I need help with Linear Optimization for a logistics problem

Supply-Chain-Analytics-Case-Study-Template.xlsx

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Model Attempt

S1 - Store 1

Problem Data

Variable

"From" DistributionDenters

SUPPLY

Travel Times (in Min)

Cambridge, OH

Dover, OH

Newark, OH

Toronto, OH

Cambridge, OH

Dover, OH

Newark, OH

Toronto, OH

Cambridge, OH

S2 - Store 2

Maximum Supply:

Bellaire, OH

105

ShippingDosts

Decision Variables (48 in total)

xIJ = # of units shipped from DistritubionDenter I to Store J

Cadiz, OH

105

"To" Stores

Minimum Demand

LogisticDost per minute per unit:

$0.85

I =D (Cambridge), D (Dover), N (Newark), T (Toronto)

Calcutta, OH

108

154

Bellaire, OH (S1)

$47.60

$74.80

$89.25

$44.20

XC1

# of units shipped from

Cambridge, OH

Bellaire, OH (S1)

J = 1(S1), 2(S2), 3(S3), 4(S4), 5(S5), 6(S6), 7(S7), 8(S8), 9(S9), 10(S10), 11(S11), 12(S12)

Coshocton, OH

104

S3 - Store 3

Cadiz, OH (S2)

$47.60

$39.95

$89.25

$32.30

XC2

# of units shipped from

Cambridge, OH

Cadiz, OH (S2)

Dover, OH

Calcutta, OH (S3)

$91.80

$73.10

$130.90

$22.95

XC3

# of units shipped from

Cambridge, OH

Calcutta, OH (S3)

Objective Function is to minimize cost:

East Palestine, OH

129

165

Coshocton, OH (S4)

$39.95

$39.10

$37.40

$88.40

XC4

# of units shipped from

Cambridge, OH

Coshocton, OH (S4)

Lisbon, OH

106

142

East Palestine, OH (S5)

$109.65

$76.50

$140.25

$40.80

XC5

# of units shipped from

Cambridge, OH

East Palestine, OH (S5)

Min:

47.6XC1 +47.6XC2 +91.8XC3 +39.95XC4 +109.65XC5 +90.1XC6 +101.15XC7 +79.9XC8 +56.95XC9 +36.55XC10 +71.4XC11 +28.05XC12 +74.8XD1 +39.95XD2 +73.1XD3 +39.1XD4 +76.5XD5 +57.8XD6 +92.65XD7 +106.25XD8 +85.85XD9 +17XD10 +63.75XD11 +56.95XD12 +89.25XN1 +89.25XN2 +130.9XN3 +37.4XN4 +140.25XN5 +120.7XN6 +61.2XN7 +120.7XN8 +22.95XN9 +68XN10 +113.9XN11 +31.45XN12 +44.2XT1 +32.3XT2 +22.95XT3 +88.4XT4 +40.8XT5 +32.3XT6 +147.05XT7 +76.5XT8 +127.5XT9 +55.25XT10 +17.85XT11 +95.2XT12

Mt. Gilead, OH

119

109

173

S4 - Store 4

Lisbon, OH (S6)

$90.10

$57.80

$120.70

$32.30

XC6

# of units shipped from

Cambridge, OH

Lisbon, OH (S6)

New Martinsville, WV

125

142

Mt. Gilead, OH (S7)

$101.15

$92.65

$61.20

$147.05

XC7

# of units shipped from

Cambridge, OH

Mt. Gilead, OH (S7)

Newark, OH

140

New Martinsville, WV (S8)

$79.90

$106.25

$120.70

$76.50

XC8

# of units shipped from

Cambridge, OH

New Martinsville, WV (S8)

Pataskala, OH

101

150

Pataskala, OH (S9)

$56.95

$85.85

$22.95

$127.50

XC9

# of units shipped from

Cambridge, OH

Pataskala, OH (S9)

Toronto, OH

140

S5 - Store 5

Uhrichsville, OH (S10)

$36.55

$17.00

$68.00

$55.25

XC10

# of units shipped from

Cambridge, OH

Uhrichsville, OH (S10)

Uhrichsville, OH

Weirton, WV (S11)

$71.40

$63.75

$113.90

$17.85

XC11

# of units shipped from

Cambridge, OH

Weirton, WV (S11)

Subject To:

XC1 + XC2 + XC3 + XC4 + XC5 + XC6 + XC7 + XC8 + XC9 + XC10 + XC11 + XC12 ≤ 45

Weirton, WV

134

Zanesville, OH (S12)

$28.05

$56.95

$31.45

$95.20

XC12

# of units shipped from

Cambridge, OH

Zanesville, OH (S12)

XD1 + XD2 + XD3 + XD4 + XD5 + XD6 + XD7 + XD8 + XD9 + XD10 + XD11 + XD12 ≤ 15

Zanesville, OH

112

Total

XN1 + XN2 + XN3 + XN4 + XN5 + XN6 + XN7 + XN8 + XN9 + XN10 + XN11 + XN12 ≤ 15

S6 - Store 6

XD1

# of units shipped from

Dover, OH

Bellaire, OH (S1)

XT1 + XT2 + XT3 + XT4 + XT5 + XT6 + XT7 + XT8 + XT9 + XT10 + XT11 + XT12 ≤ 15

What do I know?

Optimized Model

XD2

# of units shipped from

Dover, OH

Cadiz, OH (S2)

There is a total minimum supply of 81 pallets per day that need to be delivered

Pallet Order Distribution

XD3

# of units shipped from

Dover, OH

Calcutta, OH (S3)

XC1 + XD1 + XN1 + XT1=5 Bellaire, OH (S1) SUPPLY

Each truck can deliver a maximum of 26 pallets

Decision Variables:

How many deliveries can be made from each distribution center.

0.06

XD4

# of units shipped from

Dover, OH

Coshocton, OH (S4)

XC2 + XD2 XN2 XT2=5 Cadiz, OH (S2)

Each truck only has 11 hours to deliver everything

S7 - Store 7

Objective:

Maximize deliveries and minimize cost.

0.06

XD5

# of units shipped from

Dover, OH

East Palestine, OH (S5)

XC3 + XD3 XN3 XT3=9 Calcutta, OH (S3)

11 hours *60 mins = 660 mins

Contraints:

Limited to a budget of $1065.90

0.11

XD6

# of units shipped from

Dover, OH

Lisbon, OH (S6)

XC4 + XD4 XN4 XT4=4 Coshocton, OH (S4)

660 mins * $0.85 = $561.00

Each truck has a maximum budget of $561.00

0.05

XD7

# of units shipped from

Dover, OH

Mt. Gilead, OH (S7)

XC5 + XD5 XN5 XT5=10 East Palestine, OH (S5)

3 truck * 660 mins = 1980 mins

Each truck can carry a maximu of 26 pallets

0.12

XD8

# of units shipped from

Dover, OH

New Martinsville, WV (S8)

XC6 + XD6 = XN6 = XT6=8 Lisbon, OH (S6) # of units shipped from

maximum budget of 3 trucks = $1683.00

S8 - Store 8

There is a minimum demand of 81 pallets every day

0.10

XD9

# of units shipped from

Dover, OH

Pataskala, OH (S9)

XC7 + XD7 = XN7 = XT7=9 Mt. Gilead, OH (S7) # of units shipped from

Total travel time for all 5 routs is 1254 minutes or 20.9 hours per day which is $1065.90

0.11

XD10

# of units shipped from

Dover, OH

Uhrichsville, OH (S10)

XC8 + XD8 = XN8 = XT8=3 New Martinsville, WV (S8) # of units shipped from

0.04

XD11

# of units shipped from

Dover, OH

Weirton, WV (S11)

XC9 + XD9 = XN9 = XT9=10 Pataskala, OH (S9) # of units shipped from

0.12

XD12

# of units shipped from

Dover, OH

Zanesville, OH (S12)

XC10 + XD10 = XN10 = XT10=5 Uhrichsville, OH (S10) # of units shipped from

S9 - Store 9

0.06

XC11 + XD11 = XN11 = XT11=7 Weirton, WV (S11) # of units shipped from

0.09

XN1

# of units shipped from

Newark, OH

Bellaire, OH (S1)

XC12 + XD12 = XN12 = XT12=6 Zanesville, OH (S12) # of units shipped from

0.07

XN2

# of units shipped from

Newark, OH

Cadiz, OH (S2)

XN3

# of units shipped from

Newark, OH

Calcutta, OH (S3)

XIJ ≥ 0

S10 - Store 10

XN4

# of units shipped from

Newark, OH

Coshocton, OH (S4)

XN5

# of units shipped from

Newark, OH

East Palestine, OH (S5)

XN6

# of units shipped from

Newark, OH

Lisbon, OH (S6)

XN7

# of units shipped from

Newark, OH

Mt. Gilead, OH (S7)

S11 - Store 11

XN8

# of units shipped from

Newark, OH

New Martinsville, WV (S8)

XN9

# of units shipped from

Newark, OH

Pataskala, OH (S9)

XN10

# of units shipped from

Newark, OH

Uhrichsville, OH (S10)

XN11

# of units shipped from

Newark, OH

Weirton, WV (S11)

S12 - Store 12

XN12

# of units shipped from

Newark, OH

Zanesville, OH (S12)

XT1

# of units shipped from

Toronto, OH

Bellaire, OH (S1)

XT2

# of units shipped from

Toronto, OH

Cadiz, OH (S2)

XT3

# of units shipped from

Toronto, OH

Calcutta, OH (S3)

XT4

# of units shipped from

Toronto, OH

Coshocton, OH (S4)

XT5

# of units shipped from

Toronto, OH

East Palestine, OH (S5)

XT6

# of units shipped from

Toronto, OH

Lisbon, OH (S6)

XT7

# of units shipped from

Toronto, OH

Mt. Gilead, OH (S7)

XT8

# of units shipped from

Toronto, OH

New Martinsville, WV (S8)

XT9

# of units shipped from

Toronto, OH

Pataskala, OH (S9)

XT10

# of units shipped from

Toronto, OH

Uhrichsville, OH (S10)

XT11

# of units shipped from

Toronto, OH

Weirton, WV (S11)

XT12

# of units shipped from

Toronto, OH

Zanesville, OH (S12)

Template

	Problem Data		L1	L2	L3	L4
			"From" Distribution Centers				Travel Times (in Min)	Cambridge, OH	Dover, OH	Newark, OH	Toronto, OH
			Cambridge, OH	Dover, OH	Newark, OH	Toronto, OH	Cambridge, OH	0	45	62	90
		Maximum Supply:	45	15	15	15	Bellaire, OH	56	88	105	52
			Shipping Costs				Cadiz, OH	56	47	105	38
	"To" Stores	Minimum Demand		Logistic Cost per minute per unit:	$0.85		Calcutta, OH	108	86	154	27
A	Bellaire, OH	5	$47.60	$74.80	$89.25	$44.20	Coshocton, OH	47	46	44	104
B	Cadiz, OH	5	$47.60	$39.95	$89.25	$32.30	Dover, OH	45	0	81	81
C	Calcutta, OH	9	$91.80	$73.10	$130.90	$22.95	East Palestine, OH	129	90	165	48
D	Coshocton, OH	4	$39.95	$39.10	$37.40	$88.40	Lisbon, OH	106	68	142	38
E	East Palestine, OH	10	$109.65	$76.50	$140.25	$40.80	Mt. Gilead, OH	119	109	72	173
F	Lisbon, OH	8	$90.10	$57.80	$120.70	$32.30	New Martinsville, WV	94	125	142	90
G	Mt. Gilead, OH	9	$101.15	$92.65	$61.20	$147.05	Newark, OH	62	81	0	140
H	New Martinsville, WV	3	$79.90	$106.25	$120.70	$76.50	Pataskala, OH	67	101	27	150
I	Pataskala, OH	10	$56.95	$85.85	$22.95	$127.50	Toronto, OH	90	81	140	0
J	Uhrichsville, OH	5	$36.55	$17.00	$68.00	$55.25	Uhrichsville, OH	43	20	80	65
K	Weirton, WV	7	$71.40	$63.75	$113.90	$17.85	Weirton, WV	84	75	134	21
L	Zanesville, OH	6	$28.05	$56.95	$31.45	$95.20	Zanesville, OH	33	67	37	112
	Total	81
							What do I know?
	Optimized Model						There is a total minimum supply of 81 pallets per day that need to be delivered
						Pallet Order Distribution	Each truck can deliver a maximum of 26 pallets
	Decision Variables:	How many deliveries can be made from each distribution center.				0.06	Each truck only has 11 hours to deliver everything
	Objective:	Maximize deliveries and minimize cost.				0.06	11 hours *60 mins = 660 mins
	Contraints:	Limited to a budget of $1065.90				0.11	660 mins * $0.85 = $561.00
		Each truck has a maximum budget of $561.00				0.05	3 truck * 660 mins = 1980 mins
		Each truck can carry a maximu of 26 pallets				0.12	maximum budget of 3 trucks = $1683.00
		There is a minimum demand of 81 pallets every day				0.10	Total travel time for all 5 routs is 1254 minutes or 20.9 hours per day which is $1065.90
						0.11
		Truck 1 - T1	Truck 2 - T2			0.04
						0.12
						0.06
	Pallets delivered	81 ≤ Total Ordered ≤ 90				0.09
	Truck capacity	T1 ≤ $561				0.07
		T2 ≤ $561
		A	≥90
		A+B+C+…=> L
		.06A-.06B-.11C-.05D-.12E-.10F-.11G-.04H-.12I-.06J-.09K-.07L ≤ 90