Question

Companies A, B, and C supply components to three plants (F, G, and H) via two crossdocking facilities (D and E). It costs $4 to ship from D regardless of final destination and $3 to ship to E regardless of supplier. Shipping to D from A, B, and C costs $3, $4, and $5, respectively, and shipping from E to F, G, and H costs $10, $9, and $8, respectively. Suppliers A, B, and C can provide 200, 300 and 500 units respectively and plants F, G, and H need 350, 450, and 200 units respectively. Crossdock facilities D and E can handle 600 and 700 units, respectively. Logistics Manager, Jack Beauregard, had previously used ʺChain of Foolsʺ as his supply chain consulting company, but now turns to you for some solid advice. A. Sketch the network for this problem and label all nodes and arrows with the appropriate information. B. What is the objective function for the linear programming formulation of this problem? C. How many supply-side constraints are there? Write the supply-side constraints. D. How many intermediate-side (transshipment) constraints are there? Write the intermediate-side (transshipment) constraints. E. How many demand-side constraints are there? Write the demand-side constraints.

Answer 1

We sketch the network diagram as shown below:

Let the units shipped from Company A to Crossdocking Facility D be Xad, Crossdocking facility D to Plant F be Xdf and so on. Hence, we get decision variables as Xad, Xae, Xbd, Xbe, Xcd, Xce, Xdf, Xdg, Xdh, Xef, Xeg and Xeh.

Total cost = 3*Xad + 3*Xae + 4*Xbd + 3*Xbe + 5*Xcd + 3*Xce + 4*Xdf + 4*Xdg + 4*Xdh + 10*Xef + 9*Xeg + 8*Xeh.

We have to minimize this cost

Total Supply = 200 + 300 + 500 = 1000

Total Demand = 350 + 450 + 200 = 1000

Total Demand = Total capacity. hence, we will get "=" in all constraints. However, capacity for D + E = 600 + 700 = 1300. hence, we will get "<=" sign for these constraint related to crossdocking facility.

We get Supply constraints for Companies as:

Xad + Xae = 200

Xbd + Xbe = 300

Xcd + Xce = 500

We get Demand Constraints for Plant as:

Xdf + Xef = 350

Xdg + Xeg = 450

Xdh + Xeh = 200

We get Constraint for Cross-docking facilities as:

Xad + Xbd + Xcd = Xdf + Xdg + Xdh

Xae + Xbe + Xce = Xef + Xeg + Xeh

Also, based on handling capacity for cross docking facilities:

Xad + Xbd + Xcd <= 600

Xae + Xbe + Xce <= 700

Xad, Xae, Xbd, Xbe, Xcd, Xce, Xdf, Xdg, Xdh, Xef, Xeg, Xeh. >= 0..................Non-negativity constraints as no. of units shipped cannot be negative

As seen from above,

A) The sketch is shown above

B) The objective function for the LP formulation is:

Minimize Total Cost C = 3*Xad + 3*Xae + 4*Xbd + 3*Xbe + 5*Xcd + 3*Xce + 4*Xdf + 4*Xdg + 4*Xdh + 10*Xef + 9*Xeg + 8*Xeh

C) No. of supply-side constraints = 3 nos.

They are:

Xad + Xae = 200

Xbd + Xbe = 300

Xcd + Xce = 500

D) No. of intermediate-side (transshipment) constraints = 4 nos.

They are:

Xad + Xbd + Xcd = Xdf + Xdg + Xdh

Xae + Xbe + Xce = Xef + Xeg + Xeh

Xad + Xbd + Xcd <= 600

Xae + Xbe + Xce <= 700

E) No. of demand-side constraints = 3 nos.

They are:

Xdf + Xef = 350

Xdg + Xeg = 450

Xdh + Xeh = 200

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