Question

Develop a linear programming model for this problem. (Do Not Solve The Problem Warehouse City E City. E City G City H Warehouse Supply 0.53 0.21 0.52 0.41 4000 B 0.31 0.38 0.41 0.29 6000 0.56 0.32 0.54 0.33 4000 D 0.42 0.55 0.34 0.52 5500 City Demand 3.400 2,000 6.500 5.750 Based on the data provided, write your answers to the following questions: 1. Write the objective function for this model 2. Write the supply constraint for Warehouse A. 3. Write the city demand constraints for all four cities. Write the variables as per the following example: The variable of shipping from warehouse B to City His Xbh Let Xi = each combination of warehouse i and city

Answer 1

Let the no. of units shipped from Warehouse A to City E be Xae, Warehouse A to City f be Xaf and so on. Hence, we get the decision variables as Xae, Xaf, Xag, Xah, Xbe, Xbf, Xbg, Xbh, Xce, Xcf, Xcg, Xch, Xde, Xdf, Xdg, Xdh

Total cost = 0.53*Xae + 0.21*Xaf + 0.52*Xag + 0.41*Xah + 0.31*Xbe + 0.38*Xbf + 0.41*Xbg + 0.29*Xbh + 0.56*Xce + 0.32*Xcf + 0.54*Xcg + 0.33*Xch + 0.42*Xde + 0.55*Xdf + 0.34*Xdg + 0.52*Xdh

We have to minimize this total cost. Hence, we get the objective function as:

Minimize Total Cost C = 0.53*Xae + 0.21*Xaf + 0.52*Xag + 0.41*Xah + 0.31*Xbe + 0.38*Xbf + 0.41*Xbg + 0.29*Xbh + 0.56*Xce + 0.32*Xcf + 0.54*Xcg + 0.33*Xch + 0.42*Xde + 0.55*Xdf + 0.34*Xdg + 0.52*Xdh

Total Supply = 4000 + 6000 + 4000 + 5500 = 19,500

Total Demand = 3400 + 2000 + 6500 + 5750 = 17,650‬

Total Demand < Total Supply. Hence, this is an unbalanced problem. We will have "<=" sign in supply constraints

Subject to Supply Constraints:

Xae + Xaf + Xag + Xah <= 4000................Constraint for supply from Warehouse A

Xbe + Xbf + Xbg + Xbh <= 6000................Constraint for supply from Warehouse B

Xce + Xcf + Xcg + Xch <= 4000................Constraint for supply from Warehouse C

Xde + Xdf + Xdg + Xdh <= 5500................Constraint for supply from Warehouse D

Xae + Xbe + Xce + Xde = 3400................Constraint for Demand at City E

Xaf + Xbf + Xcf + Xdf = 2000...................Constraint for Demand at City F

Xag + Xbg + Xcg + Xdg = 6500................Constraint for Demand at City G

Xah + Xbh + Xch + Xdh = 5750................Constraint for Demand at City H

Xae, Xaf, Xag, Xah, Xbe, Xbf, Xbg, Xbh, Xce, Xcf, Xcg, Xch, Xde, Xdf, Xdg, Xdh >= 0........Non-negativity constraints as no. of units cannot be negative

The above formulation is complete.

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