Every day, secret, over-worked employees at Wayne Enterprises work two 6-hour shifts producing Batmobiles. Theseshifts are selected from 12 am - 6 am, 6 am - 12 pm, 12 pm - 6 pm, and 6 pm - 12 am. The following table depicts thenumber of employees needed during each shift:Shift Time Period123412 am - 6 am6 am - 12 pm12 pm - 6 pm6 pm - 12 amNumber of Employees Required13211711Bruce Wayne pays all employees working two consecutive shifts, $12 per hour (Note: employees scheduled to work12 am - 6 am and 6 pm - 12 am would be considered consecutive). Employees whose shifts are not consecutive arepaid $18 per hour for the inconvenience.Formulate an LP that can be used to minimize the cost of meeting the daily workforce demands for Batmobileproduction.To formulate the model, use the following decision variables:Xij= workers working shifts i and j where i, j = {1, 2, 3, 4} Of the following options for the objective function, which is correct for this LP?O min 2 =minmax 2 = 24(x12 + X14 + X23 + X34 + X13 + X24)min z = 216(x12 + x14 + X23 + X34) + 144(X13 + X24)* = 144(x12 + x14 + X23 + X34) +216(x13 + X24)216(x12 + X14 + X23 + X34) + 144(x13 + X24)12 Σmaxmaxz =24Σ 1Σ=1&ij*Wi+1+1+ 36ΣΣ=1&ij*Wi+23+2•j=1* Ξ4i=1Σ=1 Wij * Wi+1,j+1 + 18 Σ 1 Σ=1ij * Xi+2,j+2i=1

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Of the following options for the objective function, which is correct for this LP? O O min z = 24 Σ²=1 Σ¼=1 Xij * Xi+1,j+1 + 36 Σi=1 Σj=1 Xij * Xi+2,j+2 2= = 24(X12 + X14 + X23 + X34 +X13 + X24) min Z= 216(x12 + x14 + X23 + X34) + 144(x13 + X24) 2 = 144(x12 + x14 + X23 +X34) + 216(x13 + X24) 216(x12 +14+x23 + x34) +144(x13 + x24) max z = 12²₁₁ Σj=1 Xij * Xi+1,j+1 + 18 Σi=1 Σj=1 Xij * Xi+2,j+2 =1 max min max z =

Of the following options for the objective function, which is correct for this LP? O O min z = 24 Σ²=1 Σ¼=1 Xij * Xi+1,j+1 + 36 Σi=1 Σj=1 Xij * Xi+2,j+2 2= = 24(X12 + X14 + X23 + X34 +X13 + X24) min Z= 216(x12 + x14 + X23 + X34) + 144(x13 + X24) 2 = 144(x12 + x14 + X23 +X34) + 216(x13 + X24) 216(x12 +14+x23 + x34) +144(x13 + x24) max z = 12²₁₁ Σj=1 Xij * Xi+1,j+1 + 18 Σi=1 Σj=1 Xij * Xi+2,j+2 =1 max min max z =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Every day, secret, over-worked employees at Wayne Enterprises work two 6-hour shifts producing Batmobiles. These
shifts are selected from 12 am - 6 am, 6 am - 12 pm, 12 pm - 6 pm, and 6 pm - 12 am. The following table depicts the
number of employees needed during each shift:
Shift Time Period
1
2
3
4
12 am - 6 am
6 am - 12 pm
12 pm - 6 pm
6 pm - 12 am
Number of Employees Required
13
21
17
11
Bruce Wayne pays all employees working two consecutive shifts, $12 per hour (Note: employees scheduled to work
12 am - 6 am and 6 pm - 12 am would be considered consecutive). Employees whose shifts are not consecutive are
paid $18 per hour for the inconvenience.
Formulate an LP that can be used to minimize the cost of meeting the daily workforce demands for Batmobile
production.
To formulate the model, use the following decision variables:
Xij
= workers working shifts i and j where i, j = {1, 2, 3, 4}

Of the following options for the objective function, which is correct for this LP?
O min 2 =
min
max 2 = 24(x12 + X14 + X23 + X34 + X13 + X24)
min z = 216(x12 + x14 + X23 + X34) + 144(X13 + X24)
* = 144(x12 + x14 + X23 + X34) +216(x13 + X24)
216(x12 + X14 + X23 + X34) + 144(x13 + X24)
12 Σ
max
max
z =
24Σ 1Σ=1&ij*Wi+1+1+ 36
ΣΣ=1&ij*Wi+23+2
•j=1
* Ξ
4
i=1
Σ
=1 Wij * Wi+1,j+1 + 18 Σ 1 Σ=1ij * Xi+2,j+2
i=1

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Follow-up Questions

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Follow-up Question

Please identify all of the correct constraints for the model.
ΟΣ=1&i
U
Π
U
Π
Π
Π
=
Π
(13 + 21 + 17 + 11) Vj
214 Χ 224 Χ 234
X13 + x23 +34 ≤ 17
11
X14X24 + X34 ≥ 11
212 Χ 213 Χ 214 = 13
X14X24X34 ≤ 11
€12 Χ 223 Χ 224
=
21
x12 +13 + 14 ≥ 13
X12 + x23 + x24 ≤ 21
213 Χ 223 Χ 34 17
x13x23x34 ≥ 17
ΟΣ=1&ij < (13 + 21 + 17 + 11) Va
X12 + x23 + x24 ≥ 21
ΟΣ=1 €i; > (13 + 21 + 17 + 11) Vj
x12 + x13 + 14 ≤ 13

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Follow-up Question

Which statement represents the correct sign restriction for our decision variable?
It is not necessary to include a sign restriction.
X12, 14, X23, X34 0
X13, X240
Xij ≤0 Vi, j
xįj urs (unrestricted) Vi, j
xij≥0 Vi, j

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