Problem 6-09 The Ace Manufacturing Company has orders for three similar products
ID: 361941 • Letter: P
Question
Problem 6-09
The Ace Manufacturing Company has orders for three similar products:
Product Orders(units)
A 2000
B 500
C 1200
Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are as follows:
Machine Capacity (units)
1 1500
2 1500
3 1000
Product
Machine A B C
1 $1.00 $1.20 $0.90
2 $1.30 $1.40 $1.20
3 $1.10 $1.00 $1.20
Use the transportation model to develop the minimum cost production schedule for the products and machines. Show the linear programming formulation.
The linear programming formulation and optimal solution are shown.
Let xij = Units of product j on machine i.
Q. What is the ansers the chart above for units ? ( I know costs already and total is $3990)
Optimal Solution Units Cost 1-A $ 1-B $ 1-C $ 2-A $ 2-B $ 2-C $ 3-A $ 3-B $ 3-C $ Total $ 3990Explanation / Answer
The problem is to produce the 3 products with minimum cost.
Demand of 3 products is:
These products can be supplied by 2 machines which have a supply capacity constraint of specific units as given below:
Machine 1: 1500 units
Machine 2: 1500 units
Machine 3: 1000 units
Since unit cost of producing products on each machine varies, Production is first done on machine with minimum cost:
Machines
Products
A
B
C
1
$1
$1.2
$0.9
2
$1.3
$1.4
$1.2
3
$1.1
$1
$1.2
To minimize cost, cost of producing each unit from each machine is calculated using transportation model and minimum cost production schedule. Optimal solution will be to have a production of maximum units from a machine that costs the least .
Minimum cost = $1*xA1+$1.2*xB1+$0.9*xC1+$1.3*xA2+$1.4*xB2+ $1.2*xC2+$1.1*xA3+$1*xB3+$1.2*xC3
Demand for products =
XA1+XA2+XA3= 2000
xB1+xB2+xB3=500
XC1+XC2+XC3 = 1200
A Demand supply table is created where maximum product allocation for production is done to machine with least cost. This step then continues and remaining products are assigned to respective machines so as to maintain minimum cost.
Machines
Products
A
B
C
Supply
1
$1
$1.2
1200
$0.9
1500
2
$1.3
$1.4
0
$1.2
1500
3
$1.1
$1
0
$1.2
1000
Demand
2000
500
1200
Minimum cost is of XC1 ($0.9) so the maximum production unit of 1200 is assigned to this field and that entire demand of C is provided by Machine 1 only.
Machines
Products
A
B
C
Supply
1
300
$1
0
$1.2
1200
$0.9
1500
2
$1.3
0
$1.4
0
$1.2
1500
3
$1.1
500
$1
0
$1.2
1000
Demand
2000
500
1200
Similarly next lower cost is $1 so remaining supply is assigned to each cell. Requirement of 500 B is assigned to machine 3 and Requirement of 300 A is assigned to machine 1 (Since 1200 C units are already being produced at machine 1)
Machines
Products
A
B
C
Supply
1
300
$1
0
$1.2
1200
$0.9
1500
2
1200
$1.3
0
$1.4
0
$1.2
1500
3
500
$1.1
500
$1
0
$1.2
1000
Demand
2000
500
1200
Similarly remaining demand of A is taken first from least cost machine 3 (500 units) and then rest 1200 from machine 2.
Cost for each combination is taken and total comes as $3990.
Machines
Products
A
B
C
1
$1
$1.2
$0.9
2
$1.3
$1.4
$1.2
3
$1.1
$1
$1.2