#1 West Fuel produces a special fuel system component at its three plants. The c
ID: 3046079 • Letter: #
Question
#1 West Fuel produces a special fuel system component at its three plants. The company currently has orders from four customers. After considering relevant costs, West Fuel can expect the following per-unit profit for each plant-customer alternative. Plant 1 Plant 2 Plant 3 Customer 1 Customer 2 Customer 3 Customer 4 $20 $19 $16 $17 $17 $15 816 $17 The manufacturing capacities during the current production period are: Plant 1, 5,000 units; Plant 2, 2,500 units; Plant 3, 4,000 units. The customer demands are: Customer 1, 2,000 units; Customer 2, 3,500 units; Customer 3, 4,500 units; Customer 4, 3,000 units. Develop a transportation model that WestFuel can use to determine how many units each plant should ship to each customer, with the goal of maximizing total profit. To properly formulate the model, compute the total production capacity and the total demand, and incorporate this information into the model as needed. You do not need to solve the LP. #2 A local television station has three weekday time slots for which it is trying to choose a programming lineup. The station's programming manager has developed a list of five available program choices. Each program may air, at most, one time. Using previous data, the station's financial manager has projected the advertising revenue for each program in each time slot, as given in the table below. 2:30-3:00 p.m. 3:00-3:30 p.m. 3:30-4:00 p.m. Judge Julie The Bite Wheel of Winning Dr. Rozz Connect Hollywood $7,400 $8,100 $8,300 S7,300 S6,500 $7,000 $5,600 $5,500 $7,100 $5,700 $7,600 86,200 S6,600 6,300 8, 100 Formulate a binary programming model that can be used to determine the schedule of prograns that will maximize the total projected advertising revenue. You do not ned to solve the LPExplanation / Answer
#1. The given data in terms of per unit profit in $ along with customer demands and plant capacity is expressed as follows:
Customer
Plant
1
2
3
4
Capacity
1
16
17
16
20
1500
2
17
15
14
19
2500
3
18
16
17
20
4000
4
0
0
0
0
5000
Demand
2000
3500
4500
3000
13000
The given problem was unbalanced and therefore dummy plant 4 is added with capacity 5000 units so that revised problem is now balanced.
As transportation algorithm is developed for minimization, therefore let us obtain another matrix by subtracting all profit values from Maximum Profit = 20 $
Customer (Cij)
Plant
1
2
3
4
Capacity
1
4
3
4
0
1500
2
3
5
6
1
2500
3
2
4
3
0
4000
4
20
20
20
20
5000
Demand
2000
3500
4500
3000
13000
Let us write the matrix of transportation allocation in terms of unknowns X’s as follows:
Customer
Plant
1
2
3
4
Capacity
1
X11
X12
X13
X14
1500
2
X21
X22
X23
X24
2500
3
X31
X32
X33
X34
4000
4
X41
X42
X43
X44
5000
Demand
2000
3500
4500
3000
13000
Now our object function is Z = Cij*Xij
With conditions,
X11+X12+X13+X14=1500
X21+X22+X23+X24=2500
X31+X32+X33+X34=4000
X41+X42+X43+X44=5000
X11+X21+X31+X41=2000
X12+X22+X32+X42=3500
X13+X23+X33+X43=4500
X14+X24+X34+X44=3000
Xij 0
#2.
The given data in terms of revenue per advertising revenue in $ of five programs as follows:
Time Slots
Program
2.30 to 3.00
3.00 to 3.30
3.30 to 4.00
4.00 to 4.30
4.30 to 5.00
1
7400
7000
7600
0
0
2
8100
5600
6200
0
0
3
8300
5500
6600
0
0
4
7300
7100
6300
0
0
5
6500
5700
8100
0
0
The given problem was unbalanced and therefore dummy time slots viz. 4.00 to 4.30 and 4.30 to 5.00 is added so that revised problem is now balanced.
Algorithm for binary programming problem or assignment problem developed for minimization, therefore let us obtain another matrix by subtracting all revenue values from Maximum Profit = 8300 $
Time Slots (Cij)
Program
2.30 to 3.00
3.00 to 3.30
3.30 to 4.00
4.00 to 4.30
4.30 to 5.00
1
900
1300
700
8300
8300
2
200
2700
2100
8300
8300
3
0
2800
1700
8300
8300
4
1000
1200
2000
8300
8300
5
1800
2600
200
8300
8300
Let us write the matrix of binary allocation in terms of unknowns X’s as follows:
Time Slots
Program
2.30 to 3.00
3.00 to 3.30
3.30 to 4.00
4.00 to 4.30
4.30 to 5.00
1
X11
X12
X13
X14
X15
2
X21
X22
X23
X24
X25
3
X31
X32
X33
X34
X35
4
X41
X42
X43
X44
X45
5
X51
X52
X53
X54
X55
Where each Xij is either 1 if program at that time slot is broadcasted otherwise 0.
Now our object function is Z = Cij*Xij
With conditions,
X11+X12+X13+X14+X15=1
X21+X22+X23+X24+X25=1
X31+X32+X33+X34=X35=1
X41+X42+X43+X44+X45=1
X51+X52+X53+X54+X55=1
X11+X21+X31+X41+X51=1
X12+X22+X32+X42+X52=1
X13+X23+X33+X43+X53=1
X14+X24+X34+X44+X54=1
X15+X25+X35+X45+X55=1
Xij 0
Customer
Plant
1
2
3
4
Capacity
1
16
17
16
20
1500
2
17
15
14
19
2500
3
18
16
17
20
4000
4
0
0
0
0
5000
Demand
2000
3500
4500
3000
13000
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