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Shadow PricesShadow Pricesthe marginal value of a one- unit increase in the right-hand side of the corresponding constraintdual valuesShadow Prices are also referred to asoptimal objective value is the shadow price.if the right-hand side of a constraint is increased by one unit, then the increase in theeach constraint in your optimization problemThere is one shadow price forThe shadow price has economic implicationsnet loss on that investment.if the cost of increasing capacity by one unit is larger than the shadow price, then you will experience aWalk through examplesIf i increase the capacity of this step by one unit, the $4.375 represents the increase in profit you would receive from one more hour of cutting and dyeingThe shadow price for the cutting and dyeing constraint is 4.375. What does this mean?$4.375If you increase the time available in cutting and dyeing by one hour, how much will your profitThese constraints are not binding, i.e. they're not bottlenecks. Increasing capacity in these areas will not increase overall capacityThe shadow prices for both sewing/inspection and packaging are zero, what does this mean?Only increasing capacity at our bottlenecks will increase our optimal solutionfinishing has a higher shadow price and therefore is more profitable
-finishing is the more critical of the two bottlenecks, this is where you should put your money first in increasing capacityCutting dyeing shadow price: $4.375
Finishing shadow price: $6.9375
which would be more profitable?Example 2:For each unit increase in risk tolerance, the expected annual return will increase by .015 thousands of dollarsThe shadow price of risk tolerance LHS is .015, What does an investor have to gain if they are willing to increase their risk tolerance?nothing, not a bottleneck, there is no shadow priceWhat will be the effect if an investor decides to increase their maximum investment in the Internet fund?
internet fund with shadow price of 0expected annual return will decrease by .03
5.1-.03=5.07If an investor's total funds available for investment decreases by $1000, what is the expected decrease in the annual return?
total funds LHS shadow price is .03Example 360,000 + (500 x 60) = 90,000Management has decided to increase your budget to $1500. How will this influence audience exposure?
audience exposure initial: $60,000
500 units
$60 shadow price for budgetthe constraints are not binding, I am already putting more than 250 in, increasing those values wouldn't increase any profitWhy are the shadow prices zero for the minimum allocations to newspaper and radio?INTRODUCTION TO PROJECT SCHEDULING AND MANAGEMENT1. what is the total time to complete the project
2. what are the scheduled start and finish dates for each specific activity
3. which activities are "critical" and must be completed exactly as scheduled to keep the project on schedule
4. how long can "noncritical" activities be delayed before they cause an increase in the total project completion time?Project mgt methods answer questions such asactivities
precedence relationshipsTo begin answering these questions, we will represent a project as a graph, where nodes represent ______ and arcs represent _____among activitiesadd up all of the activity timesThe longest time to complete the project is if weHowever, we can do things simultaneously and save timeWe create a network view chart to determine the schedule that minimizes flow timeEarly Start, Early Finish, Late Start, Late Finishfor each actitivity, we define four start/finish timesearly start (ES)equal to the largest of the early finish times for all its immediate predecessorsearly finish (EF)equal to the early start time plus the activity time for that activitylate finish (LF)equal to the smallest of the latest start times for all activities that immediately follow the activityLate start (LS)equal to the late finish time less the activity time for that activityforward passThe early start and early finish times are determined via a _______ through the project networkbackward passThe late finish and late start times are determined via a ______ through the project networkslack timethe time by which the activity can be delayed w/o delaying the entire projectlate start - early startslack time =critical path of a project networkthe sequence of activities that have zero slack time.
If an activity on the _____ is delayed then the completion time of the entire project will also be delayedwhich activities the manager needs to focus onCritical path tells the manager1. We have not given any consideration to uncertainty and variability
2. We have not considered resource constraints; we assume we have infinite resourcesTwo weaknesses of the Network Viewif we introduce uncertainty, the flow time could vary as well
-we have to consider activities with small slack times and high variability as potentially impacting our completion time1. We have not given any consideration to uncertainty and variabilitywe may be doing 4 activities at once, do we have the resources to do all of this?2. We have not considered resource constraints; we assume we have infinite resourcesminimize project completion time (flow time)goal of project managementshows us the critical path
-these are the activities that if delayed, will delay the entire projectWhat is the managerial benefit of the early and late start schedules?PROJECT MANAGEMENT WITH UNCERTAIN ACTIVITY TIMESoptimistic time
most probable time
pessimistic timeOften, when planning a project, we can only estimate the duration of an activity. Accounting for variability in activity times is important and can impact how a project is managed. To do this we provide three time estimates for each activity:optimistic time (a)the minimum activity time if everything progresses ideallymost probable time (M)the most probable activity time under normal conditionspessimistic time (b)the maximum activity time if significant delays are encounteredGiven these 3 time estimates for an activity, we can then estimate the expected activity time and the variance of the activity time(a + 4m + b) /6Expected time
t =(b-a/6)^2Variance
𝞂^2 =Bc activity times are random variables, we can calculate the expected project completion time and the variance of the completion timeexpected project completion timelet T be the random variable representing project completion time. The expected value of T, E(T), is the sum of the expected activity times on the critical pathvariance of project completion timethe variance of T, 𝞂t^2, is the sum of the variances of the activity times on the critical pathWhen we calculate the expected project completion time and the variance of the project completion time, we can now calculate the probability that the project is completed by a certain dateBc the project completion time T is the sum of the random activity times on the critical path, the central limit theorem tells us that T is approximately normally distributed, with the mean and variance we defined aboveP( T≤X)
1. convert to std normal z:
Z = (T - E(T))/𝞂t
2. calculate z score: (x - E(T))/𝞂t
3. Use table to find P(Z≤ x - E(T)/𝞂tSuppose we are asked to calculate the probability that T is less than or equal to some deadline x. Follow these stepsWe do this method for ONE TIME BUSINESS PROCESSES!`project completion time which equals the sum of activity times on the critical pathT=T is normal with mean 15 and VARIANCE of 1.05 with standard dev of √1.05

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