Chapter 2 Linear programming... 2 Chapter 3 Simplex... 4 Chapter 4 Sensitivity Analysis and duality... 5 Chapter 5 Network... 8 Chapter 6 Integer

Transcription

1 目录 Chapter 2 Linear programming... 2 Chapter 3 Simplex... 4 Chapter 4 Sensitivity Analysis and duality... 5 Chapter 5 Network... 8 Chapter 6 Integer Programming Chapter 7 Nonlinear Programming Chapter 8 Decision making under uncertainty Chapter 9 Game theory Chapter 10 Markov chains Chapter 11 Deterministic dynamic programming Expanded Projects

2 Chapter 2 Linear programming 1. A firm manufactures chicken feed by mixing three different ingredients. Each ingredient contains three key nutrients protein, fat and vitamin. The amount of each nutrient contained in 1 kilogram of the three basic ingredients is summarized in the following table: Ingredient Protein (grams) Fat (grams) Vitamin (units) The costs per kilogram of Ingredients 1, 2, and 3 are $0.55, $0.42 and $0.38, respectively. Each kilogram of the feed must contain at least 35 grams of protein, a minimum of 8 grams of fat and a maximum of 10 grams of fat and at least 200 units of vitamin s. Formulate a linear programming model for finding the feed mix that has the minimum cost per kilogram. 2. For a supermarket, the following clerks are required: Days Min. number of clerks Mon 20 Tue 16 Wed 13 Thu 16 Fri 19 Sat 14 Sun 12 Each clerk works 5 consecutive days per week and may start working on Monday, Wednesday or Friday. The objective is to find the smallest number of clerks required to comply with the above requirements. Formulate the problem as a linear programming model. 3. Consider the following LP problem: 2

3 Max Z 6x 8x Subject to x 4x 16 3x 4x 24 3x 4x 12 x, x 0 (a) Sketch the feasible region. (b) Find two alternative optimal extreme (corner) points. (c) Find an infinite set of optimal solutions. 4. A power plant has three thermal generators. The generators generation costs are $36/MW, $30/MW, and $25/MW, respectively. The output limitation for the generators is shown in the table. Some moment, the power demand for this plant is 360MW, please set up an LP optimization model and find out the optimal output for each generator (with lowest operation cost). Generator Generation lower limit Generation higher limit Use the Graphical Solution to find the optimal solutions to the following LP: max z 4x1 x2 s. t. 3x1 x2 6 x1 2x2 0 x, x 0 3

4 Chapter 3 Simplex 1. Show that if ties are broken in favor of lower-numbered rows, then cycling occurs when the simplex method is used to solve the following LP: Max Z 3x x 6x 3 Subject to 9x x 9x 2x x x / 3 2 x x / x x 9x 2x 1 i 3 4 x 0( i 1,2,3,4) 2. Use the simplex algorithm to find two optimal solutions to the following LP: max z 5x1 3x2 x3 s. t. x1 x2 3x3 6 5x1 3x2 6x3 15 x, x, x Use the Big M method to find the optimal solution to the following LP: max z 5x1 x2 s. t. 2x1 x2 6 x1 x2 4 x1 2x2 5 x, x 0 4. Use the simplex algorithm to find two optimal solutions to the following LP. max z 5x1 3x2 x3 s. t. x1 x2 3x3 6 5x1 3x2 6x3 15 x, x, x For a linear programming problem: Max Z 3x 4x Subject to 2x 4x 12 3x 2x 8 x i x 5 x 0( i 1, 2) Find the optimal solution using the simplex algorithm. 4

5 Chapter 4 Sensitivity Analysis and duality 1. Consider the following linear program (LP): Max Z x 3x 2 Subject to 2x x 4 x i 2 x 0( i 1, 2) (a). Determine the shadow price for b2, the right-hand side of the constraint x2 b2. (b). Determine the allowable range to stay optimal for c1, the coefficient of x1 in the objective function Z = c1x1 + 3x2. (c). Determine the allowable range to stay feasible for b1, the right-hand side of the constraint 2x1 + x2 b1. 2. There is a LP model as following, Max Z 3x 4x Subject to x x 5 2 x 4x 12 3x 2x 8 x 0( i 1, 2) The optimal simplex tableau is i Z X1 X2 S1 S2 S /4 1/ / /2 1) Give the dual problem of the primal problem. 2) If C2 increases from 4 to 5, will the optimal solution change? Why? 3) If b2 changes from 12 to 15, will the optimal solution change? Why? 3. There is a LP model as following 5

6 Min Z 2x 3x 6x 3 Subject to x 2x x x x 3x 3 3 3x 2x 8 x j 0( j 1, 2) 1) give its dual problem. 2) Use the graphical solution to solve the dual problem. 4. You have a constraint that limits the amount of labor available to 40 hours per week. If your shadow price is $10/hour for the labor constraint, and the market price for the labor is $11/hour. Should you pay to obtain additional labor? 5. Consider the following LP model of a production plan of tables and chairs: Max 3T + 2C (profit) Subject to the constraints: 2T + C 100 (carpentry hrs) T + C 80 (painting hrs) T 40 T, C 0 (non-negativity) 1) Draw the feasible region. 2) Find the optimal solution. 3) Does the optimal solution change if the profit contribution for tables changed from $3 to $4 per table? 4) What if painting hours available changed from 80 to 100? 6. For a linear programming problem: Max Z c x c x 3x 4x Subject to x x x 4x 12 3x 2x 8 x 0( i 1, 2) i Suppose C2 rising from 4 to 5, if the optimal solution will change? Explain the reason. 7. For a linear programming problem: Max Z c x c x 3x 4x Subject to x x x 4x 12 b 3x 2x 8 x 0( i 1, 2) i Suppose b2 rising from 12 to 15, if the optimal solution will change? Explain the 6

7 reason. 8. For a linear programming problem: Max Z c x c x 3x 4x Subject to x x x 4x 12 b 3x 2x 8 x 0( i 1, 2) i Calculate the shadow price of all of the three constraints. 9. 1) Use the simplex algorithm to find the optimal solution to the model below (10 points) Max Z 5x 2x Subject to 3x x 12 x i x 5 x 0( i 1, 2) 2) For which objective function coefficient value ranges of x 1 and x 2 does the solution remain optimal? (10 points) 3) Find the dual of the model; (5 points) 4) Find the shadow prices of constraints. (5 points) 5) If x1 and x2 are all integers, using the branch-and-bound to solve it.( 15 points) 10. A factory is going to produce Products I, II and III by using raw materials A and B. The related data is shown below: Raw material product Raw material available I II III A (kg) B (kg) Profit ($) ) Please arrange production plan to make the profit maximization. (15) 2) Write the dual problem of the primal problem. (5) 3) If one more kg of raw material A is available, how much the total profit will be increased? (5) 4) If the profit of product II changes from 1 to 2,will the optimal solution change? (5) 7

8 Chapter 5 Network 1. Using the Dijkstra s Algorithm and dynamic programming method to find the shortest path of graph below (22 points). 2. During the next four months, a construction firm must complete three projects. Project 1 must be completed within three months and requires 8 months of labor. Project 2 must be completed within four months and requires 10 months of labor. Project 3 must be completed at the end of two months and requires 12 months of labor. Each month, 8 workers are available. During a given month, no more than 6 workers can work on a single job. Formulate a maximum-flow problem that could be used to determine whether all three projects can be completed on time. 3.. For the networks in the following Figure, find the maximum flow from source to sink. 4. Find the shortest path from node 1 to node 5 in the figure using network model (Dijkstra s Algorithm) 8

9 5. Use Dijkstra s algorithm to find the shortest path and its length from node 1 to 6: 6. There are several available routes between city 1 and city 6, as shown in the following figure. The number associated with each arc is the travelling cost of the arc. 1) Formulate as an MCNFP the problem of finding the minimum cost route from city 1 to city 6. 2) Find the minimum spanning tree Write the main steps of Dijkstra algorithm to find the shortest path from the start point to the terminal point. 9

10 Chapter 6 Integer Programming 1. Eastinghouse sells air conditioners. The annual demand for air conditioners in each region of the country is as follows: East, 100,000; South, 150,000; Midwest, 110,000; West, 90,000. Eastinghouse is considering building the air conditioners in four different cities: New York, Atlanta, Chicago, and Los Angeles. The cost of producing an air conditioner in a city and shipping it to a region of the country is given in Table 18. Any factory can produce as many as 150,000 air conditioners per year. The annual fixed cost of operating a factory in each city is given in the following Table. At least 50,000 units of the Midwest demand for air conditioners must come from New York, or at least 50,000 units of the Midwest demand must come from Atlanta. Formulate an integer programming whose solution will tell Eastinghouse how to minimize the annual cost of meeting demand for air conditioners. 2. Use the branch-and-bound method to find the optimal solution to the following IP: Max Z 7x 3x S. t. 2x x 9 3x 2x 13 x, x 0; x, x int eger 3. At a machine tool plant, five jobs must be completed each day. The time it takes to do each job depends on the machine used to do the job. If a machine is used at all, there is a setup time required. The relevant times are given in the following Table. The company s goal is to minimize the sum of the setup and machine operation times needed to complete all jobs. Formulate an IP. 10

11 4. Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of $16000; investment 2, an NPV of $22000; investment 3, an NPV of $12000; and investment 4, an NPV of $8000. Each investment requires a certain cash outflow at the present time: investment 1, $5000; investment 2, $7000; investment 3, $4000; and investment 4, $3000. Currently, $14000 is available for investment. Formulate an IP whose solution will tell Stockco how to maximize the PV obtained from investments One person has a backpack which can contain things with weight of 10kg and volume of 0.025m 3. there are two type of books he wants to put in the backpack, the value, weight and volume of each kind of book is shown in table, what is the number of each kind of book should be put in to maximize the value of books in the backpack. Find the optimal solution using Branch and Bound method. book Weight(kg/book) Volume Value (m 3 /book) (yuan/book) Book Book For a Integer programming problem: Max z c x c x 3x 4x S. t. x x x 4x 12 b 2 3x 2x 8 x, x 0 x, x is int eger find the optimal solution using Branch and Bound method. (The optimal solutions of all sub-problems can be found by graphical method since using the simplex algorithm is time-consuming. ) 11

12 Chapter 7 Nonlinear Programming 1. Please ascertain the convexity or concavity ,, 2 f x x x x x x x x x x x x

13 Chapter 8 Decision making under uncertainty 1. Suppose my utility function for asset position x is given by u(x) =x 2. I want to buy a painting from a market and then sell it to my client at $ 500. I have several choices:1) buy it at the first day for $ 400; 2) buy it at the second day for $ 300 (if it has not been sold) ;3) buy it at the third day for $ 260 (if it has not been sold). Each day, there is 0.6 probability that it will be sold and at the end of the third day, the paint will definitely be sold out. (18 points) a. Am I risk-averse, risk-neutral, or risk-seeking? b. What strategy maximizes my expected profit? c. What will happen if I am risk-neutral? 2. Pizza King and Noble Greek are two competing restaurants. Each must determine simultaneously whether toundertake small, medium, or large advertising campaigns.pizza King believes that it is equally likely that Noble Greek will undertake a small, a medium, or a large advertising campaign. Given the actions chosen by each restaurant,pizza King s profits are as shown in the table. For themaximin, maximax, and minimax regret criteria, determinepizza King s choice of advertising campaign. Noble Greek Choose Pizza King Chooses Small Medium Large Small $6000 $5000 $2000 Medium $5000 $6000 $1000 Large $9000 $6000 $0 3. Because of the increasing demand, of a product, a company is trying to determine which of two machines to purchase. The market research showed there is a 65% chance that the demand will increase next year, the profit with two kind of purchases is shown in the table, if they are risk-neutral, to get the maximum profit next year, use the decision tree to determine what the decision department should do? Demand increase Demand unchanged Demand Reward Purchase choice probability Purchase Machine Purchase Machine

14 4. Why the conception of utility is introduced to the decision-making analysis? What is the definition of utility? 5. You are given a choice between lottery 1 and lottery 2. You are also given a choice between lottery 3 and lottery 4. Lottery 1: A sure gain of $240 Lottery 2: 25% chance to gain $1,000 and 75% chance to gain nothing Lottery 3: A sure loss of $750 Lottery 4: A 75% chance to lose $1,000 and a 25% chance of losing nothing 84% of all people prefer lottery 1 over lottery 2, and 87% choose lottery 4 over lottery 3. 1) Explain why the choice of lottery 1 over lottery 2 and lottery 4 over lottery 3 contradicts expected utility maximization. 2) Can you explain this anomalous behavior? 6. One person s character and thinking pattern can be modeled and analyzed through observing her previous decisions. Thus her decisions can be predicts for the future choices. An urn contains 90 balls. It is known that 30 are red and that each of the other 60 is either yellow or black (probability unknown). One ball will be drawn at random from the urn. Consider the following four options: Option 1: We receive $1,000 if a red ball is drawn. Option 2: We receive $1,000 if a yellow ball is drawn. Option 3: We receive $1,000 if a yellow or black ball is drawn. Option 4: We receive $1,000 if a red or black ball is drawn. a) If a logical decision maker prefers option 1 to option 2, which option would he prefer between option 3 and option 4? b) Explain why most people prefer option 1 over option 2 and at the same time, prefer option 3 over option 4? 7. The Golden Dragon Company is developing a new bus. If Golden Dragon Company markets the product and it is successful, the company will earn a 50 million profit; if it is unsuccessful, the company will lose 35 million. In the past, similar products have been successful 60% of the time. At a cost of 5 million, the effectiveness of the new bus can be tested. If the test result is favorable, there is an 80% chance that the bus will be successful. If the test result is unfavorable, there is only a 30% chance that the bus will be successful. There is a 60% chance of a favorable test result and a 40% chance of an unfavorable test result. Determine Golden Dragon Company s optimal strategy. Also find EVSI and EVPI. 14

15 Chapter 9 Game theory 1. Solve the 0-sum game with payoff matrix below in mixed strategies (sketch the graphs). (15 points) Find the value and the optimal strategies for the two-person zero-sum game in the table Two companies try to advertise to get more customers, before the advertisement, company A has the 55% market, company B has 45%. They both have 3 advertisement strategies, company A has X1, X2 and X3, company B has Y1,Y2 and Y3, the possible choices for each company and the number of increased market of company A are shown in Table. 1) What kind of game is it? 2) What is the optimal strategy and the value to company A? Y1 Y2 Y3 X X X Two players (called Odd and Even) simultaneously choose the number of fingers (1 or 2) to put out. If the sum of the fingers put out by both players is odd, then Odd wins $1 from Even. If the sum of the fingers is even, then Even wins $1 from Odd. We consider the row player to be Odd and the column player to be Even. a) Determine whether this game has an equilibrium point. b) Allow each player to select a probability of playing each strategy: x 1 : probability that Odd puts out one finger; x 2 : probability that Odd puts out two fingers; y 1 : probability that Even puts out one finger; y 2 : probability that Even puts out two fingers; 15

16 x 1 x 0, 2 x 0, and 1 x2 1 y, 1 y 0, 2 y 0, and 1 y2 1, x Assume that Even knows the values of 1 x and 2, on a particular play of the game, y and Odd knows the values of 1 y and 2. Determine the optimal strategy for Odd and Even through a graphical solution? 5. Mo and Bo each have a quarter and a penny. Simultaneously, they each display a coin. If the coins match( both show the same side), then Mo wins both coins; if they don t match, then Bo wins both coins. Determine optimal strategies for this game. 6. Find the value and the optimal strategies for the two-person zero-sum game in Table:

17 Chapter 10 Markov chains 1. There are only three type of sugar, A, B and C. At the first purchase, a person has 20% chance to purchase A, 40% to purchase B. Given that a person last purchased A, there is a 80% chance that her next purchase will be A, 10% chance to B. Given that a person last purchased B, there is an 50% chance that her next purchase will be A, 10% chance to keep B. Given that a person last purchased C, there is an 50% chance that her next purchase will be A, 20% chance to keep C. 1) is this Stochastic Process Markov Chain? Why? 2) write the transition probability matrix. 3) What is the probability that a person will purchase A two purchases from now? 2. What is a Markov Chain? 3. A credit card company wishes to model its accounts as a Markov chain. Each account is in one of the four following states: (1) fully paid, (2) bad debt, (3) 1 to 30 days old, and (4) 31 to 60 days old. Of the accounts currently in the 1 to 30 day category, 22.5% are paid in full, 47.5% have debits and credits which cause the account to be in the same category next month, and on the other accounts nothing is paid. Of the accounts currently in the 31 to 60 day category, 30% are paid in full, enough is paid on 40% of the accounts to upgrade the status to 1 to 30 day old debt, 20% of the accounts remain in the same category next month, and on the rest nothing is paid. The company expects these patterns to continue into the future. a) Write the state transition matrix. b) There is currently $94,000 in 1 to 30 day debt, and $31,000 in 31 to 60 day debt. Showing all calculations, how much (in dollars) can be expected to end up as bad debt? 4. For any two successive days, the transition probability of rain to clear is 1/3, that X of clear to rain is 1/2, n denotes the weather state of day n(0 for clear and 1 for rain) a) Write the state transition matrix for X, 1 n n b) If December 3 is clear,what is the probability for Dec. 5 is clear and the probability for Dec. 7 is raining? 5. We Suppose that tomorrow s Jinan weather depends on the last two days of Jinan weather, as follows (1)If the last two days have been sunny, then 96% of the time, tomorrow will be sunny. (2)If yesterday was cloudy and today is sunny, then 72% of the time, tomorrow will be sunny. (3) If yesterday was sunny and 17

18 today is cloudy, then 61% of the time, tomorrow will be cloudy. (4) If the last two days have been cloudy, then 79% of the time, tomorrow will be cloudy. Use the information, model Jinan s weather as a Markov chain. 18

19 Chapter 11 Deterministic dynamic programming 1. Suggest you have a 9-oz cup and a 4-oz cup. If you want to get exactly 3 oz of milk. How can you accomplish the goal with Dynamic Programming method? 19

20 Expanded Projects Project 1. YM Unit Commitment Problem: Assumption: You are pursuing a position working as an operator in a big thermal power plant. The human resource department sent you a question as: Our company has three thermal generators whose capacities are 600MW, 300MW, and 200MW. When you are on duty, you are informed by the power grid operator that in the next time interval, the load demand will be 740 MW. You are expected to find a most efficient way (minimizing the operating cost) to meet the load demand. a) What information should be known before you can make a decision? b) There are several operating constraints for generators, please find as many of them as possible. And please tell us why this is a constraint. c) Can you set up a decision model for us? (Assuming that the information above is known, you can use some notations to form your model or you can just pick some reasonable numbers.) Project 2. YDY Economic Dispatch Problem A power system has 10 coal-fired generation units that operate together to satisfy the load demand. a) Build a LP model to describe the generation unit economic dispatch problem. b) If 8 units at most are needed to satisfy the demand, revise the LP model to describe the new problem of unit commitment. c) If we need to make an optimal operation plan for 96 time intervals, and the maximum power change between two time intervals is P for each unit, build the model. Project 3: CXD Tie Line Violation Correcting Problem When a power system operates to its angle stability boundary, active power flowing 20

21 through one certain transmission line of the system will approach to a value called transfer limit. The operator can monitor the power flow on some critical transmission lines and check whether it violates the transfer limit. Whenever there is a violation, the operator should take some measures to correct the situation. A practical measure is to identify the sending generators from the receiving ones, and then decrease output of sending generators. At the same time output of receiving generators should be increased to balance the decrease from the sending side. The control effect and cost is different for one generator to another. Assume that L is a critical transmission line with transfer limit of P Llim, for which there are N sending generators and M receiving ones. Currently, active power on L is P L0, larger than P Llim. What is the optimal correction decision that the operator should make? Sending side Receiving side P L0 Please note that the control effect is measured by linear sensitivity of unit change of output of a generator to the change of active power on L, s i = ΔP L /ΔP SGi (i=1,2,,n). Control cost for unit change of output of a generator is c i (i=1,2,,n+m). Solution: Min s.t. N N M i SGi i RGi i 1 i N 1 N i 1 c P c P s P P P i SGi L0 Llim N M N P P 0 RGi i N 1 i 1 SGi P P P P i 1,2,, N SGi min SGi SGi SGi max P P P P i N 1, N 2,, N M RGi min RGi RGi RGi max 21