Bus-221-QM.Lecture.14 - UWCENTRE · 2015-01-16 · using either the corner point method or an isocost line approach (which is analogous to the isoprofit approach in maximization problems)

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Annual Carrying Cost for Production Run Model

!  In)produc.on)runs,)setup!cost)replaces)ordering)cost.)!  The)model)uses)the)following)variables:)

Q = number of pieces per order, or production run

Cs = setup cost Ch = holding or carrying cost per unit per

year p = daily production rate d = daily demand rate t = length of production run in days

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Maximum inventory level = (Total produced during the production run) – (Total used during the production run) = (Daily production rate)(Number of days production)

– (Daily demand)(Number of days production) = (pt) – (dt)

since Total produced = Q = pt

we know pQt =

Maximum inventory

level !"

#$%

&−=−=−=pdQ

pQd

pQpdtpt 1

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Since the average inventory is one-half the maximum:

!"

#$%

&−=pdQ

12

inventory Average

and

hCpdQ!"

#$%

&−= 1

2cost holding Annual

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Annual Setup Cost for Production Run Model

sCQD

=cost setup Annual

Setup cost replaces ordering cost when a product is produced over time.

replaces

oCQD

=cost ordering Annual

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Determining the Optimal Production Quantity

By setting setup costs equal to holding costs, we can solve for the optimal order quantity

Annual holding cost = Annual setup cost

sh CQDC

pdQ

=!"

#$%

&−1

2

Solving for Q, we get

!"

#$%

&−

=

pdC

DCQh

s

1

2*

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Production Run Model

Summary of equations

!"

#$%

&−

=

pdC

DCQh

s

1

2 quantity production Optimal *

sCQD

=cost setup Annual

hCpdQ!"

#$%

&−= 1

2cost holding Annual

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Quantity Discount Models •  Quantity discounts are commonly available. •  The basic EOQ model is adjusted by adding in the

purchase or materials cost.

Total cost = Material cost + Ordering cost + Holding cost

ho CQ

CQD

DC2

cost Total ++=

where D = annual demand in units

Co = ordering cost of each order C = cost per unit

Ch = holding or carrying cost per unit per year 6-92

Quantity Discount Models •  Quantity discounts are commonly available. •  The basic EOQ model is adjusted by adding in the

purchase or materials cost.

Total cost = Material cost + Ordering cost + Holding cost

ho CQ

CQD

DC2

cost Total ++=

where D = annual demand in units

Co = ordering cost of each order C = cost per unit

Ch = holding or carrying cost per unit per year

Holding cost = Ch = IC I = holding cost as a percentage of the unit cost (C)

Because unit cost is now variable,

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Use of Safety Stock

!  If)demand)or)the)lead).me)are)uncertain,)the)exact)ROP)will)not)be)known)with)certainty.)

! To)prevent)stockouts,)it)is)necessary)to)carry)extra)inventory)called)safety!stock.!

! Safety)stock)can)prevent)stockouts)when)demand)is)unusually)high.)

! Safety)stock)can)be)implemented)by)adjus.ng)the)ROP.

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Use of Safety Stock

•  The basic ROP equation is ROP = d × L

d = daily demand (or average daily demand) L = order lead time or the number of

working days it takes to deliver an order (or average lead time)

!  A safety stock variable is added to the equation to accommodate uncertain demand during lead time

ROP = d × L + SS where

SS = safety stock

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Use of Safety Stock

Figure 6.7

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ROP with Known Stockout Costs

•  With a fixed EOQ and an ROP for placing orders, stockouts can only occur during lead time.

•  Our objective is to find the safety stock quantity that will minimize the total of stockout cost and holding cost.

•  We need to know the stockout cost per unit and the probability distribution of demand during lead time.

•  Estimating stockout costs can be difficult as there may be direct and indirect costs.

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Safety Stock with Unknown Stockout Costs

!  There)are)many)situa.ons)when)stockout)costs)are)unknown.)

!  An)alterna.ve)approach)to)determining)safety)stock)levels)is)to)use)a)service!level.)

!  A)service)level)is)the)percent)of).me)you)will)not)be)out)of)stock)of)a)par.cular)item.)

Service level = 1 – Probability of a stockout or

Probability of a stockout = 1 – Service level

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Single-Period Inventory Models

!  Some)products)have)no)future)value)beyond)the)current)period.)

!  These)situa.ons)are)called)news!vendor)problems)or)singleFperiod!inventory!models.)

!  Analysis)uses)marginal)profit)(MP))and)marginal)loss)(ML))and)is)called)marginal)analysis.)

! With)a)manageable)number)of)states)of)nature)and)alterna.ves,)discrete)distribu.ons)can)be)used.)

! When)there)are)a)large)number)of)alterna.ves)or)states)of)nature,)the)normal)distribu.on)may)be)used.!

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Marginal Analysis with Discrete Distributions

We stock an additional unit only if the expected marginal profit for that unit exceeds the expected marginal loss.

P = probability that demand will be greater than or equal to a given supply (or the probability of selling at least one additional unit).

1 – P = probability that demand will be less than supply (or the probability that one additional unit will not sell).

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Marginal Analysis with Discrete Distributions

•  The expected marginal profit is P(MP) . •  The expected marginal loss is (1 – P)(ML). •  The optimal decision rule is to stock the additional

unit if: P(MP) ≥ (1 – P)ML

!  With some basic manipulation:

P(MP) ≥ ML – P(ML) P(MP) + P(ML) ≥ ML

P(MP + ML) ≥ ML or

MPMLML+

≥P

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Steps of Marginal Analysis with Discrete Distributions

1.  Determine the value of for the problem.

2.  Construct a probability table and add a cumulative probability column.

3.  Keep ordering inventory as long as the probability (P) of selling at least one additional unit is greater than

MPMLML+

MPMLML+

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ABC Analysis

•  The purpose of ABC analysis is to divide the inventory into three groups based on the overall inventory value of the items.

•  Group A items account for the major portion of inventory costs. –  Typically about 70% of the dollar value but only 10% of the

quantity of items. –  Forecasting and inventory management must be done

carefully.

•  Group B items are more moderately priced. –  May represent 20% of the cost and 20% of the quantity.

•  Group C items are very low cost but high volume. –  It is not cost effective to spend a lot of time managing these

items.

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LP Properties and Assumptions

PROPERTIES OF LINEAR PROGRAMS 1. One objective function 2. One or more constraints 3. Alternative courses of action 4. Objective function and constraints are linear – proportionality and divisibility 5. Certainty 6. Divisibility 7. Nonnegative variables

Table 7.1

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Graphical Representation of a Constraint

!  To)produce)tables)and)chairs,)both)departments)must)be)used.)

! We)need)to)find)a)solu.on)that)sa.sfies)both)constraints)simultaneously.!

!  A)new)graph)shows)both)constraint)plots.)!  The)feasible!region)(or)area!of!feasible!solu'ons))is)

where)all)constraints)are)sa.sfied.)!  Any)point)inside)this)region)is)a)feasible)solu.on.)!  Any)point)outside)the)region)is)an)infeasible)solu.on.)

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Isoprofit Line Solution Method

•  Once the feasible region has been graphed, we need to find the optimal solution from the many possible solutions.

•  The speediest way to do this is to use the isoprofit line method.

•  Starting with a small but possible profit value, we graph the objective function.

•  We move the objective function line in the direction of increasing profit while maintaining the slope.

•  The last point it touches in the feasible region is the optimal solution.

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! A)second)approach)to)solving)LP)problems)employs)the)corner!point!method.!

!  It)involves)looking)at)the)profit)at)every)corner)point)of)the)feasible)region.)

! The)mathema.cal)theory)behind)LP)is)that)the)op.mal)solu.on)must)lie)at)one)of)the)corner!points,)or)extreme!point,)in)the)feasible)region.)

Corner Point Solution Method

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Summary of Graphical Solution Methods

ISOPROFIT METHOD 1. Graph all constraints and find the feasible region. 2. Select a specific profit (or cost) line and graph it to find the slope. 3. Move the objective function line in the direction of increasing profit (or

decreasing cost) while maintaining the slope. The last point it touches in the feasible region is the optimal solution.

4. Find the values of the decision variables at this last point and compute the profit (or cost).

CORNER POINT METHOD 1. Graph all constraints and find the feasible region. 2. Find the corner points of the feasible reason. 3. Compute the profit (or cost) at each of the feasible corner points. 4. Select the corner point with the best value of the objective function found in

Step 3. This is the optimal solution.

Table 7.4

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Solving Minimization Problems

•  Many LP problems involve minimizing an objective such as cost instead of maximizing a profit function.

•  Minimization problems can be solved graphically by first setting up the feasible solution region and then using either the corner point method or an isocost line approach (which is analogous to the isoprofit approach in maximization problems) to find the values of the decision variables (e.g., X1 and X2) that yield the minimum cost.

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Four Special Cases in LP

•  Four special cases and difficulties arise at times when using the graphical approach to solving LP problems.

–  No feasible solution –  Unboundedness –  Redundancy –  Alternate Optimal Solutions

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Four Special Cases in LP No feasible solution

–  This exists when there is no solution to the problem that satisfies all the constraint equations.

–  No feasible solution region exists. –  This is a common occurrence in the real world. –  Generally one or more constraints are relaxed until

a solution is found.

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Four Special Cases in LP Unboundedness

–  Sometimes a linear program will not have a finite solution.

–  In a maximization problem, one or more solution variables, and the profit, can be made infinitely large without violating any constraints.

–  In a graphical solution, the feasible region will be open ended.

–  This usually means the problem has been formulated improperly.

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Redundancy –  A redundant constraint is one that does not affect

the feasible solution region. –  One or more constraints may be binding. –  This is a very common occurrence in the real

world. –  It causes no particular problems, but eliminating

redundant constraints simplifies the model.

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Alternate Optimal Solutions –  Occasionally two or more optimal solutions may

exist. –  Graphically this occurs when the objective

function’s isoprofit or isocost line runs perfectly parallel to one of the constraints.

–  This actually allows management great flexibility in deciding which combination to select as the profit is the same at each alternate solution.

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Sensitivity Analysis •  Optimal solutions to LP problems thus far have been

found under what are called deterministic assumptions.

•  This means that we assume complete certainty in the data and relationships of a problem.

•  But in the real world, conditions are dynamic and changing.

•  We can analyze how sensitive a deterministic solution is to changes in the assumptions of the model.

•  This is called sensitivity analysis, postoptimality analysis, parametric programming, or optimality analysis.

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Sensitivity Analysis •  Sensitivity analysis often involves a series of what-if?

questions concerning constraints, variable coefficients, and the objective function.

•  One way to do this is the trial-and-error method where values are changed and the entire model is resolved.

•  The preferred way is to use an analytic postoptimality analysis.

•  After a problem has been solved, we determine a range of changes in problem parameters that will not affect the optimal solution or change the variables in the solution.

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1.  Define)the)project)and)all)of)its)significant)ac.vi.es)or)tasks.)2.  Develop)the)rela.onships)among)the)ac.vi.es)and)decide)

which)ac.vi.es)must)precede)others.)3.  Draw)the)network)connec.ng)all)of)the)ac.vi.es.)4.  Assign).me)and/or)cost)es.mates)to)each)ac.vity.)5.  Compute)the)longest).me)path)through)the)network;)this)is)

called)the)cri'cal!path.!6.  Use)the)network)to)help)plan,)schedule,)monitor,)and)

control)the)project.)

Six Steps of PERT/CPM

The critical path is important since any delay in these activities can delay the completion of the project.

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PERT/CPM Given the large number of tasks in a project, it is easy to see why the following questions are important:

1.  When will the entire project be completed? 2.  What are the critical activities or tasks in the project, that is,

the ones that will delay the entire project if they are late? 3.  Which are the non-critical activities, that is, the ones that

can run late without delaying the entire project’s completion?

4.  If there are three time estimates, what is the probability that the project will be completed by a specific date?

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PERT/CPM

5.  At any particular date, is the project on schedule, behind schedule, or ahead of schedule?

6.  On any given date, is the money spent equal to, less than, or greater than the budgeted amount?

7.  Are there enough resources available to finish the project on time?

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Drawing the PERT/CPM Network

!  There)are)two)common)techniques)for)drawing)PERT)networks.)

!  Ac'vityFonFnode)(AON))where)the)nodes)represent)ac.vi.es.)

!  Ac'vityFonFarc)(AOA))where)the)arcs)are)used)to)represent)the)ac.vi.es.)

!  The)AON)approach)is)easier)and)more)commonly)found)in)soIware)packages.)

!  One)node)represents)the)start)of)the)project,)one)node)for)the)end)of)the)project,)and)nodes)for)each)of)the)ac.vi.es.)

!  The)arcs)are)used)to)show)the)predecessors)for)each)ac.vity.) 12-120

Activity Times !  In)some)situa.ons,)ac.vity).mes)are)known)with)certainty.)!  The)CPM)assigns)just)one).me)es.mate)to)each)ac.vity)and)

this)is)used)to)find)the)cri.cal)path.)!  In)many)projects)there)is)uncertainty)about)ac.vity).mes.)!  PERT)employs)a)probability)distribu.on)based)on)three).me)

es.mates)for)each)ac.vity,)and)a)weighted)average)of)these)es.mates)is)used)for)the).me)es.mate)and)this)is)used)to)determine)the)cri.cal)path.)!  PERT)oIen)assumes).me)es.mates)follow)a)beta!probability!

distribu'on.!

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Activity Times

The time estimates in PERT are:

Optimistic time (a) = time an activity will take if everything goes as well as possible. There should be only a small probability (say, 1/100) of this occurring.

Pessimistic time (b) = time an activity would take assuming very unfavorable conditions. There should also be only a small probability that the activity will really take this long.

Most likely time (m) = most realistic time estimate to complete the activity

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Activity Times

Beta Probability Distribution with Three Time Estimates

Figure 12.2

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Activity Times

To)find)the)expected!ac'vity!'me)(t),)the)beta)distribu.on)weights)the)es.mates)as)follows:)

64 bmat ++

=

To compute the dispersion or variance of activity completion time, we use the formula:

2

6Variance !

"#

$%& −

=ab

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How to Find the Critical Path To)find)the)cri.cal)path,)we)need)to)determine)the)following)quan..es)for)each)ac.vity)in)the)network.)1.  Earliest!start!'me)(ES):)the)earliest).me)an)ac.vity)can)begin)

without)viola.on)of)immediate)predecessor)requirements.)2.  Earliest!finish!'me)(EF):)the)earliest).me)at)which)an)ac.vity)

can)end.)3.  Latest!start!'me)(LS):)the)latest).me)an)ac.vity)can)begin)

without)delaying)the)en.re)project.)4.  Latest!finish!'me)(LF):)the)latest).me)an)ac.vity)can)end)

without)delaying)the)en.re)project.)

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How to Find the Critical Path In the nodes, the activity time and the early and late start and finish times are represented in the following manner.

ACTIVITY t ES EF LS LF

Earliest times are computed as: Earliest finish time = Earliest start time

+ Expected activity time EF = ES + t

Earliest start = Largest of the earliest finish times of immediate predecessors

ES = Largest EF of immediate predecessors 12-126

How to Find the Critical Path

•  At the start of the project we set the time to zero. •  Thus ES = 0 for both A and B.

Start

A t = 2 ES = 0 EF = 0 + 2 = 2

B t = 3 ES = 0 EF = 0 + 3 = 3

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General Foundry’s Earliest Start (ES) and Earliest Finish (EF) times

Figure 12.4

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Latest times are computed as

Latest start time = Latest finish time – Expected activity time

LS = LF – t

Latest finish time = Smallest of latest start times for following activities

LF = Smallest LS of following activities

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•  Once ES, LS, EF, and LF have been determined, it is a simple matter to find the amount of slack time that each activity has:

Slack = LS – ES, or Slack = LF – EF

•  The activities that have no slack time are called critical activities and they are said to be on the critical path.

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Probability of Project Completion

!  The)cri'cal!path!analysis)helped)determine)the)expected)project)comple.on).me.)

!  But)varia.on)in)ac.vi.es)on)the)cri.cal)path)can)affect)overall)project)comple.on,)and)this)is)a)major)concern.)

!  PERT)uses)the)variance)of)cri.cal)path)ac.vi.es)to)help)determine)the)variance)of)the)overall)project.)

Project variance = ∑ variances of activities on the critical path

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•  We know the standard deviation is just the square root of the variance, so:

!  We assume activity times are independent and that total project completion time is normally distributed.

varianceProject deviation standardProject == Tσ

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The standard normal equation can be applied as follows:

T

Zσ

completion of date Expecteddate Due −=

570 weeks1.76

weeks15 weeks16 .=−

=

!  From Appendix A we find the probability of 0.71566 associated with this Z value.

!  That means the probability this project can be completed in 16 weeks or less is 0.716.

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Probability of General Foundry Meeting the 16-week Deadline

Figure 12.8

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Four Steps of the Budgeting Process

1.  Iden.fy)all)costs)associated)with)each)of)the)ac.vi.es)then)add)these)costs)together)to)get)one)es.mated)cost)or)budget)for)each)ac.vity.)

2.  In)large)projects,)ac.vi.es)can)be)combined)into)larger)work)packages.)A)work!package)is)simply)a)logical)collec.on)of)ac.vi.es.)

3.  Convert)the)budgeted)cost)per)ac.vity)into)a)cost)per).me)period)by)assuming)that)the)cost)of)comple.ng)any)ac.vity)is)spent)at)a)uniform)rate)over).me.)

4.  Using)the)ES)and)LS).mes,)find)out)how)much)money)should)be)spent)during)each)week)or)month)to)finish)the)project)by)the)date)desired.)

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Monitoring and Controlling Project Costs

•  Costs are monitored and controlled to ensure the project is progressing on schedule and that cost overruns are kept to a minimum.

•  The status of the entire project should be checked periodically.

•  The following table shows the state of the project in the sixth week.

•  It can be used the answer questions about the schedule and costs so far.

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Monitoring and Controlling Project Costs

The value of work completed, or the cost to date for any activity, can be computed as follows:

The activity difference is also of interest:

Value of work completed = (Percentage of work complete)

x (Total activity budget)

Activity difference = Actual cost – Value of work completed

A negative activity difference is a cost underrun and a positive activity difference is a cost overrun.

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Project Crashing

•  Projects will sometimes have deadlines that are impossible to meet using normal procedures.

•  By using exceptional methods it may be possible to finish the project in less time than normally required at a greater cost.

•  Reducing a project’s completion time is called crashing.

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Project Crashing !  Crashing)a)project)starts)with)using)the)normal!'me)to)create)

the)cri.cal)path.)!  The)normal!cost)is)the)cost)for)comple.ng)the)ac.vity)using)

normal)procedures.)!  If)the)project)will)not)meet)the)required)deadline,)

extraordinary)measures)must)be)taken.)!  The)crash!'me)is)the)shortest)possible)ac.vity).me)and)will)require)

addi.onal)resources.)!  The)crash!cost)is)the)price)of)comple.ng)the)ac.vity)in)the)earlier^

than^normal).me.!

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Four Steps to Project Crashing

1.  Find the normal critical path and identify the critical activities.

2.  Compute the crash cost per week (or other time period) for all activities in the network using the formula:

Crash cost/Time period = Crash cost – Normal cost Normal time – Crash time

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Four Steps to Project Crashing

3.  Select the activity on the critical path with the smallest crash cost per week and crash this activity to the maximum extent possible or to the point at which your desired deadline has been reached.

4.  Check to be sure that the critical path you were crashing is still critical. If the critical path is still the longest path through the network, return to step 3. If not, find the new critical path and return to step 2.

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Other Topics in Project Management

! Subprojects)! For)extremely)large)projects,)an)ac.vity)may)be)made)of)several)smaller)sub^ac.vi.es)which)can)be)viewed)as)a)smaller)project)or)subproject)of)the)original).)

! Milestones)! Major)events)in)a)project)are)oIen)referred)to)as)milestones)and)may)be)reflected)in)GanQ!charts)and)PERT)charts)to)highlight)the)importance)of)reaching)these)events.)

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Other Topics in Project Management

! Resource)Leveling)! Resource!leveling)adjusts)the)ac.vity)start)away)from)the)early)start)so)that)resource)u.liza.on)is)more)evenly)distributed)over).me.))

! SoIware)! There)are)many)project)management)soIware)packages)on)the)market)for)both)personal)computers)and)larger)mainframe)machines.)

! Most)of)these)create)PERT)charts)and)GanW)charts)and)can)be)used)to)develop)budget)schedules,)adjust)future)start).mes,)and)level)resource)u.liza.on.)

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WAITING LINES !  Queuing!theory)is)the)study)of)wai'ng!lines.!!  It)is)one)of)the)oldest)and)most)widely)used)

quan.ta.ve)analysis)techniques.)!  The)three)basic)components)of)a)queuing)process)are)

arrivals,)service)facili.es,)and)the)actual)wai.ng)line.)!  Analy.cal)models)of)wai.ng)lines)can)help)managers)

evaluate)the)cost)and)effec.veness)of)service)systems.)

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Waiting Line Costs ! Most)wai.ng)line)problems)are)focused)on)finding)the)ideal)

level)of)service)a)firm)should)provide.)!  In)most)cases,)this)service)level)is)something)management)can)

control.)! When)an)organiza.on)does)have)control,)they)oIen)try)to)

find)the)balance)between)two)extremes.)

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Waiting Line Costs !  There)is)generally)a)trade^off)between)cost)of)providing)

service)and)cost)of)wai.ng).me.)!  A)large!staff)and)many)service)facili.es)generally)results)in)high)levels)

of)service)but)have)high)costs.)!  Having)the)minimum)number)of)service)facili.es)keeps)service!cost)

down)but)may)result)in)dissa.sfied)customers.)

!  Service)facili.es)are)evaluated)on)their)total!expected!cost)which)is)the)sum)of)service!costs)and)wai'ng!costs.!

!  Organiza.ons)typically)want)to)find)the)service)level)that)minimizes)the)total)expected)cost.)

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Characteristics of a Queuing System

! There)are)three)parts)to)a)queuing)system:)1.  The)arrivals)or)inputs)to)the)system)(some.mes)referred)

to)as)the)calling!popula'on).)2.  The)queue)or)wai.ng)line)itself.)3.  The)service)facility.)

! These)components)have)their)own)characteris.cs)that)must)be)examined)before)mathema.cal)models)can)be)developed.)

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Arrival)Characteris.cs)have)three)major)characteris.cs:))size,)paQern,)and)behavior.!

! The)size)of)the)calling)popula.on)can)be)either)unlimited)(essen.ally)infinite))or)limited)(finite).)

! The)paWern)of)arrivals)can)arrive)according)to)a)known)paWern)or)can)arrive)randomly.!!  Random)arrivals)generally)follow)a)Poisson!distribu'on.))

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Behavior)of)arrivals)! Most)queuing)models)assume)customers)are)pa.ent)and)will)wait)in)the)queue)un.l)they)are)served)and)do)not)switch)lines.)

! Balking)refers)to)customers)who)refuse)to)join)the)queue.)! Reneging)customers)enter)the)queue)but)become)impa.ent)and)leave)without)receiving)their)service.)

! That)these)behaviors)exist)is)a)strong)argument)for)the)use)of)queuing)theory)to)managing)wai.ng)lines.)

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Wai.ng)Line)Characteris.cs)! Wai.ng)lines)can)be)either)limited)or)unlimited.!! Queue)discipline)refers)to)the)rule)by)which)customers)in)the)line)receive)service.)! The)most)common)rule)is)firstFin,!firstFout)(FIFO).)! Other)rules)are)possible)and)may)be)based)on)other)important)characteris.cs.)

! Other)rules)can)be)applied)to)select)which)customers)enter)which)queue,)but)may)apply)FIFO)once)they)are)in)the)queue.)

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Service Facility Characteristics – Basic queuing system configurations:

•  Service systems are classified in terms of the number of channels, or servers, and the number of phases, or service stops.

•  A single-channel system with one server is quite common.

•  Multichannel systems exist when multiple servers are fed by one common waiting line.

•  In a single-phase system, the customer receives service form just one server.

•  In a multiphase system, the customer has to go through more than one server.

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13-151

Identifying Models Using Kendall Notation

•  D. G. Kendall developed a notation for queuing models that specifies the pattern of arrival, the service time distribution, and the number of channels.

•  Notation takes the form:

!  Specific letters are used to represent probability distributions. M = Poisson distribution for number of occurrences D = constant (deterministic) rate G = general distribution with known mean and variance

Arrival distribution

Service time distribution

Number of service channels open

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Identifying Models Using Kendall Notation

•  A single-channel model with Poisson arrivals and exponential service times would be represented by:

M/M/1 •  If a second channel is added the notation would read:

M/M/2 •  A three-channel system with Poisson arrivals and

constant service time would be M/D/3

•  A four-channel system with Poisson arrivals and normally distributed service times would be

M/G/4

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Single-Channel Model, Poisson Arrivals, Exponential Service Times (M/M/1)

1.  The average number of customers or units in the system, L:

λµλ−

=L

2.  The average time a customer spends in the system, W:

3.  The average number of customers in the queue, Lq:

λµ −=

1W

)( λµµλ−

=2

qL

13-154


4.  The average time a customer spends waiting in the queue, Wq:

)( λµµλ−

=qW

5.  The utilization factor for the system, ρ, the probability the service facility is being used:

µλ

ρ =

13-155


6.  The percent idle time, P0, or the probability no one is in the system:

µλ

−= 10P

7.  The probability that the number of customers in the system is greater than k, Pn>k:

1+

> !"

#$%

&=

k

knP µλ

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Multichannel Queuing Model with Poisson Arrivals and Exponential Service Times (M/M/m)

•  Equations for the multichannel queuing model: •  Let

m = number of channels open λ = average arrival rate µ = average service rate at each channel

1.  The probability that there are zero customers in the system is:

λµ

λµµ

µλ

µλ

>

−#$

%&'

(+))*

+

,,-

.#$

%&'

(=

∑−=

=

m

mm

mn

P mmn

n

n for 11

11

0

0

!!

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13-157

Multichannel Model, Poisson Arrivals, Exponential Service Times (M/M/m)

2.  The average number of customers or units in the system

µλ

λµµλλµ

+−−

= 021P

mmL

m

)()!()/(

3.  The average time a unit spends in the waiting line or being served, in the system

λµλµµλµ LPmm

Wm

=+−−

=1

1 02)()!()/(

13-158

Multichannel Model, Poisson Arrivals, Exponential Service Times (M/M/m)

4.  The average number of customers or units in line waiting for service

µλ

−= LLq



λµq

q

LWW =−=

1

µλ

ρm

=

13-159

Constant Service Time Model (M/D/1)

! Constant)service).mes)are)used)when)customers)or)units)are)processed)according)to)a)fixed)cycle.)

! The)values)for)Lq,)Wq,)L,)and)W)are)always)less)than)they)would)be)for)models)with)variable)service).me.)

!  In)fact)both)average)queue)length)and)average)wai.ng).me)are)halved)in)constant)service)rate)models.)

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1. Average length of the queue

2. Average waiting time in the queue

)( λµµλ−

=2

2

qL

)( λµµλ−

=2qW

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3. Average number of customers in the system

4. Average time in the system

µλ

+= qLL

µ1

+= qWW

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Finite Population Model (M/M/1 with Finite Source)

Equa.ons)for)the)finite)popula.on)model:)Using)λ)=)mean)arrival)rate,)µ)=)mean)service)rate,)and)N)=)size)of)the)popula.on,)the)opera.ng)characteris.cs)are:))1.  Probability)that)the)system)is)empty:)

∑=

"#

$%&

'−

=N

n

n

nNN

P

0

01

µλ

)!(!

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13-163


2. Average length of the queue:

( )01 PNLq −"#$

%&' +

−=λµλ

4. Average waiting time in the queue:

λ)( LNL

W qq −=

3. Average number of customers (units) in the system:

( )01 PLL q −+=

13-164


5. Average time in the system:

µ1

+= qWW

6. Probability of n units in the system:

( )NnP

nNNP

n

n ,...,,!

!10 for 0 =!

"

#$%

&−

=µλ

13-165

Some General Operating Characteristic Relationships

•  Certain relationships exist among specific operating characteristics for any queuing system in a steady state.

•  A steady state condition exists when a system is in its normal stabilized condition, usually after an initial transient state.

•  The first of these are referred to as Little’s Flow Equations: L = λW (or W = L/λ)

Lq = λWq (or Wq = Lq/λ)

!  And

W = Wq + 1/µ 13-166

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