Question

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Problem 4-19 (Algorithmic) The Silver Star Bicycle Company will be manufacturing both men's and women's models of its Easy-Pedal 10-speed bicycles during the next two months. Management wants to develop a production schedule indicating how many bicycles of each model should be produced in each month. Current demand forecasts call for 160 men's and 135 women's models to be shipped during the first month and 210 men's and 160 women's models to be shipped during the second month. Additional data are shown: Production Labor Requirements (hours) Current Manufacturing Model Assembly Inventory Costs Men's $130 1.5 2.0 30 Women's $100 1.3 1.3 40 Last month the company used a total of 10000 hours of labor. The company's labor relations policy will not allow the combined total hours of labor (manufacturing plus assembly) to increase or decrease by more than 100 hours from month to month. In addition, the company charges monthly inventory at the rate of 2% of the production cost based on the inventory levels at the end of the month. The company would like to have at least 35 units of each model in inventory at the end of the two months. a. Establish a production schedule that minimizes production and inventory costs and satisfies the labor-smoothing, demand, and inventory requirements. What inventories will be maintained, and what are the monthly labor requirements? If required, round your answers to the nearest whole number. Production schedule: of to produce men's model month n's in mon TI 2 X 210 165 95 195 X of the women's model in month 2 units of the women's model in month 1, and Total Cost $ 78120 Inventory Schedule VMen's X Women's 80 0 Month 1 V Men's Women's 35 35 Month 2 Labor Levels V hours 1000 Previous month X hours 983 Month 1 Vhours 1083 Month 2 b. If the company changed the constraints so that monthly labor increases and decreases could not exceed 50 hours, what would happen to the production schedule? If required, round your answers to the nearest whole number. Production schedule: x of the men's model in month 2, The optimal solution is to produce of the men's model in month 1, 218 158 95 units of the women's model in month 1, and of the women's model in month 2 195 Total Cost $ 78139 How much will the cost increase? 19

Answer 1

1.

Decision variables

Let

x11 =Number of men's model in month 1

x21 = Number of women's model in month 1

x12 = Number of men's model in month 2

x22 = Number of women's model in month 2

s11 = men's model inventory units at end of month 1

s21 = women's model inventory units at end of month 1

s12 = men's model inventory units at end of month 2

s22 = women's model inventory units at end of month

Objective

Minimize cost

Z=( 130x11 + 100x21 + 130x12 + 100x22) +

  

Constraints

Satisfy Demand in month 1

30+ x11 - s11 = 160

or

x11 - s11 = 130             ...1

40 + x21 - s21 = 135

or

x21 - s21 = 95   ...2

Satisfy Demand in month 2

s11 + x12 - s12 = 210               ...3

s21 + x22 - s22 = 160                ...4

Ending inventory at the end of month 2 constraints

s12 ≥ 35 ...5

s22 ≥ 35 ...6

Labor Hours constraints (month 1)

In the previous month 100 hrs are used .so in month 1, this can be either

• 1000-100=900 or
• 1000+100=1100

3.5 x11 + 2.6 x21 ≥ 900 ...7

3.5 x11 + 2.6 x21 ≤ 1100 ...8

Labour Hours constraints (month 2)

In the previous month (month 1), labour hrs used = 3.5 x11 + 2.6 x21

So in month 2,

The decrease can be up to 100

3.5 x11 + 2.6 x21 - 3.5 x12 - 2.6 x22 ≤ 100 ...9

The increase   can be up to 100

3.5 x12 + 2.6 x22 -3.5 x11 - 2.6 x21 ≤100

Or

-3.5 x11 - 2.6 x21 + 3.5 x12 + 2.6 x22 ≤ 100 ...10

Non-negativity constraint

x11, x12, x21, x22, s11, s12, s21, s22 > 0 ...11

solution

Production Schedule	Men's Model	Women's Model
Month 1	210.0	95.0
Month 2	165.0	195.0
Invnetory Schedule	Men's Model	Women's Model
Month 1	80.0	0.0
Month 2	35.0	35.0

labor hrs
Month 1	983.0
Month 2	1083

Security Warning Automatic update links has been disabled Options.. G9 H C E F G K 1 2 3 month 2 4 month 1 5 x11 x21 x22 x12 6 s11 s21 s12 s22 9 10 11 (130°G6+100H6+13016+100 J6)+(2.6 G9- 12 Obective 13 14 Constraints 15 1 -G6-G9 130 16 2 H6-H9 95 H9+J6- J9 18 160 19 5 35 20 6 19 35 21 -3.5*G6+2.6*H6 7 900 22 R 3.5*G6+2.6#H6 = 3.5"G6+2.6"H6-3.5 16-2,6 16 3.5*G6-2.6 H6+3.5*16+2.6 *J6 1100 23 100 24 10 100 25

, Clean min z Connections Properties Reapply From From From From Other Refresh Model Solve Existing Connections Filter Text to Remove Data Consolidate What-If Group Ungroup Subtotal Advanced Edit Links Columns Duplicates Validation Access Web All: Analysis Tex X t External Data OpenSolver Model Connections Op AutoModel What is AutoModel? Security Warning Automatic update of links has been disabled Options.. AutoModel is a feature of OpenSolver that tries to automatically determine the problem you are trying to optimise by observing the structure of the spreadsheet. It will turn its best quess into a Solver model, which you can then edit in this window. S10 objective Cell:SG$12 C target value: C maximise E P G H J minimise 1 Variable Cells: 2 SG$6: $)56.SG $9:$3$9 3 month 1 month 2 5 x11 x21 x12 22 6 Constraints: 7 G$15:4G$18 SI$15:$1S18 $G$19:$G$21> $I$ 19 : $I $21 $I$22:$I$24 s11 s21 s12 s22 9 10 G46:$1$6>-0 11 SG$9:$3$9>-0 Cancel 0.0 Add constraint 12 Obective 13 14 Constraints Delete selected constraint 0.0= 130 16 Make unconstrained variable cels non-negative 17 0.0= 210 Show named ranges in constraint list 18 0.0 4 160 19 0.0 2 35 Sensitivity Analysis List sensitivity analysis on the same sheet with top left cel: 20 0.0 2 6 35 updating any previous output sheet Con a new sheet 21 7 0.0 2 900 Output sensitivity analysis: 22 0.0 s 1100 Solver Engine 23 0.0 s Current Solver Engine: CBC 100 Solver Engine.. 24 10 0.0 s 100 25 Show model after saving Clear Model Save Model Options.. Cance 26

Paste U a Merge & Center Format Painter Alignment Clipboard Font Options... Security Warning Automatic update of links has been disabled S10 F H J K L M N 1 2 3 month 1 4 month 2 5 x11 x21 x12 x22 210.4 95.0 164.6 195.0 7 8 s11 s21 s12 s22 80.4 0.0 35.0 35.0 10 11 78119.9 12 Obective 13 14 Constraints 15 1 130.0 130 16 95.0 95 17 210.0 3 210 18 4 160.0 160 19 5 35.0 2 35 20 35.0 2 35 21 983.2 2 7 900 22 8 983.2 1100 23 9 -100.0 100 24 100.0 s 10 100 25 N sth LL

as

2

2.

For the new policy

The RHS of constraints 7,8,9,10 Is changed to 950, 1050, 50, and 50 respectively.

Production Schedule	Men's Model	Women's Model
Month 1	218	95
Month 2	158	195.0

Cost increase =78138.5 – 78119.9=$18.6

AV GEdit Links Advar Access Web Text Sources Connections All Get External Data Connections Sort & Filter Security Warning Automatic update of links has been disabled Options... R25 E G H K N 4 month 1 month 2 5 x11 x21 x12 x22 6 217.5 95.0 157.5 195.0 7 8 s21 s12 s11 s22 9 87.5 0.0 35.0 35.0 10 11 78138.5 12 Obective 13 14 Constraints 15 1 130.0 130 16 2 95.0 95 17 3 210.0 210 18 160.0 160 4. 19 35.0 2 35 20 35.0 2 35 21 1008.3 2 7 950 22 8 1008.3 s 1050 23 -50.0 s 50 24 10 50.0 s 50 n ト 00の9

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