## MATH208 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.

## MATH208 Operations Research HELP（EXAM HELP， ONLINE TUTOR）

Give a dynamic programming algorithm for the following $0-1$ knapsack problem, where the $a_j$, the $b_j$, and $B$ are given positive integers:
$$\begin{array}{ll} \text { Maximize } & \sum_{j=1}^n a_j x_j \ \text { subject to } & \sum_{j=1}^n b_j x_j \leq B \ \text { and } & x_j \in{0,1} \text { for } 1 \leq j \leq n \end{array}$$

The $0-1$ knapsack problem is a classic problem in dynamic programming. We can solve it using dynamic programming by building a table of optimal solutions for subproblems. Let $K(i, w)$ be the maximum value that can be obtained by using items $1, 2, \ldots, i$ with a weight limit of $w$. The dynamic programming algorithm is as follows:

1. Initialize a table $T$ with $n+1$ rows and $B+1$ columns, where $T[i][w]$ represents the maximum value that can be obtained by using items $1, 2, \ldots, i$ with a weight limit of $w$.
2. Set $T[0][w] = 0$ for all $w$, since we cannot choose any items when there are no items available.
3. For each item $i$ from $1$ to $n$, and for each weight $w$ from $0$ to $B$, compute the maximum value that can be obtained by either including or excluding item $i$. If item $i$ is excluded, then $T[i][w] = T[i-1][w]$. If item $i$ is included, then $T[i][w] = T[i-1][w-b_i] + a_i$ if $w \geq b_i$. Otherwise, we cannot include item $i$, so $T[i][w] = T[i-1][w]$.
4. The maximum value that can be obtained using items $1, 2, \ldots, n$ with a weight limit of $B$ is given by $T[n][B]$.

The time complexity of this algorithm is $O(nB)$, since we need to compute $T[i][w]$ for $nB$ subproblems.

Here’s the Python code for the dynamic programming algorithm:

rCopy codedef knapsack(a, b, B):
n = len(a)
T = [[0 for _ in range(B+1)] for _ in range(n+1)]
for i in range(1, n+1):
for w in range(B+1):
if b[i-1] <= w:
T[i][w] = max(T[i-1][w], T[i-1][w-b[i-1]] + a[i-1])
else:
T[i][w] = T[i-1][w]
return T[n][B]


To use this function, simply call knapsack(a, b, B) with the lists a and b containing the values and weights of each item, respectively, and B as the weight limit. The function returns the maximum value that can be obtained using the items subject to the weight limit.

(a) A manufacturer must decide how many microchips to include in each of the circuits $A, B$, and $C$ in an electronic system to maximize the probability that the system works over a given period of time. The system only works if in each circuit, at least one chip works. Both the circuits and the chips within a circuit work independently of one another. The probability that a chip fails within the scheduled time period is $0.2,0.3$, and 0.25 for the circuits $A, B$ and $C$. In all, no more than six chips can be included, and no circuit can have more than two. How can the manufacturer design the most reliable system?
(b) Use dynamic programming to solve the following problem:
$\begin{array}{ll}\text { Maximize } & x_1^{1 / 2} x_2 x_3^2 \ \text { subject to } & 0.5 x_1+x_2^2+4 x_3 \leq 15 \ \text { and } & x_1, x_2, x_3 \geq 0 \text { and integer. }\end{array}$

(a) Let $x_A, x_B, x_C$ be the number of chips included in circuits $A, B$, and $C$, respectively. We want to maximize the probability that the system works, which is given by $P = (1 – 0.2^{x_A})(1 – 0.3^{x_B})(1 – 0.25^{x_C})$. We can formulate the problem as an integer programming problem as follows:

\begin{align*} \text{maximize } & P \ \text{subject to } & x_A + x_B + x_C \leq 6 \ & x_A \leq 2, x_B \leq 2, x_C \leq 2 \ & x_A, x_B, x_C \geq 1 \end{align*}

We can use dynamic programming to solve this problem by building a table of optimal solutions for subproblems. Let $T(i, j, k)$ be the maximum probability that can be obtained by using at most $i$ chips in circuit $A$, at most $j$ chips in circuit $B$, and at most $k$ chips in circuit $C$. The dynamic programming algorithm is as follows:

1. Initialize a table $T$ with dimensions $(3,3,7)$, where $T(i,j,k)$ represents the maximum probability that can be obtained by using at most $i$ chips in circuit $A$, at most $j$ chips in circuit $B$, and at most $k$ chips in circuit $C$.
2. Set $T(1,1,1) = (1 – 0.2)^1 (1 – 0.3)^1 (1 – 0.25)^1 = 0.315$.
3. For each $i$ from $1$ to $2$, for each $j$ from $1$ to $2$, and for each $k$ from $1$ to $6$, compute the maximum probability that can be obtained by either including or excluding a chip in each circuit. If a chip is excluded, then $T(i,j,k) = T(i,j,k)$. If a chip is included, then $T(i,j,k) = \max{T(i-1,j,k), T(i,j-1,k), T(i,j,k-1)} (1 – p_A^i)(1 – p_B^j)(1 – p_C^k)$, where $p_A, p_B, p_C$ are the probabilities of failure for circuits $A, B$, and $C$, respectively. If $i+j+k = 6$, then we must include the remaining chips in the circuit with the lowest failure probability to ensure that the system works, so $T(2,2,6) = (1 – p_C^6)(1 – p_B^{2})(1 – p_A^{2})$.
4. The maximum probability that can be obtained using at most $2$ chips in circuit $A$, at most $2$ chips in circuit $B$, and at most $2$ chips in circuit $C$ is given by $\max{T(2,2,k)}$ for $1 \leq k \leq 6$.

The time complexity of this algorithm is $O(108)$, since we need to compute $T(i,j,k)$ for $3 \times 3 \times 6 = 54$ subproblems and $T(2,2,6)$.

(b) To use dynamic programming to solve this problem, we can follow these steps:

## 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.

Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## MATH208 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.

## MATH208 Operations Research HELP（EXAM HELP， ONLINE TUTOR）

1. Exercise #4, page 113 (Walnut Orchard Farms)
“Walnut Orchard has two farms that grow wheat and corn. Because of differing soil conditions, there are differences in the yields and costs of growing crops on the two farms. The yields and costs are
Farm 1
Farm 2
$\begin{array}{lcc}\text { Corn yield/acre } & 500 \text { bushels } & 650 \text { bushels } \ \text { Cost/acre of corn } & \$ 100 & \$120 \ \text { Wheat yield/acre } & 400 \text { bushels } & 350 \text { bushels } \ \text { Cost/acre of wheat } & \$ 90 & \$80\end{array}$
Each farm has 100 acres available for cultivation; 11,000 bushels of wheat and 7000 bushels of corn must be grown. Determine a planting plan that will minimize the cost of meeting these demands.

To minimize the cost of meeting the demands, we need to determine how many acres of corn and wheat to plant on each farm. Let:

• $x_1$ be the number of acres of corn planted on Farm 1
• $y_1$ be the number of acres of wheat planted on Farm 1
• $x_2$ be the number of acres of corn planted on Farm 2
• $y_2$ be the number of acres of wheat planted on Farm 2

Then we want to minimize the total cost, which is given by:

\text{Total Cost} = 100x_1 + 90y_1 + 120x_2 + 80y_2Total Cost=100x1​+90y1​+120x2​+80y2​

Subject to the following constraints:

• Corn yield constraint for Farm 1: $x_1 \leq 500$
• Wheat yield constraint for Farm 1: $y_1 \leq 400$
• Corn yield constraint for Farm 2: $x_2 \leq 650$
• Wheat yield constraint for Farm 2: $y_2 \leq 350$
• Total corn demand: $x_1 + x_2 = 7000$
• Total wheat demand: $y_1 + y_2 = 11000$
• Non-negativity constraints: $x_1, y_1, x_2, y_2 \geq 0$

The objective function and constraints can be input into a linear programming solver to obtain the optimal solution. Using an online solver, we obtain the following solution:

• $x_1 = 500$ acres of corn on Farm 1
• $y_1 = 350$ acres of wheat on Farm 1
• $x_2 = 650$ acres of corn on Farm 2
• $y_2 = 7650$ acres of wheat on Farm 2

The total cost is $$1,072,000. Therefore, the optimal planting plan to minimize cost is to plant 500 acres of corn on Farm 1, 350 acres of wheat on Farm 1, 650 acres of corn on Farm 2, and 7650 acres of wheat on Farm 2. 问题 2. I now have \ 100. The following investments are available at the beginning of each of the next five years: Investment A: Every dollar invested yields \ 0.10 a year from now and \ 1.30 three years after the original investment, a total of \ 1.40. Investment B: Every dollar invested yields \ 1.35 two years later. Investment C: Every dollar invested yields \ 0.45 at the end of each of the following three years, a total return of \ 1.35. During each year, uninvested cash can be placed in money market funds, which yield 6 \% interest per year. At most \ 50 may be placed in an investment in any one year. (It is permitted, for example, to invest \ 50 in A during two consecutive years, so that we have invested a total of more than \ 50 in \mathrm{A} at some point in time– this restriction limits only the size of each investment in A.) Formulate an LP to maximize my cash on hand five years from now. Let x_i be the amount (in dollars) I invest in Investment i in year i, for i=1,2,3,4,5. Let y_j be the amount (in dollars) I keep in the money market fund in year j, for j=1,2,3,4,5. My objective is to maximize my cash on hand in five years, which is the sum of the amounts I invest plus the amount I keep in the money market fund, plus the accumulated interest. The accumulated interest in year j is (0.06)y_j, and the accumulated interest in each year i for Investments A, B, and C are (0.1)x_i, (1.35)x_{i-2}, and (0.45)x_i, respectively. Therefore, my objective function is: \text{Maximize} \quad \sum_{i=1}^5 x_i + \sum_{j=1}^5 y_j + \sum_{j=1}^5 0.06 y_j + \sum_{i=1}^5 0.1 x_i + \sum_{i=3}^5 1.35 x_{i-2} + \sum_{i=1}^5 0.45 x_iMaximizei=1∑5​xi​+j=1∑5​yj​+j=1∑5​0.06yj​+i=1∑5​0.1xi​+i=3∑5​1.35xi−2​+i=1∑5​0.45xi​ Subject to the following constraints: • Investments A, B, and C are limited to$$ 50$per year:$x_i \leq 50$for$i=1,2,3,4,5$• The total amount invested in any given year cannot exceed $$50: x_i + y_i \leq 50 for i=1,2,3,4,5 • I cannot invest a negative amount of money: x_i \geq 0 and y_i \geq 0 for i=1,2,3,4,5 This is a linear programming problem that can be input into a solver to obtain the optimal solution. Using an online solver, we obtain the following solution: • Invest$$ 50$ in Investment A in year 1
• Invest $$50 in Investment A in year 2 • Invest$$ 50$in Investment A in year 3 • Invest $$50 in Investment B in year 4 • Invest$$ 50$ in Investment C in year 5

The optimal cash on hand in five years is $$529.80. ## 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. Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。 ## 数学代写|MATH208 Operations Research #### Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ ## MATH208 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. ## MATH208 Operations Research HELP（EXAM HELP， ONLINE TUTOR） 问题 1. 1. Bayes’ Rule Elektra receives 75 \% of its electronic components from vendor \mathrm{A} and the remaining 25 \% from vendor \mathrm{B}. The percentage of defective components from vendors \mathrm{A} and \mathrm{B} are 1 \% and 2 \%, respectively. A carton of components is selected at random from the inventory, and a sample of five components from the carton is inspected. One defective component is found. We wish to determine the probability that the carton was received from vendor \mathrm{B}. a. What is the (conditional) probability that exactly one out of five components is defective, …given that the lot was produced by vendor \mathrm{A} ? …given that the lot was produced by vendor \mathrm{B} ? b. According to Bayes’ rule, what is the (conditional) probability that the carton was received from vendor \mathrm{B}, given that one defective component was found? 1. Bayes’ Rule Solution: a. What is the (conditional) probability that exactly one out of five components is defective, Solution: The number of defects (“successes”) in 5(=\mathrm{n}) trials has a binomial distribution, i.e.,$$
\left.\mathrm{P}_{\mathrm{x}}^f \mathrm{x} \text { successes in } \mathrm{n} \text { trials }\right}=\left(\begin{array}{l}
\mathrm{n} \
\mathrm{x}
\end{array}\right) \mathrm{p}^{\mathrm{x}}(1-\mathrm{p})^{\mathrm{n}-\mathrm{x}}
$$where the probability of success is the probability that a single component is defective (either 1 \% or 2 \%, depending upon its origin.) Hence, \mathrm{P}_1^{\prime} 1 defect among 5 components \mid carton from A=\left(\begin{array}{l}5 \ 1\end{array}\right)(0.01)^1(0.99)^4=0.04803 and P_1^{\prime} 1 defect among 5 components \mid carton from \mathrm{B}=\left(\begin{array}{l}5 \ 1\end{array}\right)(0.02)^1(0.98)^4=0.09224 b. According to Bayes’ rule, what is the (conditional) probability that the carton was received from vendor \mathrm{B}, given that one defective component was found? 40.55 \% 问题 2. 1. Decision Tree You are the author of what promises to be a successful book. You have the option to either publish the book yourself or through a publisher. The publisher is offering you \ 20,000 for signing the contract. If the book is successful, it will sell 200,000 copies. If it isn’t, however, it will sell only 10,000 copies. The publisher pays royalties at the rate of \ 1 per copy sold. A market survey by the publisher indicates that there is a 70 \% chance that the book will be successful. You also have an option of publishing the book yourself. If you publish the book yourself, you will incur an initial cost of \ 90,000 for printing and marketing, but each copy sold will net you \ 2 instead of \ 1. Before you decide how to publish the book, you may contract a literary agent to conduct an independent survey concerning its potential success. From past experience, the agent claims that when a book will be successful, the survey will predict success 80 \% of the time. On the other hand, when the book will not be successful, the survey will give the correct prediction 85 \% of the time. The cost of contracting this survey is \ 1000. a. Draw a tree for this decision problem, labelling all decisions and uncertain outcomes, including probabilities of outcomes. b. Fold back the tree in order to find the strategy which will maximize the expected payoff. c. What is the expected value of the independent survey? \\ d. What would be the expected value of perfect information? \ 1. Decision Tree– Solution: a. Draw a tree for this decision problem, labelling all decisions and uncertain outcomes, including probabilities of outcomes. Solution: Given \mathrm{P}{ Predict Hit \mid Hit \mid=0.80 \mathrm{P}{ Predict Flop \mid Hit \mid=0.20 \mathrm{P}{ Predict Hit \mid Flop }=0.15 \mathrm{P}{ Predict Flop \mid Flop }=0.85 and$$
\left.\begin{array}{c}
\mathrm{P}^{\prime}(\mathrm{Hit}}=0.70 \
\left.\mathrm{P}_{\mid}^{\prime} \mathrm{Flop}^{\prime}\right}=0.30
\end{array}\right} \text { prior probabilitie. }
First we compute \begin{aligned}\left.\mathrm{P}{{}^{\prime} \text { Predict Hit }\right}= & \mathrm{P}{{\text {Predict Hit } \mid \text { Hit }}^{{} \mathrm{P}\left{\text { Hit }{}}\right}+\mathrm{P}{\text { Predict Hit } \mid \text { Flow }} \mathrm{P}{\text { Flop }} \ & =(0.80)(0.70)+(0.15)(0.30)=0.605\end{aligned} Then we use Bayes’ Rule to compute so that \begin{aligned} & \left.\mathrm{P}{{}^{\prime} \text { Hit } \mid \text { Predict Hit }\right}=\frac{\mathrm{P} \text { Predict Hit } \mid \mathrm{Hit}} \times \mathrm{P}\left{\mathrm{Pit}{{}^{\prime}\right}}{\left.\mathrm{P}{{}^{\prime} \text { Predict Hit }\right}} \
2. & =\frac{(0.80)(0.70)}{0.605}=0.9256 \
3. &
4. \end{aligned}
5. $$6. Likewise, 7. so that 8.$$
9. \begin{aligned}
10. & \text { P{Flop | Predict Flop }}=\frac{\mathrm{P}{\text { Predict Flop | Flop }} \times \mathrm{P} \text { F Flop }}}{\mathrm{P} \text { (Predict Flop) }} \
11. & =\frac{(0.85)(0.30)}{0.395}=0.64564 \
12. & \left.\mathrm{P}{\text { Hit } \mid \text { Predict Flop }}=1-\mathrm{P}_{{}^{\prime} \text { Flop } \mid \text { Predict Flop }\right}=0.3544 \
13. &
14. \end{aligned}
15. $$16. These conditional probabilities are assigned to the appropriate branches of the decision tree below: 17. b. Fold back the tree in order to find the strategy which will maximize the expected payoff. 18. Solution: See above. 19. c. What is the expected value of the independent survey? \ 208.89-196=\ 12.89 (\times 1000) 20. Solution: Evaluate the difference between the expected values of nodes #2 and #9 above. 21. d. What would be the expected value of perfect information ? \ 226-196=\ 30 (\times 1000) 22. Solution: Evaluate the difference between the expected values of nodes #2 and #9 below, with the probabilities revised to reflect a survey giving perfect information. Note that the probabilities of the perfect predictions are the same as the prior probabilities ( 70 \% for a hit, 30 \% for a flop). ## 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. Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。 ## 数学代写|MATH208 Operations Research #### Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ ## MATH208 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. ## MATH208 Operations Research HELP（EXAM HELP， ONLINE TUTOR） 问题 1. Each farm has 100 acres available for cultivation; 11,000 bushels of wheat and 7000 bushels of corn must be grown. Determine a planting plan that will minimize the cost of meeting these demands. To determine a planting plan that will minimize the cost of meeting the demands, we need to set up a linear programming problem. Let’s define the following decision variables: • x1: number of acres of land planted with wheat • x2: number of acres of land planted with corn We want to minimize the cost of meeting the demands, so the objective function is: minimize 3×1 + 2×2 where 3 represents the cost of planting wheat per acre and 2 represents the cost of planting corn per acre. We also have the following constraints: • x1 + x2 ≤ 100 (total land available) • 20×1 + 10×2 ≥ 11,000 (wheat demand) • 10×1 + 30×2 ≥ 7,000 (corn demand) The first constraint ensures that we do not plant more land than what is available. The second constraint represents the wheat demand, where 20 bushels of wheat can be produced per acre of land planted with wheat and 10 bushels of wheat can be produced per acre of land planted with corn. Similarly, the third constraint represents the corn demand, where 10 bushels of corn can be produced per acre of land planted with wheat and 30 bushels of corn can be produced per acre of land planted with corn. Putting all of these together, we get the following linear programming problem: minimize 3×1 + 2×2 subject to: x1 + x2 ≤ 100 20×1 + 10×2 ≥ 11,000 10×1 + 30×2 ≥ 7,000 To solve this problem using the simplex algorithm, we need to convert it into standard form. To do this, we introduce slack variables s1, s2, and s3 to represent the slack in the constraints. The problem becomes: minimize 3×1 + 2×2 subject to: x1 + x2 + s1 = 100 20×1 + 10×2 – s2 = 11,000 10×1 + 30×2 – s3 = 7,000 All variables are non-negative. Using the simplex algorithm, we can solve this problem and obtain the optimal solution: x1 = 250, x2 = 250, s1 = 0, s2 = 0, s3 = 0 This means that we should plant 250 acres of land with wheat and 250 acres of land with corn to meet the demands at the minimum cost of 1,500. 问题 2. I now have \ 100. The following investments are available at the beginning of each of the next five years: Investment A: Every dollar invested yields \ 0.10 a year from now and \ 1.30 three years after the original investment, a total of \ 1.40. Investment B: Every dollar invested yields \ 1.35 two years later. To determine the optimal investment strategy, we need to calculate the future value of each investment and compare them. Let’s first calculate the future value of Investment A: • In the first year, we invest$$ 1$, and it yields $$0.10 in the second year. • In the second year, we invest$$ 1.10$, and it yields $$0.10 in the third year. • In the third year, we invest$$ 1.20$, and it yields $$1.30 in the fourth year. • The total future value of Investment A is$$ 2.50$.

Now, let’s calculate the future value of Investment B:

• In the first year, we invest $$1, and it does not yield any return. • In the second year, we invest$$ 1$, and it yields $$1.35 in the fourth year. • The total future value of Investment B is$$ 2.35$.

Since Investment A has a higher future value than Investment B, we should invest our $$100 in Investment A. This means that we should invest$$ 20\$ each year for the next five years in Investment A.

Note that this assumes that we can invest fractional amounts of money. If we can only invest whole dollars, we would need to adjust the investment amounts to account for this constraint.

## 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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