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ECON2326 Operations Research课程简介

Operations Research utlizes mathematical modeling, techniques, and algorithms, to most appropriately allocate resources and meet goals. An alternative title for the course might be “Mathematical Theory of Management, Control and Decision Making” . In general the course is intended for students with interests in applied math, statistics, economics, engineering and science. A second semester of the course devoted to probabilistic techniques will be offered in the spring semester.

PREREQUISITES 

Linear programming is a widely-used optimization technique that involves linear objective functions and linear constraints. The simplex algorithm is a widely-used algorithm for solving linear programming problems, and it involves moving from one feasible solution to another in an iterative fashion until an optimal solution is found. Sensitivity analysis involves examining the effects of changes in the objective function coefficients and constraint values on the optimal solution.

Dual problems are closely related to linear programming problems and involve the optimization of a dual objective function subject to dual constraints. The dual problem is used to provide information about the original problem, including bounds on the optimal objective value and information about the shadow prices of the constraints.

Integer programming is a type of linear programming that involves additional constraints that require the variables to take on integer values. Network models involve the modeling of complex systems using networks, such as transportation or communication systems. Dynamic programming is a powerful optimization technique that involves breaking down a problem into smaller subproblems and solving each subproblem in a recursive manner.

Finally, the KKT conditions are a set of necessary conditions for an optimization problem to have an optimal solution. These conditions involve the first-order conditions for optimality and the complementary slackness conditions, and they are used to analyze optimization problems and to derive insights about their solutions.

ECON2326 Operations Research HELP（EXAM HELP， ONLINE TUTOR）

(a) To determine the optimal order quantity, we can use the economic order quantity (EOQ) formula:

$EOQ = \sqrt{\frac{2DS}{H}}$

where:

• D is the annual demand, which is 200 copies per month x 12 months = 2400 copies per year
• S is the fixed ordering cost, which is €35
• H is the holding cost per unit per year, which is 24% of the purchase cost per unit. Since the purchase cost is €15, the holding cost per unit per year is €15 x 24% = €3.60

Plugging in the values, we get:

$EOQ = \sqrt{\frac{2 \times 2400 \times 35}{3.60}} \approx 374.16$

Rounding up to the nearest whole number, the optimal order quantity is 375.

To calculate the minimum annual ordering and holding costs, we can use the following formulas:

$Annual ordering cost = \frac{DS}{EOQ} \times S$

$Annual holding cost = \frac{EOQ}{2} \times H$

Plugging in the values, we get:

$Annual ordering cost = \frac{2400 \times 35}{375} \approx 2240$

$Annual holding cost = \frac{375}{2} \times 3.60 \approx 675$

Therefore, the minimum annual ordering and holding costs are €2,240 and €675, respectively.

(b) If the book can only be ordered by multiples of 50, we need to find the order quantity that is closest to the EOQ and also a multiple of 50. The closest multiple of 50 to the EOQ of 374.16 is 350.

The annual ordering cost for an order quantity of 350 is:

$Annual ordering cost = \frac{2400 \times 35}{350} = 2400$

The annual holding cost can be calculated using the formula:

$Annual holding cost = \frac{350}{2} \times 3.60 = 630$

Therefore, the total annual ordering and holding costs for an order quantity of 350 are €2,400 + €630 = €3,030.

The deviation from the theoretical minimum cost is:

$\frac{3030 – 2915}{2915} \times 100% \approx 3.95%$

So the cost increase due to ordering in multiples of 50 is approximately 3.95%.

(c) If the lead time is half a month, we need to adjust the EOQ formula to include the additional demand during the lead time. The modified EOQ formula is:

$EOQ = \sqrt{\frac{2DS}{H} \times \frac{P}{P+L}}$

where:

• P is the number of periods in a year, which is 12
• L is the lead time in months, which is 0.5

Plugging in the values, we get:

$EOQ = \sqrt{\frac{2 \times 2400 \times 35}{3.60} \times \frac{12}{12+0.5}} \approx 382.99$

Rounding up to the nearest multiple of 50, the optimal order quantity is 400.

To determine when to place an order, we need to calculate the reorder point. The reorder point is the level of inventory at which a new order should be placed. It can be calculated using the following formula:

$Reorder point = D \times \frac{L+P}{P} = 2400 \times \frac{0.5+12}{12}

To determine the optimal order quantity, we can use the economic order quantity (EOQ) formula:

$EOQ = \sqrt{\frac{2DS}{H}}$

where:

• D is the annual demand, which is 3500 pens per year
• S is the fixed ordering cost, which is €25 per order
• H is the holding cost per unit per year, which is 18% of the purchase cost per unit.

We need to consider two different purchase costs: €7.00 if at least 350 pens are ordered, and €7.50 otherwise. Let’s first calculate the EOQ for the lower purchase cost of €7.00:

$EOQ_1 = \sqrt{\frac{2 \times 3500 \times 25}{0.18 \times 350}} \approx 279.72$

Rounding up to the nearest whole number, the optimal order quantity for the lower purchase cost is 280.

To check whether we should order at the lower purchase cost, we need to calculate the total cost of ordering 280 pens at €7.00:

$Total cost_1 = \frac{DS}{EOQ_1} \times S + \frac{EOQ_1}{2} \times H_1 + DP_1$

where:

• D is the annual demand, which is 3500 pens per year
• S is the fixed ordering cost, which is €25 per order
• EOQ_1 is the optimal order quantity for the lower purchase cost of €7.00, which is 280 pens
• H_1 is the holding cost per unit per year for the lower purchase cost, which is 18% of €7.00 = €1.26 per pen per year
• P_1 is the purchase cost per pen, which is €7.00
• D is the annual demand, which is 3500 pens per year

Plugging in the values, we get:

$Total cost_1 = \frac{3500 \times 25}{280} \times 25 + \frac{280}{2} \times 1.26 + 3500 \times 7 = 6325$

Now let’s calculate the EOQ for the higher purchase cost of €7.50:

$EOQ_2 = \sqrt{\frac{2 \times 3500 \times 25}{0.18 \times 350}} \approx 264.58$

Rounding up to the nearest whole number, the optimal order quantity for the higher purchase cost is 265.

To check whether we should order at the higher purchase cost, we need to calculate the total cost of ordering 265 pens at €7.50:

$Total cost_2 = \frac{DS}{EOQ_2} \times S + \frac{EOQ_2}{2} \times H_2 + DP_2$

where:

• D is the annual demand, which is 3500 pens per year
• S is the fixed ordering cost, which is €25 per order
• EOQ_2 is the optimal order quantity for the higher purchase cost of €7.50, which is 265 pens
• H_2 is the holding cost per unit per year for the higher purchase cost, which is 18% of €7.50 = €1.35 per pen per year
• P_2 is the purchase cost per pen, which is €7.50
• D is the annual demand, which is 3500 pens per year

Plugging in the values, we get:

$Total cost_2 = \frac{3500 \times 25}{265} \times

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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