• url: thoughts/.../Conversion-Factors
• description: Conversion Factors

marginal value change.

extra net value obtained for one more item

$$\Delta NV = NV(x+1) - NV(x)$$ Lien vers l'original

• url: thoughts/.../Optimization
• description: Optimization

model-based

• conclusions from the model of the system

Components:

• decision variables
• constraints
• objectives
• functions: mathematical function that determines the objective as a function of decision variable

$$\begin{align*} \min_{x} \phi = f(x) & &\leftarrow &\space \text{Objective function} \\\ \text{s.t} & &\leftarrow &\space \text{Constraints} \\\ h(x) = 0 & &\leftarrow &\space \text{Equality constraints} \\\ g(x) \leq 0 & &\leftarrow &\space \text{Inequality constraints} \\\ x_{lb} \leq x \leq x_{ub} & &\leftarrow &\space \text{Bounds} \end{align*}$$

decision variables

discrete.

limited to a fixed or countable set of values

$$x_{\mathcal{D}} \mid a \in \mathcal{I} = \lbrace 1, 2, 3, 4, 5 \rbrace$$

continuous.

can take any value within a range

$$x_{\mathcal{C}} \subset \mathcal{R}$$

constraints

• physical limitations: cannot purchase negative raw materials

• model assumptions: assumptions about the system

domain of a definition

a decision upper and lower bounds ($x^{\mathcal{U}}$ and $x^{\mathcal{L}}$)

Properties

• Active/binding: $\exists \space x^{*} \mid g(x^{*}) = 0$
• Inactive: $\exists \space x^{*} \mid g(x^{*}) < 0$

graphing models

feasible set of an optimization model

The collection of decision variables that satisfy all constraints

$$\mathcal{S} \triangleq \lbrace x : g(x) \leq 0, h(x) = 0, x^L \leq x \leq x^U \rbrace$$

outcomes

optimal value

the optimal value $\phi^{*}$ is the value of the objective at the optimum(s)

$$\phi^{*} \triangleq \phi(x^{*})$$

Constraints satisfy, but it is not binding

Linear optimization problems

$$\begin{aligned} \underset{x_1,x_2}{\min} \space \phi &= 50x_1 + 37.5x_2 \\ &\text{s.t} \\\ 0.3x_1 + 0.4x_2 &\geq 2000 \\\ 0.4x_1 + 0.15x_2 &\geq 1500 \\\ 0.2x_1 + 0.35x_2 &\leq 1000, \\\ x_1 &\leq 9000 \\\ x_2 &\leq 6000 \\\ x_i &\geq 0 \end{aligned}$$

See also Linear Optimization

Lien vers l'original

Linear optimization:

• url: thoughts/.../Linear-Optimization
• description: Linear Optimization

\begin{align*}
\min_{x} \phi = c^\mathbf{T} \mathcal{x} & &\leftarrow &\space \text{Objective function} \\\
\text{s.t} & &\leftarrow &\space \text{Constraints} \\\
A_h \mathcal{x} = \mathcal{b}_h & &\leftarrow &\space \text{Equality constraints} \\\
A_g \mathcal{x} \leq \mathcal{b}g \leq 0 & &\leftarrow &\space \text{Inequality constraints} \\\
\mathcal{x}_{lb} \leq \mathcal{x} \leq \mathcal{x}_{ub} & &\leftarrow &\space \text{Variable Bounds}
\end{align*}

Lien vers l'original

• scope, cost, time to maximize quality

WBS (work breakdown structure): hierarchical decomposition of the total scope of work

CPM (critical path method): determine the longest path through the network, the critical path, and the shortest time to complete the project

crashing a project means using additional resources to shorten a specific task

market equilibrium: where supply and demand curves intersect, quantity demanded equals quantity supplied. shift to right: greater demand, higher price, higher quantity. shift to left: lower demand, lower price, lower quantity. factors of production: land, labour, capital, entrepreneurship determinants of demand:

• price: quantity demanded $Q_d$ falls when price $P$ rises and vice versa
• prices of related goods: substitutes and complements determinants of supply:
• price: quantity supplied $Q_s$ rises when price $P$ rises and vice versa
• factors of productions
• fiscal policies, taxes, regulation

elasticity: how responsive quantity demanded or supplied is to a change in price.

Surplus when $Q_s > Q_d$, shortage when $Q_s < Q_d$.

Elasticity of demand: $E_d = \frac{\% \Delta Q_d}{\% \Delta P} = \frac{\mid \frac{P}{Q_D} \mid}{\mid \frac{dP}{dQ_D} \mid}$

Elasticity of supply: $E_s = \frac{\% \Delta Q_s}{\% \Delta P} = \frac{\mid \frac{P}{Q_S} \mid}{\mid \frac{dP}{dQ_S} \mid}$

higher slope corresponds to lower elasticity: inelastic, lower slope corresponds to higher elasticity: elastic

Demand elasticity: $E_D <1$ means if price increases by 5% then demand will decrease by less than 5%, inelastic. $E_D >1$ means if price increases by 5% then demand will decrease by more than 5%, elastic.

taxes

arbitrary lower the equilibrium quantity,

price seen by consumers vs. suppliers changes depends on relative elasticities of demand and supply: more price change will end up on consumer side

quantities change depends on total elasticities of demand and supply: more elastic means more quantity change.

subsidies

arbitrary increase the equilibrium quantity,

price seen by consumers vs. suppliers changes depends on relative elasticities of demand and supply: more price change will end up on consumer side

quantities change depends on total elasticities of demand and supply: more elastic means more quantity change.

invisible hand of the market: self-interest of individuals leads to the best outcome for society as a whole, in a free market economy, as rational actors are motivated by incentives.

perfect competition: wheat (control of price none, low barrier to entry, high # of producers, products are identical) monopolistic competition: restaurants (control of price low, low barrier to entry, high # of producers, products are similar) oligopoly: airlines (control of price high, high barrier to entry, few producers, products are similar) monopoly: utilities (control of price high, high barrier to entry, one producer, unique product)

game theory, most notable The Prisoner’s Dilemma

anti-trust legislation: prevent monopolies, promote competition, protect consumers

behavioural economics: + psychology to look at reasons people make irrational decisions

“bounded rationality”: you don’t have perfect information, and understand there’s an opportunity cost to get it

law of demand and ultimatum game: people will pay less for a good if they can get it elsewhere for less, even if they value it more than the price they pay.

Cooperation: R. Axelrod’s The Evolution of Cooperation propose a “strategy”, what you do dependent on what the other person does.

PPF (production possibility frontier): trade-offs between two goods, given a fixed amount of resources.

risk aversion: people prefer a certain outcome to a risky one, even if the expected value of the risky one is higher. ⇒ assume that the given investment is loss, then calculate based on margin gains

income, corporate, property, sales

personal income tax: progressive tax rate corporate tax: flat tax rate, regardless of income level → net income: subtracting expenses from gross income.

profit on investments will be tax. If yields loss, then offset the loss against the profits from another to pay less tax overall.

optimization strategies: minimize liabilities, timing of expenditures → incorporate into financial models, do sensitivity analysis

before-tax MARR: set MARR high enough to include taxes that need to be paid ⇒ for investment’s gross profit after-tax MARR: if tax is explicitly accounted for in the cash flows of the project, then MARR should be lower ⇒ for final investment decisions

incentives: tax credits, tax reliefs, programs to encourage certain activities depreciation: due to use-related physical loss, technological obsolescence, functional loss, market fluctuation.

Deprecation is a non-cash expense, but reduces the taxable income of a business. Can deduct annually by spreading the cost of an asset over its useful life.

affects NPV (net present value), IRR (internal rate of return), and payback period calculation

Market value: actual value of the asset can be sold for, estimated Book value: deprecated value of the asset, using a depreciation model Salvage value: estimated value of the asset at the end of its useful life

value calculations

Depreciation in year $n$ $D(n)$ is the decline in book value over that year: $BV(n) = BV(n-1) - D(n)$

Salvage value $SV$ is the book value at object’s EOL: $SV = BV(N) = MV(0) - \sum_{n=1}^{N} D(n)$

Straight-line depreciation

spreads uniformly over useful life, SLD of a period $D_{\text{sl}}(n) = \frac{\text{Purchase price}-\text{Salvage value after N periods}}{\text{N periods of useful life}}$.

book value at end of $n^{th}$ year: $BV_{\text{sl}}(n) = P - n \times \frac{P-S}{N}$

Declining-balance depreciation

different assets are classified into classes: $D_{\text{db}}(n) = BV_{\text{db}}(n-1) \times d (\text{depreciation rate})$, such that book value at the end of a period $BV_{\text{db}}(n)$ is $BV_{\text{db}}(n) = P(1-d)^n$

given salvage value $S$ and period of useful life $N$, depreciation rate $d = 1 - \sqrt[N]{\frac{S}{P}}$

Sum-of-years-digits depreciation

$D_{\text{syd}}(n) = \frac{N-n+1}{\sum_{i=1}^{N} i} \times (P-S)$

Unit of production depreciation

$D_{\text{uop}}(n) = \frac{\text{units produced of period}}{\text{life in \# of units}} \times (P - S)$

assumes a SLD but vs. # of units rather than time.