In this problem we have n jobs j1, j2, ..., jn, each has an associated deadline d1, d2, ..., dn and profit p1, p2, ..., pn. Profit will only be awarded or earned if the job is completed before the deadline. We assume that each job takes 1 unit of time to complete. The objective is to earn maximum profit when only one job can be scheduled or processed at any given time. Provide the pseudocode of an algorithm to find the sequence of jobs to do with the maximum total profit. Also describe the main idea of your algorithm using plain language. [Hint: You can select the jobs in a greedy way. You can use the following example to help your analysi

In this problem we have n jobs j1, j2, ..., jn, each has an associated deadline d1, d2, ..., dn and profit p1, p2, ..., pn. Profit will only be awarded or earned if the job is completed before the deadline. We assume that each job takes 1 unit of time to complete. The objective is to earn maximum profit when only one job can be scheduled or processed at any given time. Provide the pseudocode of an algorithm to find the sequence of jobs to do with the maximum total profit. Also describe the main idea of your algorithm using plain language. [Hint: You can select the jobs in a greedy way. You can use the following example to help your analysi

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter4: The Simplex Algorithm And Goal Programming

Section4.10: Matrix Generators, Lingo, And Scaling Of Lps

Problem 4P

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6. Consider a modification to the rod-cutting problem in which, in addition to a value pi for each rod, there is handling cost ci that is one and a half times the length of the rod cut plus a flat fee of 3 (i.e., the handling cost to cut a rod of length 5 is 7.5+3 =10). The revenue generated is the sum of the value of the pieces cut minus the sum of handling costs of the cuts. Provide a dynamic programming approach to solve this problem given an initial rod of length n with potential cut lengths of 1, 2, 5, and 7 [Adapted from ITA, pg. 370, 15.1-3] length of cut (i) 1 2 5 7 value (pi) 5 7 30 45 handling cost (ci) 4.5 6 10.5 13.5
Consider the following Job Scheduling problem. We have one machine and a setof n jobs {1, 2, . . . , n} to run on this machine, one at a time. Each job has a start time sifinish time fi and profit pi where 0 ≤ si < fi < ∞ and pi > 0. Two jobs i and j are compatible if the intervals [si, fi)and [sj , fj ) do not overlap. The goal is to find a set A of mutually compatible jobs with the maximumtotal profit, i.e.,P j∈A pj is maximized.Consider the following two greedy choices. For each one, determine whether it is a “safe” greedy choicefor this Job Scheduling problem. If your answer is yes, prove the “Greedy-choice property”. If your answeris no, please give a counterexample and show that the greedy choice will not lead to an optimal solution. a.Greedy choice 1: Always select a job with the earliest finish time that is compatible with allpreviously selected activities.b.Greedy choice 2: Always select a job with the highest profit per time unit (i.e., pi/(fi − si))that is…
South Africans and Botswanaans enjoy Chakalaka and Beer. Suppose that these are the only two products they consume and produce with Chakalaka being relatively more labour intensive and Beer being relatively more capital intensive in the production process ( αb=0.8 and αc=0.2). Suppose that, as a result of the Southern African Customs Union, all impediments or barriers to trade are lifted, such that these countries can freely trade both goods without any costs. Assuming that an Entrepreneur knows the price of both these goods and both goods are produced, using the Learner diagram describe how the entrepreneur chooses the cost-minimising combinations of inputs in the production of both goods based on the neoclassical trade theory.
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One important factor in inventory control is measured using the variance of the daily demand for the product. A management scientist has developed the optimal order quantity, assuming that the variance is equal to 250. Recently, the company has experienced some inventory problems, which induced the operation manager to doubt the assumption. To examine the problem, the manager recorded the daily demands for 12 random days. The data is in Table 1. 188 212 176 213 182 203 185 214 213 209 199 229 By assuming the data of daily demands is normally distributed, can we conclude that the operation manager’s doubt is true? Use the confidence interval approach.
4It has been observed by management that some faculty member at the university demonstratesa lackadaisical attitude toward teaching. They seldom go to class yet at the end of each month they receive full salary. Management hasdecided that GHs200, 300, 400, and 500 will be deducted from a faculty’s salary if he/she offends once, twice, third time and forth time respectively in a month. This means that if a faculty member offends once in a month GHs200 will be deducted, if a faculty member offends twice in a month,GHs500 (i.e. 200+300) will be deducted, if a faculty member offends three times in a month GHs900(i.e. 200+300+400), will be deducted and if a faculty member offends four times in a month GHs1,400 (i.e. 200+300+400+500). Assuming all faculty members are ona flat salary rate of GHs 2000.a.Write a program to request for the names of four faculty members and number times he/she has absented him/herself from class. One of them should have absented him/herself once, another twice,…
It has been observed by management that some faculty members at theuniversity demonstrate a lackadaisical attitude toward teaching. They seldom go to class yet at the end of each month they receive full salary. Management has decided that GHs 200, 300, 400, and 500 will be deducted from a faculty’s salary if he/she offends once, twice, third time and forth time respectively in a month. This means that if a faculty member offends once in a month GHs200 will be deducted, if a faculty member offends twice in a month, GHs 500 (i.e. 200+300) will be deducted, if a faculty member offends three times in a month GHs 900 (i.e. 200+300+400), will be deducted and if a faculty member offends four times in a month GHs 1,400 (i.e. 200+300+400+500). Assuming all faculty members are on a flat salary rate of GHs 2000.a. Write a program to request for the names of four faculty members and number times he/she has absented him/herself from class in a month.One of them should have absented him/herself…
Recall Pigou’s example discussed in class, where there are two roads that connect a source, s, and destination, t. The roads have different travel costs. Fraction x1 of the traffic flow on route 1, and the remainder x2 on route 2. Here consider the following scenario. • The first road has “infinite” capacity but is slow and requires 1 hour travel time, T1 = 1. • The second road always requires at least 15 mins, which then increases as a function of traffic density, T2 = 0.25 + 0.75x2. If drivers act in a “selfish” manner – the user optimal scenario – all the traffic will flow on the second path, as one is never worse off. Worst case scenario for path 2, both paths take one hour. So no one is incentivized to change their behavior. 1. Assume user optimal behavior, and calculate τ the expected travel time per car. 2. If instead we could control the flows, we could minimize the expected travel time. Using the expression in part (a), calculate the optimal allocation of flows x¯1 and ¯x2…
It has been observed by management that some faculty member at the university demonstrates a lackadaisical attitude toward work. They seldom go to class yet at the end of each month they receive full salary. Management has decided that GHs 200, 300, 400, and 500 will be deducted from a faculty’s salary if he/she offends once, twice, third time and forth time respectively in a month. This means that if a faculty member offends once in a month GHs 200 will be deducted, if a faculty member offends twice in a month, GHs 500 (i.e. 200+300) will be deducted, if a faculty member offends three times in a month GHs 900 (i.e. 200+300+400), will be deducted and if a faculty member offends four times in a month GHs 1,400 (i.e. 200+300+400+500). Assuming all faculty members are on a flat salary rate of GHs 2500.a. Write a program to request for the names of four faculty members and number times he/she has absented him/herself from class. One of them should have absented him/herself once, another…
It has been observed by management that some faculty member at the university demonstrates a lackadaisical attitude toward work. They seldom go to class yet at the end of each month they receive full salary. Management has decided that GHs 200, 300, 400, and 500 will be deducted from a faculty’s salary if he/she offends once, twice, third time and forth time respectively in a month. This means that if a faculty member offends once in a month GHs 200 will be deducted, if a faculty member offends twice in a month, GHs 500 (i.e. 200+300) will be deducted, if a faculty member offends three times in a month GHs 900 (i.e. 200+300+400), will be deducted and if a faculty member offends four times in a month GHs 1,400 (i.e. 200+300+400+500). Assuming all faculty members are on a flat salary rate of GHs 2500.a. Write a program to request for the names of four faculty members and number times he/she has absented him/herself from class. One of them should have absented him/herself once, another…
2. A safe has 5 locks v, w, y and z; all of which must be unlocked for the safe to open. The keys to the locks are distributed among five executives in the following manner.Mr. A has keys for locks v and x. Mr. B has keys for locks v and y. Mr. C has keys for locks w and y. Mr. D has keys for locks x and z. Mr. E has keys for locks v and z.a) Determine the minimal number of executives required to open the safe.b) Find all the combinations of executives that can open the safe; write and expression f(A, B, C, D, E) which specifies when the safe can be opened as a function of what executives are present.c) Who is the essential executive?
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