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-- About Operation Research
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MOTIVATION: When considering complex content, visualization using graphs can help intuitive understanding. This will be confirmed through a simple example.
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6.1.1 Revolving Payment:
- A type of loan. It is a mechanism whereby the customer borrows money from a financial institution or other source and repays it.
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6.1.2 Break-even point:
- One of the management indicators. The point at which SALES and COST are equal.
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6.1.1 Revolving Payment:
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$inter_a$ : anual intersest$inter_m$ : monthly interest$repay$ : repayment (per month)$debt$ : debt balance (last month) -
Paying for the interest of this month:
$repay_i = int(debt * inter_m)$ -
*Paying for the principal of this month:
$ret = repay - repay_i$ -
*Debt balance at the end of this month:
$debt = debt - ret$
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- 6.1.1 Revolving Payment:
- Left one shows the upward convexity of the graph as the remaining balance changes. Therefore, after 30 months, the rate of decline in the principal gradually increases.
- Right one shows the percentage of cash payments out of the monthly fixed payment gradually becomes larger.
- Making a partial or lump-sum repayment in the middle of the repayment period will exponentially reduce the repayment period.
- 6.1.2 Break-even point:
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- fixed cost : rent and human resource cost...
varible cost: raw material costs, purchase costs, transportation costs...
nonlinear varible cost : a typical sales environment (e.g., weather, traffic conditions...)
cost : fixed cost + varible cost + nonlinear varible cost
- fixed cost : rent and human resource cost...
varible cost: raw material costs, purchase costs, transportation costs...
nonlinear varible cost : a typical sales environment (e.g., weather, traffic conditions...)
cost : fixed cost + varible cost + nonlinear varible cost
- 6.1.2 Break-even point:
calculate break-even point:
At the intersection（break-even point） on the left side, a loss occurs if the value is below and a profit occurs if the value is above
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6.2.1 Mathematical expression:
- Objective function: $$ \text{arg} \min_{x, y} f(x, y) = -4x + y $$
- Constraint condition: $$ \text{subject to:} \begin{cases} 0 \leq x \leq 3 \ 0 \leq y \leq 1 \ x + y \leq 2 \end{cases} $$
- 6.2.2 Can use PuLP:
- 6.2.3 Simulation sample: Please Check -> MM_LP.ipynb
- 6.3.1 What is inventory management:
- Inventory :The term is used to refer to goods (merchandise, materials, etc.) stored in the warehouses of stores and factories
- Excessive inventory cost occurs when inventory is too large for demand.
- If inventory is too small relative to demand, Inventory shortage cost arise at the time of insufficient inventory. -> Periodical ordering system and Order point system
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6.3.2 Economic Order Quantity(EOQ):
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$Q$ : Order quantity for one time$C_{year}$ : Annual maintenance cost per item [yen/(item/year)]$C_{one}$ : Ordering cost per order [yen]
$D$ : Demand per year [pcs/year]$$Q^* = \sqrt{\frac{2 C_{\text{one}} D}{C_{\text{year}}}}$$
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- 6.3.3 Order point system: This is a method of inventory management in which a certain amount of orders are placed when the inventory level drops to a certain level.
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$O_p$ : Ordering point$L$ : Lead time$\bar d$ : demand for one day$S$ : Safety stock$k(\alpha)$ : Safety factor $ \sigma^2$ :Dispersion of demand$$O_P = L \bar{d} + S $$ $$O_P = L \bar{d} + k(\alpha) \sigma \sqrt{L} $$
- 6.3.4 Periodocal ordering system: This is a method in which the order cycle (also called the order interval) is predetermined and the order is placed.
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$T$ : Ordering cycle interval$Q$ : Order quantity$Q^*$ : EOQ$L$ : Lead time$k(\alpha)$ : Safety factor $ \sigma^2$ :Dispersion of demand$x_k$ : Inventory level when k Almost the same as the previous method. The quantity in stock at the next time$x_{k+1}$ is as follows:$$x_{k+1} = x_k + Q = (T + L) \bar{d} + \left( k(\alpha) \sigma \sqrt{T + L} \right)$$
- 6.3.5 Sample of simulation:
-> check MM_Inventory.ipynb
- It can be seen that the inventory volume is smaller in the order placement method.
- 6.4.1 What is queueing:
- Queueing are the phenomenon of customers waiting at the ATMs of banks, ticket sales windows, supermarket registers, etc.
- 6.4.2 Kendall's notation:
- If the customer arrival process is M, the service time is M, and the number of services (number of windows) is C, the waiting process model is expressed as M|M|C. -> Today's simulation will all based on M|M|1
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6.4.3 Poisson Arrival Process and Arrival Time Distribution:
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The following equation shows that the number of customers
$k$ arriving in one unit of time follows a Poisson distribution.$$P(k) = e^{-\lambda} \frac{\lambda^k}{k!}$$ -
$\lambda$ :Average number of passengers per hour$1/\lambda*$ : Average time interval between arrivals
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In case of arrival of a poisson, the interval of arrival time of the poisson follows an exponential distribution. The probability that the next customer will arrive within T hours after one customer's arrival is:
$$P_{\text{arr}}(T) = 1 - P^c(T) = 1 - e^{-\lambda T}$$
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6.4.3 Poisson Arrival Process and Arrival Time Distribution:
->Check MM_Poisson.ipynb
The larger $\lambda$ is considered shorter for the time distribution until the next guest arrives.
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6.4.4 Exponential service time distribution:
- If the service time T per person follows an exponential distribution, the probability that the service will be completed in T hours or less is expressed as:
$$P_{\text{srv}}(T) = 1 - e^{-\mu T}$$ -
$\mu$ :Average number of persons served(per unit time)$1/\mu*$ : Average hours of service(per person)
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- Highly variable service times are undesirable because they lead to longer wait times and inconsistent customer experiences.
- If the service time T per person follows an exponential distribution, the probability that the service will be completed in T hours or less is expressed as:
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6.4.5 Evaluation indicators:
- Utilization rate:
$\rho = \lambda/\mu$ - Average number of persons in the system:
$L = \lambda/\mu(1-\rho)$ - Average number of persons in the queue:
$L_q = L-\rho$ - Average waiting time in the system:
$W = L/\lambda$ - Average waiting time in the queue:
$W_q = L_q/\lambda$ - Little's law: the relationship between the number of customers and waiting time $$ L = \lambda W$$ $$ L_q = \lambda W_q$$
- Utilization rate:
- 6.4.6 Use SimPy to do the simulation:
- SimPy is a discrete-event simulation that can be run on Python, and is a type of event-driven programming.
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- 6.4.6 Use SimPy to do M|M|1 simulation:
-> Check MM_MM1.ipynb
- The vertical axis shows the number of waiting customers, and the range on the horizontal axis in the left figure is simulation time. We can see that a maximum of 16 people are waiting in line (left graph). We can use it to studing about what effect does the maximum waiting line have on the level of customer satisfaction? How many more windows will you need to achieve a good balance between cost, customer satisfaction and brand?...
- 6.4.6 Use SimPy to do M|M|1 simulation:
-> Check MM_MM1.ipynb
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