Question

Poppy Seed Pharmaceuticals (PSP) has a contract with the Viral Control Center (VCC) for delivering varying amounts of a fungal culture to be used in the development of a vaccine for Foot-in-Mouth Disease, over the course of the next year. To satisfy the terms of this contract, PSP must deliver 40, 100, 60 and 80 kilograms of this fungal culture at the end of each of the next four quarters, respectively, to VCC. The fungus from which the culture is produced only grows naturally and plentifully (all year round) in a remote forest area in Sarawak, Malaysia and cannot be grown anywhere else. This fungus would have to be purchased from an autonomous local Poonan tribe in Sarawak. Two months before each contractstipulated delivery, a batch of the culture can be produced and held in a frozen state over an indefinite period of time. For each batch of the culture produced, PSP would incur a fixed cost of $200 (regardless of the batch size). The unit variable production cost of a kilogram of this fungal culture will vary depending on when the culture is grown. It is estimated that in quarter1 this cost would be $5/Kg., $7/Kg. in quarter 2, $6/Kg. in quarter 3 and $5/Kg. in quarter 4. Any amount of this biological material produced in any quarter, but not delivered at the end of that period, to be carried in inventory for delivery in the future, would incur a holding cost of $2/Kg./quarter. PSP is attempting to develop a minimum cost batch production plan of this fungal culture for satisfying its delivery obligations. Draw a network diagram for this problem, clearly explaining the meanings of the nodes and the arcs in your representation. With reference to this network, formulate (i.e. clearly define all variables, construct the objective function and all constraints, showing all known numerical parameter values) a linear optimization model, in order to find the optimal culture production schedule for PSP. Do not solve.

Answer 1

1)

Network diagram is following:

Ndes P1, P2, P3, P4 represent production in respective quarter

D1, D2, D3, D4 represent demand in respective quarter

V1, V2, V3, V4 represent ending inventory of respective quarter

2)

Linear Programming model is formulated as under:

Decision variables:

Let

Xi = 1, if a batch is produced in quarter i, otherwise Xi = 0

Pi = Quantity (kg) to be produced in quarter i

Vi = Inventory (kg) at the end of quarter i

Objective function:

Minimize 200X1+200X2+200X3+200X4+5P1+7P2+6P3+5P4+2V1+2V2+2V3+2V4

s.t.

Constraints:

P1-280X1 <= 0   (total demand of four quarters = 40+100+60+80 = 280. So, maximum production required in any quarter is 280 kg)

P2-280X2 <= 0

P3-280X3 <= 0

P4-280X4 <= 0

P1-V1 = 40

P2-V2+V1 = 100

P3-V3+V2 = 60

P4-V4+V3 = 80

Pi, Vi >= 0

Xi = {0,1}