Winter_IE6315(HW3)_2024_SS
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Wayne State University *
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Industrial Engineering
Date
Apr 3, 2024
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WSU –
Industrial and Systems Engineering ISE 6315 Production Systems Instructor: Jeremy L. Rickli Assignment #3: News Vendor, Basestock, (Q,r) Department of Industrial and Systems Engineering Wayne State University Teams:
Teams of two (maximum –
no teams great than two allowed) Points:
10 points possible Deliverables:
Homework Submission in PDF format (submitted to Canvas –
Lastname(s)_HW1.pdf). Write all equations and derivations in Word using equation editor for one extra point (use this assignment file as a template for font size and formatting). 1.
(5) Slaq Computer Company manufactures notebook computers. The economic lifetime of a particular model is only four to six months, which means that Slaq has very little time to make adjustments in production capacity and supplier contracts over the production run. For a soon-
to-be-introduced notebook, Slaq must negotiate a contract with a supplier of motherboards. Because supplier capacity is tight, this contract will specify the number of motherboards in advance of the start of the production run. At the time of contract negotiation, Slaq has forecasted that demand for the new notebook is normally distributed with a mean of 10,000 units and a standard deviation of 3,500 units. The net profit from a notebook sale is $650 (note that this already includes the cost of the motherboard, as well as all other material, production, and shipping costs). Motherboards cost $250 and have no salvage value (i.e., if they are not used for this particular model of notebook, they will have to be written off) –
(Chapter 2 - #11).
a.
(5) Use the news vendor model to compute the optimal order quantity of motherboards that balances the cost of lost sales and the cost of excess material. b.
(5) Now assume that the distribution follows an exponential distribution with a mean of 0.0001. What is the optimal order quantity now?
Mean(μ) = 10000 units Standard Deviation(σ) = 3500 unit
Shortage Cost (C
s
) = 650 Overage cost (C
o
) = 250 Now, G(Q) would be 𝐺(𝑄) =
𝑐
𝑠
𝑐
𝑜
+ 𝑐
𝑠
𝐺(𝑄) = 650
650 + 250
= 0.72
By using Standard table (z-value table) to find
(0.58) = 0.72 Hence, z = 0.58 and Quantity (Q*) = Mean(μ) + Z * Standard Deviation (
σ)
= 10000 + 0.58*3500 = 12,030 units (answers may vary based on rounding or table used for z value)
WSU –
Industrial and Systems Engineering ISE 6315 Production Systems Instructor: Jeremy L. Rickli a.
(5) Comment on the appropriateness of the news vendor model for this capacity planning situation. What factors are not considered that might be important?
F 𝐺(𝑄) = 1 − 𝑒
−𝑄
∗
10000
=
𝑐
𝑠
𝑐
𝑜
+ 𝑐
𝑠
1 − 𝑒
−𝑄
∗
10000
= 0.72
ln 𝑒
−𝑄
∗
10000
= 0.28
𝑄
∗
= −10000 ∗ ln 0.28
𝑄
∗
= 12730
2.
Consider that a manufacturer has an inventory of two replacement parts for two pieces of equipment that often fails. Part 1 costs $175 and has demand of 8 per month. Part 2 costs $20 and has demand of 25 per month (a month is 30 days). The lead time for Part 1 is 30 days and for Part 2 is 15 days. Answer the following questions below using this information. a.
(2) Assume demand is Poisson distributed, what base stock is required for Part 1 and Part 2 so that a fill rate of 98% is achieved (show your table and submit your excel file for the Poisson distribution). Calculate the expected backorders and on-hand inventory for each part given this Q* values and a service level of 98% b.
(3) Assume holding cost is 3% per month and order costs are $5. What is the Q* using EOQ for each part? Is there a difference in the expected backorder level and on-hand inventory? If different, explain. c.
(4) Assume backorder cost per unit per month is $10, recalculate reorder points. Now, using Q values from part b compare average on-hand inventory and backorder level. d.
(1) Explain how you would modify the model(s) if lead time is variable. A.
WSU –
Industrial and Systems Engineering ISE 6315 Production Systems Instructor: Jeremy L. Rickli B. Part 1
Part 1
Demand
8
Demand
25
Lead time
30
Lead time
15
Lambda
8
Lambda
12.5
sigma
2.828427125
sigma
3.535533906
theta
8
theta
12.5
x
p(x)
G(x)
B(x)
x
p(x)
G(x)
B(x)
0
0.000335463
0.000335
8
0
3.72665E-06
3.73E-06
12.5
1
0.002683701
0.003019
7.000335
1
4.65832E-05
5.03E-05
11.5
2
0.010734804
0.013754
6.003355
2
0.000291145
0.000341
10.50005
3
0.028626144
0.04238
5.017109
3
0.001213103
0.001555
9.500395
4
0.057252288
0.099632
4.059489
4
0.003790948
0.005346
8.50195
5
0.091603662
0.191236
3.159121
5
0.009477369
0.014823
7.507296
6
0.122138215
0.313374
2.350357
6
0.019744519
0.034567
6.522118
7
0.139586532
0.452961
1.663731
7
0.03525807
0.069825
5.556686
8
0.139586532
0.592547
1.116692
8
0.055090734
0.124916
4.626511
9
0.124076917
0.716624
0.70924
9
0.076514908
0.201431
3.751427
10
0.099261534
0.815886
0.425864
10
0.095643635
0.297075
2.952859
11
0.072190206
0.888076
0.24175
11
0.108685949
0.405761
2.249933
12
0.048126804
0.936203
0.129826
12
0.11321453
0.518975
1.655694
13
0.029616495
0.965819
0.066028
13
0.108860125
0.627835
1.174669
14
0.016923711
0.982743
0.031848
14
0.09719654
0.725032
0.802505
15
0.009025979
0.991769
0.014591
15
0.080997117
0.806029
0.527536
16
0.00451299
0.996282
0.00636
16
0.063278998
0.869308
0.333565
17
0.00212376
0.998406
0.002642
17
0.046528675
0.915837
0.202873
18
0.000943893
0.99935
0.001047
18
0.03231158
0.948148
0.11871
19
0.000397429
0.999747
0.000397
19
0.021257618
0.969406
0.066858
20
0.000158971
0.999906
0.000144
20
0.013286011
0.982692
0.036264
21
6.05606E-05
0.999967
5.02E-05
21
0.00790834
0.9906
0.018956
22
2.2022E-05
0.999989
1.68E-05
22
0.004493375
0.995094
0.009556
23
7.65984E-06
0.999996
5.4E-06
23
0.002442052
0.997536
0.00465
24
2.55328E-06
0.999999
1.67E-06
24
0.001271902
0.998808
0.002186
r =
14
r = 20
20
R = r + 1 =
15
R = r + 1 = 21
21
backorder =
0.014591
backorder =
0.018956
on-hand inventory =
7.014591
on-hand inventory =
8.518956
Part 1
Part 2
Theta
8
12.5
Part cost
175
20
Holding cost
5.25
0.6
Order cost
5
5
Q*
3.9036
14.43376
Part 1
Part 2
Q*(rounding)
4
14
15
21
Backorder
4.059489
0.802505
0.014591
0.018956
On-hand invent
0.059489
2.302505
7.014591
8.518956
From part a
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WSU –
Industrial and Systems Engineering ISE 6315 Production Systems Instructor: Jeremy L. Rickli The order quantity is decreased when holding and order costs are incorporated in the calculation. From part a to b, the on-hand inventory decreases and backorder increases. It is due to the reason that holding cost has a good weight on choosing the order amount. C. Both parts decreased backorder quantity. It happened due to the backorder cost assigned into the function. Because backorder cost is higher than holding cost, backorder level remains very low compared to the on-hand inventory. Part 1
Part 2
Theta
8
12.5
Holding cost
5.25
0.6
Backorder cost
10
10
b/(h+b)
0.655738
0.943396226
Z inverse
0.400858
1.583939238
R*
9.133798
18.10007088
R* (rounding)
9
18
backorder
0.70924
0.118710136
on-hand inventory
1.70924
5.618710136