Name:______ID:______

Operations Management I 73-331-91 (Distance)

Winter 2003 Final Exam Solution

Tuesday, April 15, 8:30 – 11:30 a.m.

Instructor: Mohammed Fazle Baki

Aids Permitted: Calculator, straightedge, and 3 one-sided formula sheets.

Time available: 3 hours

Instructions:

  • This exam has 32 pages including this cover page, 1 blank page and 8 pages of Table
  • It’s not necessary to return the tables and unused blank pages
  • Please be sure to put your name and student ID number on each odd numbered page
  • Show your work

Grading:

Question / Score / Question / Score
1 / /15 / 2 / /8
3 / /15 / 4 / /10
5 / /12 / 6 / /6
7 / /6 / 8 / /8
9 / /8 / 10 / /4
11 / /8 / Total / /100

Question 1: (15 points) Circle the most appropriate answer

1.1Jidoka

  1. prevents defects
  2. lights signal quality problem
  3. is the authority to stop production line
  4. makes problems visible

1.2 Which of the following is false about lot sizing heuristic procedures?

  1. Lot for lot minimizes carrying cost
  2. Lot for lot maximizes ordering cost
  3. Silver-Meal heuristic and least unit cost perform best if the costs change over time
  4. Silver-Meal heuristic and least unit cost perform best if the costs do not change over time

1.3Material requirements planning is used for

  1. dependent demand and for assembly
  2. dependent demand and for both assembly and manufacturing
  3. independent demand and for assembly
  4. independent demand and for both assembly and manufacturing

1.4We find an optimal policy subject to some service constraint because it’s difficult to

  1. estimate a penalty cost,
  2. estimate lead time
  3. find an optimal without any service constraint
  4. find an optimal without any service constraint

1.5Backorder is charged if

  1. production is initiated by customer order
  2. production is initiated by demand forecast
  3. the excess demand is lost because the customer goes elsewhere
  4. the excess demand is backlogged and fulfilled in a future period

1.6A rotation cycle policy is used

  1. to solve the lot sizing problem with capacity constraint
  2. if multiple products are manufactured using the same production facility
  3. to minimize hiring and firing when seasonal products are manufactured
  4. if the unit production time decreases due to learning and experience

1.7Consider the all unit discount schedule. If a larger quantity is ordered,

  1. both the unit cost and total cost may decrease
  2. unit cost may decrease but the total cost increases
  3. both the unit cost and total cost may increase
  4. total cost may decrease but the unit cost increases

1.8The following assumption of the EOQ model is not used in the finite production rate model:

  1. Demand and cost parameters are known and fixed.
  2. Shortages are not permitted.
  3. The inventory increases instantaneously at one point of time when an order is received.
  4. There is no price discount or resource constraint
  5. The following are some characteristics of the EOQ model cost curve
  1. Total annual holding cost is the same as total annual ordering cost
  2. The total annual cost does not change much from the optimal value if the order quantity is near the EOQ value
  3. Both of the above
  4. None of the above

1.10Consider a seasonal demand series with an increasing trend. Compare multiplicative and additive forecasting series.

  1. There will be more fluctuation in the multiplicative series in the later years.
  2. There will be more fluctuation in the additive series in the later years.
  3. Both series will demonstrate the same fluctuation every year.
  4. Any of a, b, or c can be true.

1.11Which of the following is not a part of the firing cost?

  1. Severance pay
  2. The costs of a decline in worker morale
  3. The potential for decreasing the size of the labor pool in the future
  4. The time and cost to advertise positions

1.12The coefficient of correlation,

  1. varies from -100 to 100
  2. varies from -1 to +1
  3. demonstrates a weak or no relationship if is too small
  4. demonstrates a weak or no relationship if is too large

1.13Which of the following is false?

  1. Forecasts are usually wrong
  2. Long-term forecasts are less accurate than short-term forecasts
  3. Aggregate forecasts are less accurate than disaggregate forecasts
  4. Aggregate forecasts are more accurate than disaggregate forecasts

1.14Consider learning curve. Consider the straight line that best fits points. Which of the following is true?

  1. The intercept, is an estimate of the time required by the first unit.
  2. The log of intercept, is an estimate of the time required by the first unit.
  3. If the slope is less, then the learning is faster.
  4. If the slope is more, then the learning is faster.

1.15Economies of scale can be represented as where, is a constant and

Question 2: (8 points)

A start-up firm has kept careful records of the time required to manufacture its product, a shutoff valve used in gasoline pipelines.

Cumulative Number of Units Produced / Number of Hours Required for Next Units
5 / 26.2
15 / 14.9
25 / 11.5
75 / 6.5
  1. (2 points) Compute the logarithms of the numbers in each column.

Cumulative Number of
Units Produced
u / Number of Hours Required
For the Next Unit
Y(u) / Ln(u) / Ln(Y(u))
5 / 26.2 / 1.60943791 / 3.265759411
15 / 14.9 / 2.7080502 / 2.701361213
25 / 11.5 / 3.21887582 / 2.442347035
75 / 6.5 / 4.31748811 / 1.871802177
  1. (4 points) Estimate the time required to produce the first unit and the appropriate percentage learning curve that fits these data. Use an exact least squares fit of the logarithms computed in part a.

i / x
ln(u) / y
ln(y(u) / xy / x^2
1 / 1.609437912 / 3.26575941 / 5.256037009 / 2.590290394
2 / 2.708050201 / 2.70136121 / 7.315421776 / 7.333535892
3 / 3.218875825 / 2.44234704 / 7.861611828 / 10.36116158
4 / 4.317488114 / 1.87180218 / 8.08148365 / 18.64070361
Sum / 11.85385205 / 10.2812698 / 28.51455426 / 38.92569147
Average / 2.963463013 / 2.57031746

Time required to produce the first unit,

Rate of learning,

  1. (2 points) Estimate the time required to produce 130th unit.

= -slope = 0.5145

hours

Question 3: (15 points)

A popular brand of tennis shoe has had the following demand history by quarters over a two-year period.

Quarter
2001 / Demand / Quarter
2002 / Demand
1 / 35 / 1 / 47
2 / 50 / 2 / 58
3 / 55 / 3 / 64
4 / 45 / 4 / 56
  1. (6 points) Determine the seasonal factors for each quarter by the method of centered moving averages

N = / 4 / The demand is quarterly, there are 4 quarters in each year.
Period / Demand / MA(4) / Centered
MA / (B/D)
Ratio
A / B / C / D / E
1 / 35 / 49 / 0.714285714
2 / 50 / 49 / 1.020408163
3 / 55 / 47.75 / 1.151832461
4 / 45 / 46.25 / 50.25 / 0.895522388
5 / 47 / 49.25 / 52.375 / 0.897374702
6 / 58 / 51.25 / 54.875 / 1.056947608
7 / 64 / 53.5 / 53.625 / 1.193473193
8 / 56 / 56.25 / 53.625 / 1.044289044
Period / Seasonal
Factors / Final
Seasonal
Factors
1 / 0.80583021 / 0.8084
2 / 1.03867789 / 1.0420
3 / 1.17265283 / 1.1765
4 / 0.96990572 / 0.9731
Total / 3.98706664 / 4.0000

(Continued…)

  1. (6 points) Compute the deseasonalized demand series. Using the method of linear regression, determine the slope and intercept of the straight line that best fits the deseasonalized series.

/ Deseasonalized
Demand
/ /
1 / 43.29303212 / 43.29303212 / 1
2 / 47.98247238 / 95.96494476 / 4
3 / 46.75055139 / 140.2516542 / 9
4 / 46.24624736 / 184.9849894 / 16
5 / 58.13635741 / 290.6817871 / 25
6 / 55.65966796 / 333.9580078 / 36
7 / 54.40064162 / 380.8044913 / 49
8 / 57.5508856 / 460.4070848 / 64
Sum / 36 / 410.0198558 / 1930.345991 / 204
Average / 4.5 / 51.25248198


  1. (3 points) Predict the demand for the third quarter of 2003

Deseasonalized demand,

For the third quarter of 2002,

So, the deseasonalized demand,

To get the demand, reseasonalize,

Question 4: (10 points)

The Paris Paint Company is in the process of planning labor force requirements and production levels for the next four quarters. The marketing department has provided production with the following forecasts of demand for Paris Paint over the next year:

Quarter / Demand Forecast
(in thousands of gallons)
1 / 600
2 / 700
3 / 650
4 / 200

Assume that there are currently 250 employees with the company. Employees are hired for at least one full quarter. Hiring costs amount to $500 per employee and firing costs are $1,000 per employee. Inventory costs are $0.20 per gallon per quarter. It is estimated that one worker produces 1,750 gallons of paint each quarter. Assume that Paris currently has 300,000 gallons of paint in inventory and would like to end the year with an inventory of at least 100,000 gallons.

  1. (5 points) Determine the minimum constant workforce plan (i.e., level strategy) for Paris Paint. Assume that stock-outs are not allowed.

Quarter / Forecast
(000 gallons) / Beg/ End
Inventory
(000 gallons) / Production
Requirement
(000 gallons) / Cumulative
Production
Requirement
(000 gallons) / Cumulative
units
produced
per worker
(000 gallons) / E/F / Workers
Required
= Round up (G)
A / B / C / D / E / F / G / H
1 / 600 / 300 / 600-300=300 / 300 / 1.75 (given) / 171.4 / 172
2 / 700 / 700 / 300+700=1000 / 1.75+1.75 = 3.5 / 285.7 / 286
3 / 650 / 650 / 1000+650=1650 / 3.50+1.75 = 5.25 / 314.3 / 315
4 / 200 / 100 / 200+100=300 / 1650+300=1950 / 5.25+1.75 = 7 / 278.6 / 279

Since the maximum workers required is 315, the minimum constant workforce plan must use 315 workers. So, the number of workers to hire = 315 – 250 = 65 workers.

  1. (5 points) Determine the hiring, firing, and inventory holding cost of the plan derived in part a.

Quarter / Demand
= Forecast
(000 gallons) / Beginning Inventory
(000 gallons) / Production
(000 gallons) / Ending Inventory = Production + Beginning Inventory – Demand
(000 gallons)
1 / 600 / 300 (given) / 315(1.75)=551.25 / 551.25 + 300.00 – 600 = 251.25
2 / 700 / 251.25 / 551.25 / 551.25 + 251.25 – 700 = 102.50
3 / 650 / 102.50 / 551.25 / 551.25 + 102.50 – 650 = 003.75
4 / 200 / 003.75 / 551.25 / 551.25 + 003.75 – 200 = 355.00

Total ending inventory = (251.25+102.50+3.75+355.00) = 712.50 thousand gallons

Inventory holding cost = 712,500  0.20 = $142,500

Hiring cost = 65(500) = $32,500

Total cost = 142,500+32,500 = $175,000

Question 5: (12 points)

Suppose that Item A has a production rate of 720 items per year, unit cost of $10.00, a setup cost of $80, and a monthly demand of 30 units. It is estimated that cost of capital is approximately 15 percent per year. Storage cost amounts to 3 percent and breakage to 2 percent of the value of each item.

  1. (2 points) Compute EPQ of Item A.

EPQ=

units

  1. (3 points) What are the uptime, downtime and cycle time of Item A?

Cycle time, years

Uptime, years

Downtime, years

Item B has a production rate of 1200 items per year, a unit cost of $20.00, an ordering cost of $77.5, and a monthly demand of 25 units. Recall that the cost of capital is approximately 15 percent per year. Storage cost amounts to 3 percent and breakage to 2 percent of the value of each item.

  1. (3 points) What is the cycle time if both Items A and B are produced in a single facility?

years

  1. (4 points) What are the optimal order quantity and uptime of items A and B? Assume both items A and B are produced using the same facility.

Item A

units

Uptime = year

Item B

units

Uptime = year

Question 6: (6 points)

Green City sells a particular model of lawn mower, with most of the sales being made in the summer months. Green city makes a one-time purchase of the lawn mowers prior to each summer season at a cost of $150 each and sells each lawn mower for $225. The demand is normally distributed with a mean of 1500 and a standard deviation of 100. Find the optimal order quantity if

  1. (3 points) any lawn mower unsold at the end of summer season are marked down to $100 and sold in a special fall sale.

Purchase price – salvage value = $150-100=$50

Selling price – Purchase price = $225-150=$75

Find such that Or,

Or,

Hence, from Table A-1 (the -value for which area = 0.10)

  1. (3 points) any lawn mower unsold at the end of summer season are marked down to $25 and sold in a special fall sale.

Purchase price – salvage value = $150-25=$125

Selling price – Purchase price = $225-150=$75

Find such that Or,

Or,

Hence, from Table A-1 (the -value for which area = 0.125)

Question 7: (6 points)

The home appliance department of a large department store is planning to use a lot size-reorder point system to control the replenishment of a particular model of FM table radio. The store sells an average of 240 radios each year. The annual demand follows a normal distribution with a standard deviation of 50. The store pays $60 for each radio. The holding cost is 25 percent per year. Fixed costs of replenishment amount to $200. If a customer demands the radio when it is out of stock, the customer will generally go elsewhere. Replenishment lead-time is one month.

  1. (3 points) Find an optimal (Q,R) policy with probability(no stockout)=0.94.

Step 1:

Step 2: Find for which area on the left = =probability(no stockout) = 0.94

From Table A-1, for area = 0.94-0.50 = 0.44

From Table A-4, for

Step 3: Compute reorder point,

Hence an optimal policy is

  1. (2 points) Compute the annual holding cost resulting from the (Q,R) policy obtained in part a.

Annual holding cost, regular =

Safety stock = units

Annual holding cost, safety stock =

Annual holding cost == 600 + 336.60 = $936.60

  1. (1 point) Compute the annual ordering cost resulting from the (Q,R) policy obtained in part a.

Annual ordering cost =

Question 8: (8 points)

Consider Question 7 again. Find an optimal (Q,R) policy with fill rate = 0.98. Use the iterative method and show 2 iterations. Show your computation and summarize your results in the table below:

Iteration 1

Step 1:

Step 2:

(Table A-4)

Step 3: (Table A-4)

(Continued…)

Summary of results:

Fixed cost (K) / 200 / Note: K and h
Holding cost (h) / 15 / are input data
Mean annual demand (lambda) / 240 / input data
Lead time (tau) in years / 0.083333333 / input data
Lead time demand parameters:
mu / 20 / <--- computed
sigma / 14.43375673 / input data
Type 2 service, fill rate, beta / 0.98 / input data
Iteration 1 / Iteration 2
Step 1 / Q= / 80
Step 2 / n= / 1.6
L(z)= / 0.1108513
z= / 0.85
R= / 32.268693
Step 3 / Area on the right=1-F(z) / 0.1976625 / 0.214764
Step 4 / Modified Q= / 88.50308 / 88.66533
Step 5 / n= / 1.7700616 / 1.773307
L(z)= / 0.1226335 / 0.122858
z= / 0.79 / 0.79
R= / 31.402668 / 31.40267

Step 4:

(not near 80, more iterations are necessary)

Step 5:

(Table A-4)

Iteration 2

Step 3:

Step 4:

(same as before, stop the process after finding )

Step 5:

(Table A-4)

andconverge. An optimal policy is Q=89, R=31 (rounded to the nearest integer)

Question 9: (8 points)

Each unit of A is composed of two units of B and one unit of C. Items A, B and C have on-hand inventories of 20, 30 and 10 units respectively. Item B has a scheduled receipt of 50 units in period 1, and C has a scheduled receipt of 100 units in Period 1. Lot-for-lot (L4L) is used for Item A. Item B requires a minimum lot size of 50 units. Item C is required to be purchased in multiples of 100. Lead times are two periods for Item A, and one period for each Item B and C. The gross requirements for A are 30 in Period 3, 50 in Period 6, and 90 in Period 9. Find the planned order releases for all items to meet the requirements over the next 10 periods.

  1. (2 points) Construct a product structure tree.


  1. (2 points) Consider Item A. Find the planned order releases and on-hand units in period 10

Period

/ 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Item
A
LT=2
Q=
L4L / Gross Requirements / 30 / 50 / 90
Scheduled receipts
On hand from prior period / 20 / 20 / 20 / 0 / 0 / 0 / 0 / 0 / 0 / 0
Net
requirements / 10 / 50 / 90
Time-phased Net Requirements / 10 / 50 / 90
Planned order releases / 10 / 50 / 90
Planned order delivery / 10 / 50 / 90

(Continued…)

  1. (2 points) Consider Item B. Find the planned order releases and on-hand units in period 10.

Period

/ 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Item
B
LT=1
Q >= 50 / Gross Requirements / 20 / 100 / 180
Scheduled receipts / 50
On hand from prior period / 30 / 60 / 60 / 60 / 10 / 10 / 10 / 0 / 0 / 0
Net
Requirements / 40 / 170
Time-phased Net Requirements / 40 / 170
Planned order releases / 50 / 170
Planned order delivery / 50 / 170
  1. (2 points) Consider Item C. Find the planned order releases and on-hand units in period 10.

Period

/ 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Item
C
LT= 1
Q=
100 / Gross Requirements / 10 / 50 / 90
Scheduled receipts / 100
On hand from prior period / 10 / 100 / 100 / 100 / 50 / 50 / 50 / 60 / 60 / 60
Net
requirements / 40
Time-phased Net Requirements / 40
Planned order releases / 100
Planned order delivery / 100

Question 10: (4 points)

A single inventory item is ordered from an outside supplier. The anticipated demand for this item over the next 7 months is 10, 13, 11, 15, 14, 9, 6. Current inventory of this item is 2, and the ending inventory should be 1. Assume a holding cost of $3 per unit per month and a setup cost of $100. Assume a zero lead time. Determine the order policy for this item over the next 7 months.

Use the Silver-Meal heuristic.

Net requirements:

Months / Q / I1 / I2 / I3 / I4 / I5 / I6 / I7 / Holding Cost / Ordering Cost
1 to 1 / 8 / 0 / 100 / 100
1 to 2 / 21 / 13 / 39 / 100 / 69.5
1 to 3 / 32 / 24 / 11 / 105 / 100 / 68.33
1 to 4 / 47 / 39 / 26 / 15 / 240 / 100 / 85 stop
4 to 4 / 15 / 0 / 100 / 100
4 to 5 / 29 / 14 / 42 / 100 / 71
4 to 6 / 38 / 23 / 9 / 96 / 100 / 65.33
4 to 7 / 45 / 30 / 16 / 7 / 159 / 100 / 64.75
  1. (3 points) State your order policy:

MonthLot size to order

  1. 32
  2. 45
  1. (1 point) Using the table below, show the ending inventory that results from your order policy at the end of each month:

Month

/ 1 / 2 / 3 / 4 / 5 / 6 / 7
Gross Requirements / 10 / 13 / 11 / 15 / 14 / 9 / 6

Beginning Inventory

/ 2 / 24 / 11 / 0 / 30 / 16 / 7
Net Requirements / 8 / 15
Time-phased Net Requirements / 8 / 15
Planned order Release / 32 / 45
Planned Deliveries / 32 / 45
Ending Inventory / 24 / 11 / 0 / 30 / 16 / 7 / 1

Question 11: (8 points)

Consider Question 10 again.

(2 points) Suppose that the maximum order size is 12 per month. Does there exist a feasible solution? If there does not exist a feasible solution, what is first month when there will be a shortage?

Month / Production Requirement / Capacity / Cumulative Production Requirement / Cumulative capacity
1 / 10-2=8 / 12 / 8 / < 12
2 / 13 / 12 / 8+13=21 / < 12+12=24
3 / 11 / 12 / 21+11=32 / < 24+12=36
4 / 15 / 12 / 32+15=47 / < 36+12=48
5 / 14 / 12 / 47+14=61 / > 48+12=60 Shortage
6 / 9 / 12 / 61+9=70
7 / 6+1=7 / 12 / 70+7=77

Since the cumulative capacity is less than the cumulative production requirement in period 5, there is no feasible solution. There will be a shortage in Month 5.

(3 points) Suppose that the maximum order size is 13 per month. Use lot-shifting technique to obtain a feasible solution (without using holding and setup cost). Show your final solution in the table given below.

Month / Production Requirement / Actual Production / Production Capacity / Excess Capacity
1 / 10-2=8 / 8 9 / 13 / 5 4
2 / 13 / 13 / 13
3 / 11 / 11 13 / 13 / 2 0
4 / 15 / 15 13 / 13
5 / 14 / 14 13 / 13
6 / 9 / 9 / 13 / 4
7 / 6+1=7 / 7 / 13 / 6

First, observe that production requirement in Month 4 is 2 units more than the capacity. So, back-shift 2 units to Month 3 Again, production requirement in Month 5 is 1 unit more than the capacity. So, back-shift 1 unit to Month 1. Final production schedule is as follows:

A feasible solution obtained by lot-shifting technique:

Month / 1 / 2 / 3 / 4 / 5 / 6 / 7
Actual Production / 9 / 13 / 13 / 13 / 13 / 9 / 7

(3 points) Improve the solution obtained in Part . Assuming a maximum order size of 13 units per month and using the back-shifting technique, find another solution that has less total holding and setup cost than the solution obtained in Part . Show your final solution in the table given below.

Back-shift 7 units of Month 7?

Check if it’s better to backshift 4 units to Month 6 and 3 units to Month 1

Additional holding cost = 4(1)(3)+3(6)(3) = 66 < 100 = savings in ordering cost

Back-shift

Improved solution:

Month / 1 / 2 / 3 / 4 / 5 / 6 / 7
Actual Production / 12 / 13 / 13 / 13 / 13 / 13 / --

1