)

Simple Interest:

Simple Interest, 𝐼 = 𝑃 ∗ 𝑛 ∗ 𝑖 I = Simple Interest

P = Principal (amount borrowed or lent)

Final Amount, 𝐹 = 𝑃 + 𝐼 = 𝑃 + 𝑃𝑛𝑖 n = number of interest periods

i = simple interest rate

Compound Interest Factors: i = interest rate

n = number of interest (compounding) periods

P = present sum (initial principal amount)

A = an equal periodic amount (uniform series)

F = a future sum

(F/P): Algebraic Notation: 𝐹 = 𝑃(1 + 𝑖)𝑛 Factor Notation: 𝐹 = 𝑃(𝐹/𝑃 𝑖, 𝑛)

(P/F): Algebraic Notation: 𝑃 = 𝐹(1 + 𝑖)−𝑛 Factor Notation: 𝑃 = 𝐹(𝑃/𝐹 𝑖, 𝑛)

(F/A): Algebraic Notation: 𝐹 = 𝐴{ (1+𝑖)𝑛−1

𝑖 }

Factor Notation: 𝐹 = 𝐴(𝐹/𝐴 𝑖, 𝑛)

(A/F): Algebraic Notation: 𝐴 = 𝐹{ 𝑖

(1+𝑖)𝑛−1 }

Factor Notation: 𝐴 = 𝐹(𝐴/𝐹 𝑖, 𝑛)

(P/A): Algebraic Notation: 𝑃 = 𝐴{ (1+𝑖)𝑛−1

𝑖(1+𝑖)𝑛 }

Factor Notation: 𝑃 = 𝐴(𝑃/𝐴 𝑖, 𝑛)

(A/P): Algebraic Notation: 𝐴 = 𝑃{ 𝑖(1+𝑖)𝑛

(1+𝑖)𝑛−1 }

Factor Notation: 𝐴 = 𝑃(𝐴/𝑃 𝑖, 𝑛)

(A/G): Factor Notation: 𝐴 = 𝐴1 ± 𝐺(𝐴/𝐺 𝑖, 𝑛) (A/G): Algebraic Notation: 𝐺 ( 1

𝑖 −

𝑛

(1+𝑖)𝑛−1 )

𝐴1= Constant amount 𝐺 = Gradient amount

(P/G): Factor Notation: 𝐴1(𝑃/𝐴 𝑖, 𝑛) ± 𝐺(𝑃/𝐺 𝑖, 𝑛) (P/G): Algebraic Notation: 𝐺 ( (1+𝑖)𝑛−𝑖𝑛−1

𝑖2(1+𝑖)𝑛 )

Interest Rate Conversion:

Effective Interest Rate, 𝑖 = 𝑟

𝑐 Effective Annual Interest rate, 𝑖 = (1 +

𝑟

𝑐 )

𝑐

− 1

Nominal Interest Rate, 𝑟 = 𝑐 {(1 + 𝑖) (

1

𝑐 )

− 1}

Continuous Interest Factors:

(F/P): Algebraic Notation: 𝐹 = 𝑃𝑒 𝑟𝑛 Factor Notation: 𝐹 = 𝑃[𝐹/𝑃 𝑖, 𝑛]

(P/F): Algebraic Notation: 𝑃 = 𝐹 ( 1

𝑒 𝑟𝑛 )

Factor Notation: 𝑃 = 𝐹[𝑃/𝐹 𝑖, 𝑛]

(F/A): Algebraic Notation: 𝐹 = 𝐴 (𝑒𝑟𝑛−1)

(𝑒 𝑟−1)

Factor Notation: 𝐹 = 𝐴[𝐹/𝐴 𝑖, 𝑛]

(A/F): Algebraic Notation: 𝐴 = 𝐹 (𝑒𝑟−1)

(𝑒 𝑟𝑛−1)

Factor Notation: 𝐴 = 𝐹[𝐴/𝐹 𝑖, 𝑛]

(P/A): Algebraic Notation: 𝑃 = 𝐴 (1−𝑒 −𝑟𝑛)

(𝑒𝑟 −1)

Factor Notation: 𝑃 = 𝐴[𝑃/𝐴 𝑖, 𝑛]

(A/P): Algebraic Notation: 𝐴 = 𝑃 (𝑒 𝑟 −1)

(1−𝑒 −𝑟𝑛)

Factor Notation: 𝐴 = 𝑃[𝐴/𝑃 𝑖, 𝑛]

Continuous Interest Rate Conversion:

Effective Annual Interest rate, 𝑖 = 𝑒 𝑟 − 1 Nominal Interest Rate, 𝑟 = ln(1 + 𝑖)

Bonds:

𝑃 = 𝑘𝑉

𝑐 (𝑃/𝐴

𝑟

𝑐 , 𝑐𝑛) + 𝑉 (𝑃/𝐹

𝑟

𝑐 , 𝑐𝑛) Where: 𝑃 = 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑝𝑟𝑖𝑐𝑒, 𝑉 = 𝐹𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒, 𝑘 = 𝐵𝑜𝑛𝑑 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑢𝑎𝑙 𝑟𝑎𝑡𝑒,

𝑐 = # 𝑜𝑓 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟, 𝑛 = # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 𝑡𝑜 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦, 𝑟 = 𝑒𝑎𝑟𝑛𝑒𝑑 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑎𝑡𝑒

Depreciation Methods or Models:

In General: 𝐷𝑡 = 𝐵𝑡−1 − 𝐵𝑡 𝐵𝑡 = 𝑃 − ∑ 𝐷𝑥 P = Initial Cost = 𝐵𝑉0; 𝐵𝑡= Book Value at EOY t; 𝐷𝑡= Depreciation for year t; L = Salvage Value; n = Depreciation Life

Straight Line:

𝐷𝑡 = (𝑃 − 𝐿)

𝑛

𝐵𝑡 = 𝑃 − ∑ 𝐷𝑥 = 𝑃 − 𝑡{ (𝑃 − 𝐿)

𝑛 }

Sum of Years Digits:

𝐷𝑡 = { (𝑛 − 𝑡 + 1)

𝑛(𝑛+1)

2

} (𝑃 − 𝐿)

𝐵𝑡 = ( (𝑛 − 𝑡)

𝑛 ) (

(𝑛 − 𝑡 + 1)

(𝑛 + 1) ) (𝑃 − 𝐿) + 𝐿

Usage:

𝐷𝑡 = (𝑃 − 𝐿)𝑈𝑡

𝑈

𝐵𝑡 = 𝑃 − ∑ 𝐷𝑖

Sinking Fund:

𝐷𝑡 = (𝑃 − 𝐿)(𝐴/𝐹 𝑖, 𝑛)(𝐹/𝑃 𝑖, 𝑡 − 1) 𝐵𝑡 = 𝑃 − ∑ 𝐷𝑥 = 𝑃 − (𝑃 − 𝐿)(𝐴/𝐹 𝑖, 𝑛)(𝐹/𝐴 𝑖, 𝑡)

Declining Balance: 𝐷𝑡 = 𝑎(𝐵𝑡−1), 𝐷𝑡 = 𝑎𝑃(1 − 𝑎) 𝑡−1, 𝐵𝑡 = (1 − 𝑎)

𝑡 𝑃 where 𝑎 = 1 − √ 𝐿

𝑃

𝑛 , Switch to SL if :

𝐵𝑡−1−𝐿

𝑛−(𝑗−1) > 𝑎𝐵𝑡−1

Double Declining Balance: 𝐷𝑡 = 𝑎(𝐵𝑡−1), 𝐷𝑡 = 𝑎𝑃(1 − 𝑎) 𝑡−1

, 𝐵𝑡 = 𝑃 − ∑ 𝐷𝑖 = 𝑃(1 − 𝑎) 𝑡 where 𝑎 =

2

𝑛

Equivalent Annual Cost of Capital Recovery Plus Return:

𝐸𝐶𝑅 = (𝑃 − 𝐿)(𝐴/𝑃 𝑖, 𝑛) + 𝐿(𝑖)

Taxes:

𝑇𝑎𝑥𝑎𝑏𝑙𝑒 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝐺 − 𝐶 − 𝐷𝑡 − 𝐼 G = Gross Income C = Cost of Goods Sold

𝐷𝑡= Tax Depreciation Allowance I = Interest Paid on Debt Obligations

Tax rates:

Taxable income Tax rate

0-$50,000 15% over $0

$50,000 - $75,000 $7,500 + 25% over $50,000

$75,000 - $100,000 $13,750 + 34% over $75,000

$100,000 - $335,000 $22,250 + 39% over $100,000

$335,000 - $10 million $113,900 + 34% over $335,000

$10 million - $15 million $3,400,000 + 35% over $10 million

$15 million - $18,333,333 $5,150,000 + 38% over $15 million

>= $18,333,333 35%

Effective tax rate = 𝑡𝑎𝑥

𝑡𝑎𝑥𝑎𝑏𝑙𝑒 𝑖𝑛𝑐𝑜𝑚𝑒

MACRS Depreciation:

* Year to switch from declining balance to straight line

Measures of Merit:

NPV=∑ 𝑋𝑗 𝑛 𝑗=0 (𝑃/𝐹 𝑖, 𝑗) or: NPV=∑

𝑋𝑗 ′

(1+𝑖)𝑗 𝑛 𝑗=0

IRR: 0 = ∑ 𝑋𝑗 𝑛 𝑗=0 (𝑃/𝐹 𝑖, 𝑗) or: 0 = ∑

𝑋𝑗 ′

(1+𝑖)𝑗 𝑛 𝑗=0

Payback Period: 0 = ∑ 𝑋𝑗 ′𝑝

𝑗=0

Linear Break-Even Models:

𝑃 = (𝑠𝑉 − 𝑐𝑉 − 𝐹 − 𝐷𝑏 − 𝐼) − (𝑠𝑉 − 𝑐𝑉 − 𝐹 − 𝐷𝑡 − 𝐼)𝑇 (𝑝𝑟𝑜𝑓𝑖𝑡 𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥) 𝑃𝑏 = (𝑠𝑉 − 𝑐𝑉 − 𝐹 − 𝐷𝑏 − 𝐼) (𝑝𝑟𝑜𝑓𝑖𝑡 𝑏𝑒𝑓𝑜𝑟𝑒 𝑡𝑎𝑥) 𝐹′ = 𝐹 + 𝐷𝑏 + 𝐼 𝑖𝑓 𝐷𝑏 = 𝐷𝑡 → 𝑃 = (𝑠𝑉 − 𝑐𝑉 − 𝐹

′)(1 − 𝑇) 𝑖𝑓 𝐷𝑏 ≠ 𝐷𝑡 → 𝑃 = (𝑠𝑉 − 𝑐𝑉 − 𝐹

′)(1 − 𝑇) + (𝐷𝑡 − 𝐷𝑏 )𝑇 Where:

P = after-tax profit per unit of time V = volume of sales per unit of time

s = selling price per unit c = variable cost per unit (raw material, direct labor, direct supplies, etc.)

F = fixed costs per unit of time 𝐷𝑏 = book depreciation per unit of time I = debt interest expense per unit of time 𝐷𝑡 = tax depreciation per unit of time T = tax rate s𝑉𝑏 = gross income (revenues) per unit of time cV = variable costs per unit of time

𝑉𝑏 = 𝐹+𝐷𝑏+𝐼

𝑠−𝑐 (𝑏𝑟𝑒𝑎𝑘 − 𝑒𝑣𝑒𝑛 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛)

𝑠𝑏 = 𝑐 + ( 𝐹+𝐷𝑏+𝐼

𝑉 )

(𝑏𝑟𝑒𝑎𝑘 − 𝑒𝑣𝑒𝑛 𝑠𝑒𝑙𝑙𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒 𝑜𝑟 𝑢𝑛𝑖𝑡 𝑠𝑎𝑙𝑒𝑠 𝑝𝑟𝑖𝑐𝑒)

Cost Comparisons:

Equivalent Annual: Equivalent Annual Cost of Capital Recovery and Return (ECR): 𝐸 = (𝑃 − 𝐿)(𝐴/𝑃 𝑖, 𝑛) + 𝐿𝑖 + 𝐶)

Capitalized Cost (𝐶𝐶)𝑜𝑟 𝑃 = 𝐴 𝑖⁄ ;Where A=equivalent annual amount, i=interest rate

Present Worth (𝑃) = 𝐸(𝑃/𝐴 𝑖, 𝐿𝐶𝑀); 𝑤ℎ𝑒𝑟𝑒 𝐿𝐶𝑀 = 𝐿𝑒𝑎𝑠𝑡 𝐶𝑜𝑚𝑚𝑜𝑛 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑒 (when infinite service life and lifetimes are different), if lifetime the same, discount all cash flows for each alternative to present

Benefit-Cost Analysis: Project acceptable if: B – C ≥ 0 or B/C ≥ 1

Inflation:

𝑖 = 𝑖 ′ + 𝑓 + 𝑖 ′𝑓 ; 𝑤ℎ𝑒𝑟𝑒 𝑓: 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒, 𝑖 ′: 𝑟𝑒𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒, 𝑖 = 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒

𝑖 ′ = (𝑖 − 𝑓)

(1 + 𝑓)

𝐴$ 𝐴𝑐𝑡𝑢𝑎𝑙 𝐷𝑜𝑙𝑙𝑎𝑟𝑠 → 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 (𝑖) 𝑅$ 𝑅𝑒𝑎𝑙 𝐷𝑜𝑙𝑙𝑎𝑟𝑠 → 𝑟𝑒𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 (𝑖 ′)

Price Index: 𝐴𝑛𝑛𝑢𝑎𝑙 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑛 = 𝑖𝑛𝑑𝑒𝑥𝑛 −𝑖𝑛𝑑𝑒𝑥𝑛−1

𝑖𝑛𝑑𝑒𝑥𝑛−1 𝑥100% ; where cost index would be given

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒: 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑠ℎ𝑜𝑢𝑙𝑑 𝑏𝑒 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑒 𝑐𝑜𝑠𝑡 𝑖𝑛𝑑𝑒𝑥𝑒𝑠 𝑓𝑜𝑟 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 & 𝑓𝑢𝑡𝑢𝑟𝑒 𝑦𝑟𝑠

DPMO (Defects Per Million Opportunities):

𝐷𝑃𝑀𝑂 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑠

[𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑖𝑒𝑠 𝑓𝑜𝑟 𝑒𝑟𝑟𝑜𝑟 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡] 𝑥 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑥 1,000,000

Process Capability:

Measure of Potential Capability: 𝐶𝑝 = 𝑈𝑇𝐿−𝐿𝑇𝐿

6𝜎 ; Where UTL=Upper Tolerance Limit, LTL=Lower Tolerance Limit,

𝜎=Standard deviation

Measure of Actual Capability:  





 



 

 3

X-UTL ,

3

LTLX min=C

pk ; Where �̅� = mean (average of samples)

Cp and Cpk ≥ 1 in order for the process to be capable If Cp≠ Cpk  process not centered

Kanban:

C

SDL k

)(1  ; Where:

k=Number of Kanban card sets (a set is a card) D=Average number of units demanded over some time period L=lead time to replenish an order S=Safety stock expressed as a percentage of demand during lead time C=Container size (in number of units per container)