Assignment-2 Submission
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Saint Mary's University *
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665
Subject
Industrial Engineering
Date
Dec 6, 2023
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8
Uploaded by ConstableFrogPerson910
Assignment-2
Question:1
The sensitivity report is:
A.
How would adding four (4) hours in the edging department at Plant 1 affect costs?
Adding 4 hours in edging at plant-1 reduces the costs by 4*shadow price. ie. 4*285.71 =
1142.86. 4hrs increase is within the allowable increase.
B.
What if the furnace at Plant 1 was shut down for 30 hours. How would this affect
costs?
Shutting down of furnace at plant-1 for 30 hours would not affect the costs because, the
shadow price is $0 and 30 hrs decrease is within the allowable decrease.
C.
How would total costs change if you lost four (4) hours of the edging capacity in Plant
1, but gained two (2) hours of edging capacity in plant 2?
-4*shadow price at plant 1 + 2*shadow price at plant 2
= -4*-285.714 + 2*-857.143
The cost reduces by $571.43.
D.
Suppose both furnaces are shut down for 15 hours. How would this affect costs?
Shutting down both furnaces for 15 hours would not affect the costs because, the shadow
prices are $0 and 10 hrs decrease is within the allowable decrease for furnaces at both the
plants.
E.
Consider the objective function cost of IG at Plant 1. Would raising it $1 affect the
optimal production schedule?
No raising it $1 doesn’t affect the production schedule, because it is within the allowable limit.
F.
Consider the cost of IG at Plant 2. How would raising it $.5 per SQFT affect the optimal
production schedule?
As the increase in price is beyond the allowable limit as per sensitivity table, the optimal
production schedule can be estimated by resolving the LP. The new optimal production
schedule is:
26750 hrs. of IG at plant 1, 0 hrs. of TG at plant 1, 11250 hrs. IG at plant 2 and 23000 of TG
at plant 2.
G.
Suppose you needed to eliminate four (4) hours from the edging department at both
plants.
How much would this impact your costs?
As 4 hours are within the allowable limit as seen in the sensitivity table. The total costs
change by 4* (-285.71 - 857. 14) =$ 4571.43 increase in costs.
H.
Suppose you needed to eliminate one 8-hr shift in the edging department at both
plants in the coming week. How much would this impact your costs?
Since as 8 hours are within the allowable limit as seen in the sensitivity table. The total
costs change by 8* (-285.71 - 857. 14) = $9142.9 increase in costs.
I.
Suppose the cost of glass increases by $.10 per SQFT for all four products. Would this
affect your optimal production schedule?
The optimal production schedule would not get affected by the increase in cost of $0.1,
since the allowable price increases of all 4 products are more than $0.1.
J.
What happens if the cost of insulated (IG) glass decreases by $.15 per SQFT at Plant
1 but increases by $.15 per SQFT at Plant 2?
The overall optimum costs increase by $1425
K. Suppose the amount of IG glass needed this week increases by 1000 SQFT (via an
emergency order). How much more does each additional SQFT of IG glass cost Newfort?
How does this compare with the variable costs of IG glass (per SQFT) at each plant? Can you
explain why this is the case?
An increase in each SQFT of IG glass increases the cost per additional SQFT by an amount of
shadow price i.e., $9.7143. This is higher than the variable costs at each plant (9 and 8
respectively).
Question:2
Let the cost for the transportation routes be defined as JD, JA, JH, ND, NA, NH.
Where JD stands for cost to transport the short blocks from Juarez to Dallas, JA for Juarez to
Austin….and so on.
The objective function is:
Minimize: 3440 JD + 3380 JA + 3340 JH + 4160 ND + 4120 NA + 4080 NH
Subject to:
JD + ND >= 5000
JA + NA >= 3500
JH + NH >= 6000
JD + JA + JH <= 8000
ND + NA + NH <= 9000
JD, JA, JH, ND, NA, NH >= 0
By solving this LP in excel we get the minimum cost (optimal shipping solution) as $53780000.
This solution is obtained when:
0 units of ‘short blocks’ are transported from Juarez to Dallas.
2000 units of ‘short blocks’ are transported from Juarez to Austin.
6000 units of ‘short blocks’ are transported from Juarez to Houston.
5000 units of ‘short blocks’ are transported from New Orleans to Dallas.
1500 units of ‘short blocks’ are transported from New Orleans to Austin.
0 units of ‘short blocks’ are transported from New Orleans to Houston.
C) If 25% tariffs are imposed on all short blocks coming from Juarez, Mexico, then purchase cost
from Juarez becomes 3200*1.25 = $4000; The new Objective function is:
Minimize: 4240 JD + 4180 JA + 4140 JH + 4160 ND + 4120 NA + 4080 NH
Subject to:
JD + ND >= 5000
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JA + NA >= 3500
JH + NH >= 6000
JD + JA + JH <= 5500
ND + NA + NH <= 9000
JD, JA, JH, ND, NA, NH >= 0
By solving this LP in excel we get the minimum cost (optimal shipping solution) as $60030000.
This solution is obtained when:
0 units of ‘short blocks’ are transported from Juarez to Dallas.
0 units of ‘short blocks’ are transported from Juarez to Austin.
5500 units of ‘short blocks’ are transported from Juarez to Houston.
5000 units of ‘short blocks’ are transported from New Orleans to Dallas.
3500 units of ‘short blocks’ are transported from New Orleans to Austin.
500 units of ‘short blocks’ are transported from New Orleans to Houston.
The increased cost tariff of 25% corresponds to increase in costs of $800. The quantity 800 is
beyond the allowable limits from the sensitivity report.
Hence, we cannot use sensitivity report
without solving the LP function.
Question 3:
Let
J1 be gallons of raw milk to buy from Jersey Farm to process at facility 1
J2 be gallons of raw milk to buy from Jersey Farm to process at facility 2
H1 be gallons of raw milk to buy from Holstein Farm to process at facility 1
H2 be gallons of raw milk to buy from Holstein Farm to process at facility 2
M1F be Gallons of pasteurized milk to ship from facility 1 to Food Wholes
M2F be Gallons of pasteurized milk to ship from facility 2 to Food Wholes
M1C be Gallons of pasteurized milk to ship from facility 1 to Central Markup
M2C be Gallons of pasteurized milk to ship from facility 2 to Central Markup
C1F be Gallons of pasteurized cream to ship from Facility 1 to Food Wholes
C2F be Gallons of pasteurized cream to ship from Facility 2 to Food Wholes
C1C be Gallons of pasteurized cream to ship from Facility 1 to Central Markup
C2C be Gallons of pasteurized cream to ship from Facility 2 to Central Markup
Hence, the objective function of the problem is:
Minimize
2.78J1 + 2.69H1 + 2.63J2 + 2.79H2 + 0.35M1F + 0.45M1C + 0.40M2F + 0.37M2C +0.45C1F +
0.55C1C + 0.50C2F + 0.47C2C
Subject to:
J1 + J2 ≤ 14000
H1 + H2 ≤ 12000
M1F+M2F≥8000
M1C+M2C≥11000
C1F+C2F≥1500
C1C+C2C≥1850
M1F+M1C- 0.78J1- 0.85H1 = 0
M2F+M2C-0.80J2 - 0.83H2 = 0
C1F+C1C-0.18J1 - 0.12H1 = 0
C2F+C2C-0.16J2 - 0.10H2 = 0
J1,J2,H1,H2,M1F,M2F,M1C,M2C,C1F,C2F,C1C,C2C ≥ 0
By solving the LP in excel the cost at optimal purchase plan is
$
70292.485
This solution is obtained when:
J1 386.6
J2 13613.4
H1 9185.6
H2 0
M1F 8000
M2F 0
M1C 109.27
M2C 10890.7
C1F 1171.85
C2F 328.14
C1C 0
C2C 1850
Question 4:
The objective function is:
maximize 6C+ 12M + 9S;
The sensitivity report is
A. What if M. L. Doud could purchase another 1000 gallons of Cabernet. What impact would this
have on profits?
If Cabernet increases by 1000 gallons; Then the profit would increase by 1000*32(shadow price)
= 32000 profit increase
B. What
if the price of Syrah increased $2 per gallon. What impact would this have on M.L. Doud’s
final purchase quantities? What impact would it have on profits? (
Note the direction of the change
in the coefficient!
)
If the price of Syrah increased by $2 per gallon, The profit generated by syrah reduces by $2. As
2 is within the allowable decrease limit, the quantity doesn’t change, and the profits will reduce by
2*10000 = 20000. SO, reduction in profit by 20000.
C. What if the price of Merlot rose $4. Would this impact M.L. Doud’s purchase quantities?
The price of Merlot rose by $4, profit reduces by $4, and the new objective function becomes
6C+8M+9S; This exceeds the allowable decrease ($3) in objective coefficient. Hence the impact
cannot be found using sensitivity tables. LP should be resolved.
The new purchase quantities for this condition are:
15000 Cabernet, 19000 Merlot and 16000 Syrah.
Variable Cells
Final
Reduced
Objective
Allowable Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$B$2
C
15000
0
6
1E+30
32
$C$2
M
25000
0
12
1E+30
3
$D$2
S
10000
0
9
3
48
Constraints
Final
Shadow
Constraint Allowable Allowable
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$E$5
Min C
2.27374E-12
-38
0
6000
1125
$E$6
Min M
10000
0
0
10000
1E+30
$E$7
Min S
-9.09495E-13
-3
0
6000
3000
$E$8
C Avail
15000
32
15000
1800
15000
$E$9
M Avail
25000
0
28000
1E+30
3000
$E$10
S Avail
10000
0
16000
1E+30
6000
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D. What if the price of cabernet dropped $6 per gallon and the price of Merlot increased $2 per
gallon. Would M.L. Doud purchase different quantities? What is the impact on profits?
The profit of cabernet increases by $6 and of Merlot decreases by $2. Both these changes are
within the limits. Hence, the purchase quantities wouldn’t be different. The profits increase would
be 6*15000
–
2*25000 = $40000 (increase in profits)
E. What if the availability of Merlot and Syrah both increased by 5000 gallons. What impact would
this have on M.L. Doud’s profits?
From the sensitivity table the allowable limits for the availability of Merlot and Syrah are both
infinities. Hence, even an increase in 5000 gallons of availability of both Merlot and Syrah will
not impact the Maximum profit condition of the objective function. Hence no impact on M.L
Doud’s profits.
Question 5:
Apart from the decision variables used in question 2, 9 more decision variables must be used to
solve this LP. They are:
D1, D2, D3: Dallas to Automakers 1,2,3
A1, A2, A3: Austin to Automakers 1,2,3
H1, H2, H3: Houston to Automakers 1,2,3
The new objective function is to
3200 JD + 3200 JA + 3200 JH + 4000 ND + 4000 NA + 4000 NH + 3100 D1 + 3100 D2+ 3075
D3 + 3400 A1 + 3350 A2 + 3300 A3 + 2950 H1 + 2900 H2 + 2850 H3
Subject to constraints:
JD + ND >= 5000
JA + NA >= 3500
JH + NH >= 6000
JD + JA + JH <= 8000
ND + NA + NH <= 9000
D1 + D2 + D3 <= 6000
A1 + A2 + A3 <= 6000
H1 + H2 + H3 <= 6000
D1 + A1 + H1 >= 5500
D2 + A2 + H2 >= 4000
D3 + A3 + H3 >= 5000
All variables >= 0
By solving this LP in excel, the optimal purchasing/production/shipping plan is as follows:
Route
Units
JA
2000
JH
6000
ND
5000
NA
1500
NH
0
D1
5500
D2
500
D3
0
A1
0
A2
2500
A3
0
H1
0
H2
1000
H3
5000
The cost for this plan is $ 95725000.