Nahmias+Chapter+3+Solutions

Nahmias Chapter 3 : Answers for Selected Problems 3.9

Cum Net Demand Forecast (in 1000)

a)

Year

#Units/ Worker (in 1000)

1 2 3 4 5

Cum #Units /Worker (in 1000)

30 30 30 30 30

Net Demand Forecast (in 1000)

30 60 90 120 150

280 120 200 110 135

Min # Workers

280 400 600 710 845

10 7 7 6 6

Minimum constant work force = 10 workers b) #Units/ Worker

Year 1 2 3 4 5

Yearly Production

30 30 30 30 30

300 300 300 300 300

Cum Yearly Production

Cum Net Demand

300 600 900 1200 1500

Ending Inventory

280 400 600 710 845

20 200 300 490 655 1665

3.10

a) CH = 500, CF = 1,000, CI = .04 per package and payroll costs are $25,000 per year per worker. There are exactly 10 - 3 = 7 workers hired in year 1. Hence the total cost of the constant work force plan is (500)(7) + (.04)(1665000) + (10)(25,000)(5) = $1,320,100.

Produce 280 in periods 1 and 2 then decrease production to (845-560)/3 = 95 in period 3.

Year 1 2 3

Ideal Production 280 280 95

Number Workers 10 10 3

Actual Production 300 300 90

Cum Production 300 600 690

Cum Net Ending Demand Inv. 280 400 600

20 200 90

4 5

95 95 Totals

b)

3 3

90 90

780 870

710 845

70 25

29

405

This plan requires the hiring of 7 workers in period 1 and firing 7 workers in period 3. The cost of the plan is: Hiring and firing = (500)(7) + (100)(7) = $10,500 Payroll = (29)(25,000) = $725,000 Holding = (.04)(405,000) = $16,200 ___________________________________________________ Total Cost Number Year Workers 1 2 3 4 5 Totals

10 10 2 2 5 29

= $751,70

#Hired

#Fired

7 0 0 0 3 10

0 0 8 0 0 8

Cum Net Ending Production Demand 300 600 660 720 870

280 400 600 710 845

Inv.

.

20 200 60 10 25 315

Hiring and firing cost = (500)(10) + (1000)(8) = $13,000 Payroll cost = (29)(25,000) = $725,000 Holding cost = (.04)(315,000) = $12,600 Total cost = $750,600 Note that this is not the only solution possible. An improved solution drops the number of workers in period 2 to 5 and in period 5 to 4. The cost of this solution is slightly under $740,000.

3.11

Again assuming a constant work force size of 10 workers, we obtain Year 1 2 3 4 5

Yearly Production 300 300 300 300 300

Yearly Net Demand 300 120 200 110 135

Ending Inventory (Disposal) 0 180 100 190 165 Total = 635

Cost of plan = (7)(500) + (.20)(635,000) + (10)(5)(25,000) = $1,380,500

3.13

k = 1700/120/46 = 0.308 thousands of cookies per worker per day.

Working Month Days 1 2 3 4

Cum Cum #Units #Units Forecast Forecast *Monthly /Workers /Workers Demand Demand #Min Product (000 (000) (000) (000) Workers (000)

26 24 20 16

8.01 7.39 6.16 4.93

8.01 15.40 21.56 26.49

850 1260 510 980

850 2110 2620 3600

107 138 122 136

1105 1020 850 680

*Cum Monthly Product (000) 1105 2125 2975 3655 Total

*End Inv. (000 255 15 355 55 680

*Note: These three columns assume a work force of 138 workers in every period a) Minimum constant work force = 138 workers b) CH = 100

CF = 200

CI = 0.1 per cookie per month

Initial # workers = 100 # workers hired = 138 - 100 = 38

Beg. inv. = 0 End inv. = 680

Total cost of this plan is (100)(38) + (.1)(680,000) = $71,800

3.14

k = $60,000/250 = $240 per worker per day

#Units Working /Worker Mo Days ($000) 1 2 3 4 5 6 7 8 9 10

22 16 21 19 23 20 24 12 19 22

5.28 3.84 5.04 4.56 5.52 4.80 5.76 2.88 4.56 5.28

(A) Cum #Units /Worker ($000) 5.28 9.12 14.16 18.72 24.24 29.04 34.80 37.68 42.24 47.52

(B) Predict Cum Net Net Demand Demand ($000) ($000) 3280 3800 2200 1000 4900 6250 3750 3100 1750 1450

3280 7080 9280 10280 15180 21430 25180 28280 30030 31480

[B/A] Min # *Monthly Workers Product ($000) ($000) 622 777 656 550 627 738 724 751 711 663

4102.56 2983.68 3916.08 3543.12 4289.04 3729.60 4475.52 2237.76 3543.12 4102.56

*Cum Monthly Product ($000)

*End Invent. ($000)

4102.56 7086.24 11002.32 14545.44 18834.48 22564.08 27039.60 29277.36 32820.48 36923.04

822.56 6.24 1722.32 4265.44 3654.48 1134.08 1859.60 997.36 2790.48 5443.04

11 12

20 16

4.80 3.84

52.32 56.16

1200 1750

32680 34430

625 614

3729.60 2983.68

40652.64 43636.32

7972.64 9206.32

Total $39,874.56

*Note: These figures assume the minimum constant workforce of 777 workers each month. a) Minimum constant work force = 777 workers b) CH = 200

CF = 400

Initial # workers = 675 Workers added = 102 Total ending inventory

Beg. inv = 120000 End inv = 100000

= 39,874,560 + 100,000 = $39,974,560

Inventory costs per month = .25/12 = . 020833/$/month. Hence total hiring + inventory costs for the constant work force plan are (200)(102) + (.020833)(39,974,560) = $853,190 3.15

Mo. 1 2 3 4 5 6 7 8 9 10 11 12

#Units /Worker (000) 5.28 3.84 5.04 4.56 5.52 4.80 5.76 2.88 4.56 5.28 4.80 3.84

Predict Net Monthly Demand Min # Workers Workers Product (000) Workers Hired Fired (000) 3280 3800 2200 1000 4900 6250 3750 3100 1750 1450 1200 1750

622 990 437 220 888 1303 652 1077 384 275 251 456 Totals

CH = 200

0 368 0 0 668 415 0 425 0 0 0 205

53 0 553 217 0 0 651 0 693 109 24 0

2081

2300

3284.16 3801.60 2202.48 1000.32 4901.76 6254.40 3755.52 3101.76 1751.04 1452.00 1204.80 1751.04

Cum Monthly Product 3284.16 7085.76 9288.24 10291.44 15193.20 21447.60 25203.12 28304.88 30055.92 31507.92 32712.72 34463.76

Cum Net End Demand Invent. (000) (000) 3280 7080 9280 10280 15180 21430 25180 28280 30030 31480 32680 34430

4.16 5.76 8.24 11.44 13.20 17.60 23.12 24.88 25.92 27.92 32.72 33.76 228.72

CF = 400

Total Cost of hiring, firing, and inventory = (2081)(200) + (2300)(400) + (.020833)(328,720) = $1,343,048.20.

3.20

a) 12 12 12   * Min 200  Ht  400  Ft  208.33  It   t 1  t 1 t 1

Subject to (A)

W1 - W0 - H1 + F1 = 0 W2 - W1 - H2 + F2 = 0    W12 - W11 - H12 + F12 = 0

(B) *P1 P2

(D)

3.22

P3

= .5280W1 = .3840W2 = .5040W3

P4 P5 P6 P7 P8 P9 P10

= = = = = = =

P11

= .4800W11

P12

= .3840W12

I0 W0

(C)

P1 - I1 + I0 = 340* P2 - I2 + I1 = 380 P3 - I3 + I2 = 220

.4560W4 .5520W5 4800W6 .5760W7 .2880W8 .4560W9 .5280W10

= 12 = 675

P4 - I4 + P5 - I5 + P6 - I6 + I5 = P7 - I7 + I6 = P8 - I8 + I7 = P9 - I9 + P10 - I10 + I9

I3 = 100 I4 = 490 625 375 310 I8 = 175 = 145

P11 - I11 + I10 = 120 P12 - I12 + I11 = 165

(E) Ht, for

Ft , It, Wt, 1  t  12

Pt , 

0

a) Month 1 2 3 4 5 6

Belts(#) 2500 2800 2000 3400 3000 1600

Belts(hrs) 5000 5600 4000 6800 6000 3200

Hand(#) 1250 680 1625 745 835 375

Hand(hrs) 3750 2040 4875 2235 2505 1125

Att. Case# 240 380 110 75 126 45

Att. Total Case(hrs) Hrs. 1440 2280 660 450 756 270

10190 9920 9535 9485 9261 4595

b) Month

Working Days

1 2 3 4 5 6

22 20 19 24 21 17

Working Hours

Cum Working Hours

154 140 133 168 147 119

154 294 427 595 742 861

Cum Demand Hours

Min. Workforce (Ratio)

10190 20110 29645 39130 48391 52986

67 69 70 66 66 62

Hence the minimum constant work force size is 70. It is probably not advantageous to bring the work force level up to 70 workers. The excess demand can be absorbed by overtime or by employing part-time workers. The firm can be more flexible by keeping the number of full-time employees to a minimum. c) If a plan is to utilize only regular time employees, then excess demand must be absorbed by overtime. Since there are 46 employees with the firm, we obtain (A) Month

(B) Working Hours

1 2 3 4 5 6

154 140 133 168 147 119

(C=46*B) Regular Hours 7084 6440 6118 7728 6762 5474

Totals:

(D) Required Hours

(E=D-C) Overtime Hours

10,190 9,920 9,535 9,485 9,261 4,595

3106 3408 3417 1757 2499

39,606

14,259

Cost of plan = (39,606)(8.50) + (14,259)(14.00) = 336,651 + 199,626 = $536,277 d) Here we will determine the size of the work force necessary to meet monthly demands as closely as possible and add additional employees to meet the excess demand. (A) Month 1 2 3 4 5 6

(B) Demand Hours 10,190 9,920 9,535 9,485 9,261 4,595

(C) (D=[B/C]) (E) Working Regular Hour Ratio Workers 154 140 133 168 147 119

67 71 72 57 63 39

46 46 46 46 46 46

(F=D-E) Add'l Workers 21 25 26 11 17 0

Total:

(G=F*C) Total Hrs. Add. Emply. 3234 3500 3458 1848 2499 _______ 14,539

From part (c), regular employees cost $336,651. Additional employees cost (11)(14,539) = $159,929. Total cost = $336,651 + $159,929 = $496,580

3.23

a) The only decision variables here are the number of hours of overtime versus the number of additional employees to be hired. The objective is to minimize the total additional payroll cost. Let Ot = # overtime hours in period t. Nt = # additional employees hired in period t. 6

Min 14.00

0 t 1

t



 11.00 154N 1  140N 2  133N 3  168N 4  1147N 5  119N 6



Subject to: O1 + 154N1  10,190 - 7084 = 3106 O2 + 140N2  9920 - 6440 = 3408 O3 + 133N3  9535 - 6118 = 3417 O4 + 168N4  9485 - 7728 = 1757 O5 + 147N5  9261 - 6762 = 2499 O6 + 119N6  4595 - 5474 < 0 N1, N2,...,N6  30 3.29

a) An obvious choice here is the total number of students graduating each year. However, some academic programs are clearly more popular than others and some academic departments might require a minimum "critical mass" of faculty even if there are few student majors. b) In this case production could be measured by the total number of bookings made by the agency. If the company earns its commission on percentage basis, then the total dollar volume of bookings could be a better measure. c) Here an aggregate unit of production would be a can of fish. The firm would probably define an average can of fish by forming the weighted average of the labor hours and costs to produce each fish type where the weights would correspond to the proportion of total demand represented by each type. The procedure would be similar to that of Example 3.1.

3.32

a) Quarter 1

Unit/Worker (000) 1

Net Demand

Cum Net Dem.

300

300

Min. Work Force 300

2 3 4

1 1 1

630 220 180

930 1150 1330

465 384 333

Hence the min. constant workforce is 465 workers. The cost of the resulting plan is:

Quarter

Cum Production (000)

1 2 3 4

465 930 1395 1860

Cum Net Demand (000)

Ending Inventory (000)

300 930 1150 1330

165 0 245 530

Total

940

We must also add back in the 20,000 required to be on hand in the fourth quarter. Hence the total cost of this plan is: (1,200)(465 - 280) + (1000)(940 + 20) = $1,182,000. This is (1)(1000) since production and demand are expressed in (000) of cans. b) Quarter

Units/ Worker (000)

1 2 3 4

1 1 1 1

Predicted Net Demand Workers 300 630 220 180

300 630 220 180 Totals

Hires

Fires

20 330 0 0

0 0 410 40

350

450

Ending Inv. 0 0 0 0

Total cost = (1,200)(350) + (2500)(450) + 20,000 = 1,565,000

c) If we use the minimum number of workers required through period 3 of 1150/3 = 384, it will satisfy the conditions stated.

Quarter

Cum Prod

Cum Net Dem.

Ending Inv.

d) 1 2 3 4

384 768 1152 1536

300 930 1150 1330

84 -162 2 206

Total cost = (1,200)(384 - 280) + (1,000)(312) + (2,000)(162) = $760,800. Better than policies in parts (a) or (b).

3.38 a) 6

Min

 125 H t 1

t

 300 Ft  .75 It



Sub to: (A)

W1 - H1 + F1 = W0 = 86 W2 - W1 + H2 + F2 = 0    W6 - W5 + H6 + F6 = 0

(B)

P1 - 50.77 W1 = 0 P2 - 101.54 W2 = 0 P3 - 92.31 W3 = 0 P4 - 106.15 W4 = 0 P5 - 73.85 W5 = 0 P6 - 92.31 W6 = 0

(C)

I1 - P1 = I0 - D1 = -4000 I2 - I1 - P2 = -9300 I3 - I2 - P3 = -12200 I4 - I3 - P4 = -17600 I5 - I4 - P5 = -14000 I6 - I5 - P6 = -6300

(D)

It  1000 1  t  5 I6  3000

(E)

Ht, Ft, It, Wt, Pt  0

1  t  6.