OPRE $6302.$xh$1$: Homework assignment $2$

Problem $1($line balancing$)$:

Consider the following process:

Task Dura$$on $($in mins$)$ Descrip$$on

$15$ Review consumer$$s order

$232+36/($# workers$)$ Build top part of flow unit

$35$ Paint top part of flow unit

$430$ Build botom part of flow unit

$510$ Paint botom part of flow unit

$620$ Glue together top and botom parts of flow unit

$760$ Rest

$815$ Decorate merged flow unit

$910$ Wrap decorated merged flow unit

Consider the following facts$/$constraints:

$-$ The process is a make$-$to$-$order process

$-$ An order for one flow unit is placed exactly every $5$ minutes.

$-$ The following workers are available: A$,$ B$,$ C$,$ D$,$ E$,$ F and G$.$ Not all of them need to be employed

in the process but no other worker can be hired.

$-$ All other resources can be ignored because they are irrelevant.

$-$ The dura$$on of Task $2$ per flow unit depends on the number of workers assigned to the task

according to the formula $32+36/($# workers$).$ For example, if one worker does Task $2,$ its

dura$$on is $32+36/1=68$ minutes; if two workers $($jointly$)$ do Task $2,$ its dura$$on is $32+36/2=50$

minutes; etc.

$-$ Tasks $1$ must be performed before all other tasks.

$-$ Tasks $2-3$ are performed on the top part of the flow unit and tasks $4-5$ are performed on the

botom part of the flow unit. As such, tasks $2-3$ can take place at the same $$me as tasks $4-5.$

$-$ Task $6$ can only be once tasks $1-5$ have been completed and is done using a very specialized

machine that only worker E knows how to operate.

$-$ Task $7$ does not involve any worker;

$-$ All tasks other than Task $2$ and Task $7$ must be done by one worker only;

$-$ Task $8$ must follow Task $7$ and Task $9$ must follow Task $8$;

$-$ All workers are paid $$20$ per hour;

$-$ The selling price of a flow unit is $$300$;

$-$ One worker cannot do more than one task at the same $$me;

$-$ Assume that if two $($or more$)$ workers are in parallel for one task, they are in parallel for all the

tasks they do$.$

Draw a process flow diagram for how you would suggest organizing the process, in order to achieve the

best value for the following performance metrics:

$-$ Profit rate;

$-$ Rush order flow $$me;

$-$ Flow rate;

$-$ Average labor u$$liza$$on;

$-$ Cost of direct labor;

And provide the value of these metrics for your process flow diagram. Show all your work.

Problem $2($Process with variability$)$:

Consider a $24-$hour call center which provides tech support for a so$$ware company. There are currently

$85$ operators who can answer calls at any moment in $$me$.$ It takes, on average, $10$ minutes for an

operator to solve a customer$$s problem, with a coefficient of varia$$on of $2.$ Assume that the number of

calls per hour does not vary much throughout the day, and averages $500$ calls an hour. The standard

devia$$on of the $$me between two calls is equal to $6$ seconds. If a customer calls when all operators are

busy answering calls, he or she is put on hold. Customers on hold are then answered on a first$-$comefirst$-$served basis. Assume that there is no limit to the number of customers who can be on hold at any

moment in $$me and that callers do not get impa$$ent and always stay on hold un$$l they speak to an

operator.

$($a$)$ Say what are the values of the $5$ key parameters for this problem $($i$.$e$.,$ m$,$ p$,$ a$,$ CVa and Cvp$)$ and

check that the process is demand$-$constrained.

$($b$)$ How long does a customer spend wai$$ng on hold, on average?

$($c$)$ What is the average number of customers that are on the phone with the call center $($either on

hold or talking to an operator$)$ at any point in $$me$?$

$($d$)$ What is the average number of calls completed by one operator in one hour?

$($e$)$ What is the average number of operators who are not on the phone with a customer at a given

moment in $$me$?$

$($f$)$ What percentage of the $$me spent on the phone by a customer is spent on hold?

$($g$)$ The call center es$$mates that it costs them $$1$ per minute that a customer spends wai$$ng on

hold. Operators are paid $$20$ per hour. The call center manager is considering hiring $5$ more

operators. Is it wise?