In: Operations Management
Consider a supplier order allocation problem under multiple
sourcing,
where it is required to buy 2000 units of a certain product
from
three different suppliers. The fixed set-up cost (independent of
the
order quantity), variable cost (unit price), and the maximum
capacity
of each supplier are given in Table 5.15 (two suppliers offer
quantity
discounts).
The objective is to minimize the total cost of purchasing (fixed
plus
variable cost). Formulate this as a linear integer programming
problem.
You must define all your variables clearly, write out the
constraints to
be satisfied with a brief explanation of each and develop the
objective
function.
TABLE 5.15
Supplier Data for Exercise 5.5
Supplier Fixed cost Capacity Unit Price
1 $100 600 units $10/unit for the first 300 units
$7/unit for the remaining 300 units
2 $500 800 units $2/unit for all 800 units
3 $300 1200 units $6/unit for the first 500 units
$4/unit for the remaining 700 units
Let us first construct a table with all the information related to total capacity of each supplier, supplier fixed cost and supplier variable cost information & calculate net cost of procurement of 1 unit from each supplier. | |||||||||||
Fixed Cost | Total Capacity | Quantity of first lot offered | Variable Cost of first lot | Quantity of balance lot | Variable cost of balance lot | Net cost of procurement of 1 unit | |||||
A | B | C | D | E | F | G=(A+C*D+E*F)/B | |||||
USD | Units | Units | USD/Unit | Units | USD/Unit | USD / Unit | |||||
Supplier 1 | 100 | 600 | 300 | 10 | 300 | 7 | 5.25 | ||||
Supplier 2 | 500 | 800 | 800 | 2 | 0 | 0 | 2.63 | ||||
Supplier 3 | 300 | 1200 | 500 | 6 | 700 | 4 | 2.76 | ||||
Now let us represent Net cost of procurement of 1 unit from each supplier is represented by N1 , N2 & N3 | |||||||||||
So we have | Net cost of procurement of 1 unit | ||||||||||
USD / Unit | |||||||||||
N1 | 5.25 | ||||||||||
N2 | 2.63 | ||||||||||
N3 | 2.76 | ||||||||||
Also let us assume that quantity to be procured from Supplier 1 , Supplier 2 & Supplier 3 is represented as Q1, Q2 & Q3. | |||||||||||
Now let us formulate the objective function of linear programming which is | |||||||||||
Minimize N1*Q1+N2*Q2+N3*Q3 as the objective is to minimize the total cost of purchasing . | |||||||||||
Where the constraint is Q1+Q2+Q3=2000 units | |||||||||||
This means total quantity to be procured from all three suppliers can not exceed 2000 units. | |||||||||||
Now solving this is easier as first we will do is we will exhaust the total capacity of lowest cost supplier which is Supplier 2. | |||||||||||
So we will buy 800 units from Supplier 2. | |||||||||||
So we have | Q2 = 800 | ||||||||||
Balance to be procured is 2000-800 = 1200 units | |||||||||||
Next lowest supplier is Supplier 3 whose capacity is 1200 . So we will buy 1200 units from supplier 3. | |||||||||||
So we have Q3 = 1200 units. | |||||||||||
Now balance to be procured is 1200-1200=0 | |||||||||||
So we do not have anything to be bought from Supplier 1 . | |||||||||||
So we have Q1 =0 | |||||||||||
Now solving the linear equation we have N1*Q1+N2*Q2+N3*Q3=0*5.25+800*2.63+1200*2.76 = 5416 USD |