Question

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 unit / $10 per unit for first 300 units; $7 per unit for remaining 300 units

2 / $500 / 800 units/ $2 per unit for all 800 units

3 / $300 / 1200 units / $6 per unit for first 500 units;  $4 per unit for remaining 700 units

Reformulate the problem under the assumption that both suppliers 1 and 3 offer all units discount, as described in the following;

-Supplier 1 charges $10 per unit for orders up to 300 units and for orders more than 300 units, the entire order will be priced at $7 unit.

-Supplier 3 charges $6 per unit for orders up to 500 units and for orders more than 500 units, the entire order will be priced at $4 per unit.

Answer 1

Let,

x11=number of units purchased from supplier 1 at price of $10/unit,

x12=number of units purchased from supplier 1 at price of $7/unit,

x2=number of units purchased from supplier 2 at price of $2/unit,

x31=number of units purchased from supplier 3 at price of $6/unit,

x32=number of units purchased from supplier 3 at price of $4/unit.

yi = 1 if supplier i is selected else 0

Objective is to minimize procurement cost so objective function = Min 10x11+7x12+2x2+6x31+4x32+100y1+500y2+300y3

subject to,

x11+x12+x2+x31+x32 = 2000 (Total demand)

Sourcing constraint from supplier 1
x11 <= 300y1
x12 >= 300y1
x12 <= 300y1

Sourcing constraint from supplier 2
x2 <= 800y2

Sourcing constraint from supplier 3
x31 <=500y3
x32>= 500y3
x32<=700y3

x11,x12,x2,x31,x32,y1,y2,y3 >= 0 (non-negativity constraint)
x11,x12,x2,x31,x32 = Integer
y1,y2,y3 = binary

Solving in excel we get,

x11=number of units purchased from supplier 1 at price of $10/unit = 0

x12=number of units purchased from supplier 1 at price of $7/unit = 0

x2=number of units purchased from supplier 2 at price of $2/unit = 800

x31=number of units purchased from supplier 3 at price of $6/unit= 500

x32=number of units purchased from supplier 3 at price of $4/unit = 700

and minimized cost = 8200

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