Chapter 1+2
1. (a) Differentiate between operations management and production management. 05 Chapter 1
2(b) Write down the functions of operations management. 10 Chapter 1. (Page 4,5,6) Ans:
3. (c) Discuss different types of production systems.
-20 Chapter 1
14. Define Operations Management.
-15 Chapter 1
Operations management (OM) is defined as the design, operation, and improvement of the systems that create
and deliver the firm’s primary products and services
(slide)
Operations management is the management of processes or systems that create goods and/or provide services.
It encompasses forecasting, capacity planning, scheduling, man aging inventories, assuring quality, motivating
employees, deciding where to locate facilities etc. (book)
15. (b) "Production is a value addition process" comment on the statement with an example. 10 27. (b) Clarify
the statement, "production is a value addition process". Describe lean production and craft production
briefly. (ch-1) 15
Ans
lean production System that uses minimal amounts of resources to produce a high volume of high-quality
goods with some variety.
craft production System is a system in which highly skilled workers use simple, flexible tools to produce small
quantities of customized goods
16. (c) What are trade-offs? Why is careful consideration of trade-offs important in decision making? 10 Chapter
1 (pdf Pg 54)
Ans:
trade-offs
A balance achieved between two desirable but incompatible features
Operations managers encounter decisions that can be described as trade-off decisions. , the higher the accuracy,
the higher the cost, so it is important to weigh cost-accuracy trade-offs carefully. In selecting a piece of
equipment, a manager must evaluate the merits of extra features relative to the cost of those extra features. And
in the scheduling of overtime to increase output, the manager must weigh the value of the increased output
against the higher costs of overtime (e.g., higher labor costs, lower productivity, lower quality, and greater risk of
accidents).
Careful consideration of trade-offs important in decision making because
26. (a) Describe the nature of the operation manager's job.(ch-1) 10
Ans:
The Operations Manager role is mainly to implement the right processes and practices across the organization.
The specific duties of an Operations Manager include formulating strategy, improving performance, procuring
material and resources and securing compliance
Responsibilities
•
•
•
•
•
•
•
•
•
•
Ensure all operations are carried on in an appropriate, cost-effective way
Improve operational management systems, processes and best practices
Purchase materials, plan inventory and oversee warehouse efficiency
Help the organization’s processes remain legally compliant
Formulate strategic and operational objectives
Examine financial data and use them to improve profitability
Manage budgets and forecasts
Perform quality controls and monitor production KPIs
Recruit, train and supervise staff
Find ways to increase quality of customer service
28. (c) Contrast the terms strategies and tactics. Why is productivity important? (Ch- 2) 10 Ans:
Strategies
Tactics
strategies are the roadmaps for reaching the
destinations.
Tactics are the methods and actions used to
accomplish strategies.
Strategies provide focus for decision making.
They are more specific than strategies, and
they provide guidance and direction for arrying
out actual operations, which need the most
specific and detailed plans and decision making
in an organization
strategy is the long-term plan
Source: Book
Source: Internet
Productivity is important because-
tactics are the short-term steps that help you hit
smaller goals
38. (a) How can you differentiate manufacturing organization from service organization?
06
Ans: We can differentiate them by considering the following points
Degree of customer contact. Many services involve a high degree of customer contact then manufacturing.
Labor content of jobs. Services often have a higher degree of labor content than manufacturing jobs
do
Uniformity of inputs. Service operations are often subject to a higher degree of variability of
inputs.
Measurement of productivity. Measurement of productivity can be more difficult for
service jobs due largely to the high variations of inputs
Quality assurance. Quality assurance is usually more challenging for services due to the higher variation in
input, and because delivery and consumption occur at the same time.
Inventory. Many services tend to involve less use of inventory than manufacturing operations, so
the costs of having inventory on hand are lower than they are for manufacturing
Wages. Manufacturing jobs are often well paid, and have less wage variation than service
jobs, which can range from highly paid professional services to minimum-wage workers
Ability to patent. Product designs are often easier to patent than service designs, and some
services cannot be patented, making them easier for competitors to copy.
39. (b) What are meant by production and productivity? What factors affect productivity? 12
Ans: Production:
the production is a value addition process because in the transformation stage value is addedto the input to
enhance the quality of the output.
Productivity:
Productivity is an index that measures output (goods and services) relative to the input (labor, materials, energy,
and other resources) used to produce it. It is usually expressed as the ratio of output to input:
Productivity=Output/Input
Standardizing processes and procedures wherever possible to reduce variability Quality
differences may distort productivity measurements.
Use of the Internet can lower costs of a wide range of transactions, thereby increasing Productivity
Computer viruses can have an immense negative impact on productivity
Searching for lost or misplaced items wastes time, hence negatively affecting productivity.
Scrap rates have an adverse effect on productivity, signaling inefficient use of resources.
New workers tend to have lower productivity than seasoned workers.
Safety should be addressed. Accidents can take a toll on productivity.
A shortage of information technology workers and other technical workers hampers
the ability of companies to update computing resources, generate and sustain growth, and
take advantage of new opportunities Layoffs often affect productivity.
Labor turnover has a negative effect on productivity;
Design of the workspace can impact productivity
Incentive plans that reward productivity increases can boost productivity.
40. (c) Can you think of a business that doesn't have operations management? 07
Ans: No, I cannot think of that because operation management is an important part of business organization.
Operations management is crucial in a business organization because it helps in efficient management , control
and oversee goods, services . Operations is responsible for producing the goods or providing the services offered
by the organization. To put this into perspective, if a business organization were a car, operations would be its
engine. And just as the engine is the core of what a car does, in a business organization, operations is the core
of what the organization does. Operations management is responsible for managing that core. Hence,
operations management is the management of systems or processes that create goods and/or provide
services. Operations and supply chains are intrinsically linked and no business organization could exist without
both.
53. Define 'Operation'
Ans: Operations is that part of a business organization that is responsible for producing goods and/ or
services.
Chapter-3
Solution to Ch-3
6. (c) Suppose a software producer is about to release a new version of its popular software.
What information do you think it would take into account in forecasting initial sales?
08
Chapter 3 (Exercise 18, pdf pg 156)
Forecasting: It is a method which is used by the analysts to predict the demand It helps the
manufacturers to allocate the resources and the costs associated with the product
Forecasting can be conducted through qualitative means and by quantitative means
Information technology sector is considered as one of the most prominent sector in the service
industry. This sector comprises of the products which generally has a very short life span due to
continuous innovations Estimating the demand of a product in this sector is very difficult
because of the number of alternative availability and piracy But for a new product with no
competitor seems favorable for a company to capture the market
To estimate the initial sales of a new version of popular software, the company should take into
the account of given below information
The company should analyze the trends of the sale of the previous version of the software to
determine the forecast for the new version Through this analysis, the company would able to
know. who the potential customers are
The company should also look for the altematives of that software available in the market
Analysis of the alternatives would provide a clear picture that whether their product would meet
the customer's demand or not.
After doing the general analysis, the company should determine the purpose of the forecast
This would help the company to analyze what is to be required to conduct the forecast
The company then should analyze the technique and the budget of the forecast accordingly.
Whether a qualitative of a quantitative method would be beneficial to conduct the forecast After
deciding the technique and the budget decision, the company then should follow the plan
accordingly It should collect all the relevant data required to conduct the forecast
At the end, the company then should monitor the progress of the forecast whether the forecast
was effective or not
Thus the above mentioned steps would help the company to forecast the demand of the
software in the market
17. (a) Define forecasting. What are the limitations of quantitative forecasting techniques?
10
Chapter 3 (exercise 1) Pg 156
Similar question: What are the main advantages that quantitative techniques for forecasting
have over qualitative techniques? What limitations do quantitative techniques have?
Quantitative forecasts are mathematically computed and are based on numerical data. This
numerical data could be further used in modelling to enable the future trends to be predicted
These models do also account for trends like seasonality as well as various economic and
political conditions However there are some very complex models which are difficult for people
to understand and hence, they are averse to using them
One of the basic conditions of all simple quantitative methods is that they rely on historical data
and also on the fact about what happened in the past would continue to happen in the future.
Also they rely greatly on the accuracy of the data being used to apply the quantitative method
Quantitative methods are of two types, namely:
1)
Time-series forecasts These are a simple attempt to project the past experiences into
the future As explained earlier, they attempt to use histoncal data with the premise that the
future will be the same as the past. In some models there is an attempt to remove random
variation by the use of smoothing techniques Others try to identity patterns and apply them
without determining the cause of the patterns
2)
Associative models These are models where there is a relationship between certain
variable and the data. Because of this, these are called associative since the results use certain
causal or explanatory variables to generate a forecast. This results in a causal relationship to be
established between any occurrence and its effect on the forecast and the resulting demand
Quantitative forecasts are usually preferred over qualitative forecasts for the simple reason that
they are objective, based on data and facts and are therefore impersonal Qualitative forecasts
are usually very personal to the people involved in the forecasting and therefore are very
subjective and quite unreliable
However, there are circumstances where one does not have an option but to go in for a
qualitative forecast. This is when there is a sharp change in the market conditions and there is
no time to gather data and analyze the same. In such situation the only option is to rely on
experience and the intuitive opinion of the managers.
Quantitative methods of forecasting have a major flaw as it assumes that what happened in the
past will continue to occur in the future. Hence, by projecting historical data into the future, one
is ignoring any rapid economic or political changes that may occur These can only be done by
applying qualitative forecasting methods
Therefore the best possible alternative is a quantitative forecast that is tempered by an overlay
of a qualitative forecast
19. (c) A manager uses this equation to predict demand: y = 10+ st. Over the past eight 17
periods, demand has been as follows:
17
Is the forecast performing adequately? Explain.
Chapter 3 (Problem 33 math) pg 164
44. (Solved)(c) Choose the type of forecasting technique (survey, Delphi, averaging, seasonal
naïve, trend. or associative) that would be most appropriate for predicting.
05 chapter-3
page -156 exercise-19
i)
Demand for mother's day greeting cards.
ii)
Popularity of a new television series.
iii)
Demand for vacation on the moon,
iv)
The impact a price increase of 10 percent would have on sales of orange marmalade
v)
Demand for toothpaste in a particular super market. text book publishing company has
complied data on total annual sales of its business
(d) A text book publishing company has complied data on total annual sales of its
business texts for the preceding nine years:
15 chapter-3
page -164 exercise-31
45.
.
i) Using an appropriate model, forecast textbook sales for each of the next five
years ii) Prepare a control chart for the forecast using the original data. Use 2s limits
ili) Suppose actual sales for the next five years turn out as follows:
Is the forecast performing adequately? Explain.
An electrical contractor's records during the last five weeks indicate the number of
job requests:
07 chapter-3 page -157 exercise-4
46.
Predict the number of requests for week 5, using smoothing constant, a-0.30 and 20 for week 2
forecast.
58. The manager of a travel agency has been using a seasonally adjusted forecast to predict
demand for packaged tours. The actual and predicted values are following table:
15
(11th edition, Chapter-3, Page-163, Problem28)
i) Compute MAD for the fifth period, then update it period by period using exponential
smoothing with x = 0.3.
ii) Compute a tracking signal for periods 5 through 14 using the initial and updated MADs. If
limits of ±4 are used, what can you conclude?
Chapter-11
20. (a) What is meant by aggregate planning? Why is there a need for aggregate
planning?
Ans:
In the spectrum of production planning, aggregate planning is intermediate-range
capacity planning that typically covers a time horizon of 2 to 12 months, although in some
companies it may extend to as much as 18 months. It is particularly useful for organizations that
experience seasonal or other fluctuations in demand or capacity. The goal of aggregate planning
is to achieve a production plan that will effectively utilize the organization’s resources to match
expected demand. Planners must make decisions on output rates, employment levels and
changes, inventory levels and changes, back orders, and subcontracting in or out.
Aggregate planning is required to meet the aggregate forecast of demand in the
intermediate range. This can be for a period of 3 months to 12 months. In certain cases it can
even be as less as one month. Aggregate planning is basically required so deal with variations in
the demand over the intermediate range. This can be done by manipulating the employee
strength, output principles as well as the inventory policy. All these can be adjusted according
to the needs imposed on the system due to the variations as well as the business policies of the
firm. Aggregate planning is initially, basically a budgeting exercise which as time progresses,
becomes an exercise in trying to keep the business policies intact while trying to satisfy the
demand within the restrictive parameters of inventory, back-log and overtime/sub-contracting.
Managers usually have to consider numerous strategies before the zero in on the most optimal
one for the current scenario. The crucial thing is that these pains have to be updated as and
when the situation demands.
21. What are the inputs to master scheduling? What are the outputs?
Ans:
The master schedule is the heart of production planning and control. It determines the
quantities needed to meet demand from all sources, and that governs key decisions and
activities throughout the organization.
Inputs
The master schedule has three inputs: the beginning inventory, which is the actual quantity on
hand from the preceding period; forecasts for each period of the schedule; and customer
orders, which are quantities already committed to customers. Other factors that might need
to be taken into consideration include any hiring or firing restrictions imposed by HR, skill
levels, limits on inventory such as available space, whether items are perishable, and whether
there are some market lifetime (e.g., seasonal or obsolescence) considerations.
Outputs
The master scheduling process uses this information on a period-by-period basis to determine
the projected inventory, production requirements, and the resulting uncommitted inventory,
which is referred to as available-to-promise (ATP) inventory. Knowledge of the uncommitted
inventory can enable marketing to make realistic promises to customers about deliveries of
new orders.
47. (a) What is disaggregation? Why is it done?
Ans:
•
For the production plan to be translated into meaningful terms of production, it
is necessary to disaggregate the aggregate plan.
•
This means breaking down the aggregate plan into specific product
requirements in order to determine labor requirements (skills, size of
workforce), materials, and inventory requirements.
•
To put the aggregate production plan into operation, one must convert, or
decompose, those aggregate units into units of actual product or services that
are to be produced or offered.
•
For example, televisions manufacturer may have an aggregate plan that calls for
200 televisions in January, 300 in February, and 400 in March. This company
produce 21,26, and 29 inch TVs, therefore the 200, 300, and 400 aggregate TVs that are
to be produced during those three months must be translated into specific numbers of
TVs of each type prior to actually purchasing the appropriate materials and parts,
scheduling operations, and planning inventory requirements.
9. (b) Mention the advantages and disadvantages of chase approach and level approach of
aggregate planning Ans:
Level capacity strategy Maintaining a steady rate of regular-time output while meeting
variations in demand by a combination of options.
Capacities (workforce levels, output rates, etc.) are kept constant over the planning horizon. A
level strategy works best when inventory carrying costs and backlog costs are relatively low.
Advantage:
• Stable output rates and workforce levels.
Disadvantages:
• Greater inventory costs.
• Increased overtime and idle time.
• Resource utilizations that vary over time.
chase demand strategy
Matching capacity to demand; the planned output for a period is set at the expected demand
for that period.
Capacities (workforce levels, output rates, etc.) are adjusted to match demand requirements
over the planning horizon. A chase strategy works best when inventory carrying costs are high
and costs of changing capacity are low.
Advantages:
• Investment in inventory is low.
• Labor utilization is kept high.
Disadvantage:
• The cost of adjusting output rates and/or workforce levels.
59.
(a) What is meant by aggregate production planning? What are the inputs and
outputs of an aggregate production plans? Ans:
Aggregate planning is essentially a “big-picture” approach to planning. Aggregate planning
begins with a forecast of aggregate demand for the intermediate range.This is followed by a
general plan to meet demand requirements by setting output, employment, and finished-goods
inventory levels or service capacities. Managers might consider a number of plans, each of
which must be examined in light of feasibility and cost. If a plan is reasonably good but has
minor difficulties, it may be reworked. Conversely, a poor plan should be discarded and
alternative plans considered until an acceptable one is uncovered. The production plan is
essentially the output of aggregate planning. Aggregate plans are updated periodically, often
monthly, to take into account updated forecasts and other changes. This results in a rolling
planning horizon (i.e., the aggregate plan always covers the next 12 to 18 months).
Aggregate planning inputsand outputs:
60.
(b)Briefly explain the strategies generally followed for aggregate production planning.
10 Ans:
Aggregate planning strategies can be described as
proactive, reactive, or mixed. Proactive strategies
involve demand options: They attempt to alter
demand so that it matches capacity. Reactive
strategies involve capacity options: They attempt
to alter capacity so that it matches demand. Mixed
strategies involve an element of each of these
approaches.
61
Ans:
34. (c) Prepare a schedule where the forecast is 90 units for each of the first three periods and
70 units for each of the next three periods. The starting inventory is 20 units. The company
uses a chase strategy for determining the production lot size, expect there is anupper limit on
the lot size of 80 units. Also. The demand safety stock in 10 units. Note: The ATP quantities
are based on maximum allowable production and do not include safety stock. Committed
orders are as follows:
Solution to Ch-11
8.
(a) Why is there a need for aggregate planning? 05 Chapter 11
32 (a) What is agreement (Aggregate) planning? What is its purpose?
pg-537
05 (ch -11)
10. (c) Manager TC Downs of plum engines, a procedure of lawn mowers and leaf blowers,
must 18 develop an aggregate plan given the forecast for engine demand shown in the
table. The department has a normal capacity of 130 engines per month. Normal output has
a cost of 560 per engine. The beginning inventory is zero engine. Overtime has a cost of
$90 per engine. Develop a chase plan that matches the forecast and compute the total cost
of your plan
18 Chapter 11 (exercise 5, page 503)
22. (c) Wormwood limited, produces a variety of furniture products. The planning committee
wants to prepare an aggregate plan for the next six months using following information:
Subcontracting can handle a maximum of 10 units per month. Beginning inventory is zero.
Develop a plan that minimizes total cost. No back orders are allowed. Chapter 11 (Problem
9 math) NOT SURE!
33. (b) Manager T.C Downs of plum engines, a procedure of lawn mowers and leaf blowers,
must develop an aggregate plan given the forecast for engine demand shown in the table.
The department has a normal capacity of 130 engines per month. Normal output has a cost
of $60 per engine. The beginning inventory is zero engine. Overtime has a cost of $ 90 per
engine. (ch- 11, exercise 5, page 503)
20
Similar math solution for question 34.
34. (c) Prepare a schedule where the forecast is 90 units for each of the first three periods
and 70 units for each of the next three periods. The starting inventory is 20 units. The
company uses a chase strategy for determining the production lot size, expect there is
anupper limit on the lot size of 80 units. Also. The demand safety stock in 10 units. Note:
The ATP quantities are based on maximum allowable production and do not include safety
stock. Committed orders are as follows: (ch- 11, similar of exercise 22, page 541)
10
48.Solution
49.
Michigan Manufacturing produces a product that has a 6-month demand cycle. Each unit
requires 10 worker-hours to produce, at a labor cost of $6 per hour regular rate (or $9 per
hour over time). The total cost per unit is estimated at $200, units can be subcontracted at a
cost of $208 per unit. There are currently 20 workers employed in the subject department,
and hiring and training costs for additional workers are $300 per person, where layoff costs
are $400 per person. Company policy is to retain a safety stock equal to 20 percent of the
monthly forecast, and each month’s safety stock becomes the beginning inventory for the
next month. There are currently 50 units in stock carried at a cost of $2 per unit-month. Unit
shortage, or stock outs has been assigned a cost of $20 per unit month.
Three aggregate plans are proposed. Plan 1: Vary work force size to accommodate demand.
Plan 2: Maintain constant workforce of 20, and use over time and idle time to meet demand.
Plan 3: Maintain constant workforce of 20, and build inventory or incur stock out cost. The firm
must begin January with the 50-unit inventory on hand. Compare the costs of the three plans in
table form.
P.T.O
Chapter-16
11. (a) Briefly describe the following priority rules (0) FCFS (0) SPT ()
EDD (iv) 50 (v) Rush
First come, first served (FCFS): Jobs are processed in the order in which
they arrive at a machine or work center. Shortest processing time
(SPT): Jobs are processed according to processing time at a machine or
work center, shortest job first. Earliest due date
(EDD): Jobs are processed according to due date, earliest due date
first. Critical ratio (CR): Jobs are processed according to smallest ratio
of time remaining until due date to processing time remaining. Slack
per operation
(S/O): Jobs are processed according to average slack time (time until
due date minus remaining time to process). Compute by dividing slack
time by number of remaining operations, including the current one.
Rush: Emergency or preferred customers first
23. (a) What is meant by job shop scheduling?
Job-shop scheduling Scheduling for low-volume systems with many
variations in requirements. The characteristics of low-volume systems
(job shops) are considerably different from those of high- and
intermediate-volume systems. Products are made to order, and orders
usually differ considerably in terms of processing requirements,
materials needed, processing time, and processing sequence and
setups. Because of these circumstances, job-shop scheduling is usually
fairly complex. This is compounded by the impossibility of establishing
firm sched ules prior to receiving the actual job orders. Job-shop
processing gives rise to two basic issues for schedulers: how to
distribute the workload among work centers and what job processing
sequence to use.
12. (b) Differentiate between job-shop scheduling and flow-shop scheduling Why scheduling is fairly
complex for job shops systemis?
A group of five jobs us to be processed through a two-machine how shop Use Johnson's
rule to obtain the optimum sequence that will mununize the total completion time for thes
group of jobs Also draw the Giant chart for this sequence and calculate percentage of idle
time for each machine
arrival.
ii) For each of the methods in part (i), determine, the average job flow time, the
average
tardiness
iii)
Is one method
and the superior
number of
to jobs
the others?
at the work
Explain.
center.
The following table contains information concerning four jobs that are awaiting
37
79
24
processing at a work center.
i) Sequence the jobs using FCFS, SPT, EDD andCR. Assume the list is by order of
1
24
24
6
2.27
37
work centers.
ii) Construct a chart of the activities at the two centers and
determine each
one's idle time, assuming no other activities are involved.
Given the operation times provided:
i) Develop a job sequence that minimizes idle time at the two
Consider the following 3 machines and 5 jobs flow shop problem. Check whether
Johnson’s
rule can be extended to this problem. If so, what is the schedule and the
corresponding
makespan?
16
(1)
5
5
52
The following table contains order-dependent setup times for four-jobs. For safety
reasons,
Job C cannot follow job A, nor can job A follow job C. Determine the processing
sequence
that will minimize the total setup time
A-D-B-C
A-B-D-C
2+4+3+3 = 12
2+5+2+6 = 15
processing. sequence. using i) ·FCFS ii) SPT and iii) EQD.
For each method find the average
job flow time and the 'average job tardiness.
. Given the following information on job times and due dates,
determine the optimal
Solution to Ch-16
24. (b) What are the Gantt charts? How are they used in scheduling?
16 (exercise 3) Pg 740,741
10 Chapter
23. a) What are the basic assumptions of the assignment method of linear programming?
10
11. (a) Why are priority rules needed?
25. (c) The times required to complete each of eight jobs in a two-machine flow shop are
shown in the following table. Each job must follow the same sequence beginning with
machine A and moving to machine B. 15 Chapter 16 (Problem 10 math)
(i) Determine a sequence that will minimize make span time.
(ii) Construct a chart of the resulting sequence, and find machine B's idle time.
63. A group six jobs is to be processed through a two-machine flow shop, The first operation
involves cleaning and 'the second involving painting. Determine a sequence that will minimize the
total completion time for this group of jobs, idle time for each work center. Processing times arc
as follows:
(11th edition, Chapter-16, Page-754, Example-04)
-
36 37 51sim 52 64
12. Why scheduling is fairly complex for job shops systemis?
10
50. What are the main decision areas of job-shop scheduling?
Job-shop processing consists of 2 main decision areas for schedulers: how to distribute the
workload among work centers and what job processing sequence to use.
Loading: Loading refers to the assignment of jobs to processing (work) centers. Loading
decisions involve assigning specific jobs to work centers and to various machines in the work
centers.
Sequencing Determining the order in which jobs at a work center will be processed
45. (d) A text book publishing company has complied data on total annual sales of its business
texts for the preceding nine years:
15
.
i) Using an appropriate model, forecast textbook sales for each of the next five years
ii) Prepare a control chart for the forecast using the original data. Use 2s limits
ili) Suppose actual sales for the next five years turn out as follows:
Is the forecast performing adequately? Explain.
=
9×2756.1−45×503
=
9∗285−45×45
=4.01
503 − 4.01 × 45
= 35.833
9
The equation: 𝑦 = 𝑎 + 𝑏𝑥
𝑦 = 35.833 + 4.01𝑥
Here, x = number of years, so forecast for next five years is –
Period
Forecast(𝒚 = 𝟑𝟓. 𝟖𝟑𝟑 + 𝟒. 𝟎𝟏𝒙)
10
75.98
11
79.99
12
84.02
13
88.04
14
92.05
Formulase = Demand - Forecast
|e| = Absolute error
e2 = e*e
MSE (F1) = Total of e2 / (Total time period-1)
= 15.1665 / (9-1)
= 1.89
s = √1.89
= 1.376
Control limits are : 0± 2(1.376)
= 0 ± 2.75