a) Formulate this problem (assume there are ?? warehouses and ?? customers). b)

Question

a) Formulate this problem (assume there are ?? warehouses and ?? customers). b) Then define an example, where there are two warehouses and three customers. a) Formulate this problem (assume there are ?? warehouses and ?? customers). b) Then define an example, where there are two warehouses and three customers. Problem 1: In modeling distribution systems, decisions must be made about tradeoffs between transportation costs and costs for operating distribution centers. As an example, suppose that a manager must decide which of n warehouses to use for meeting the demands of m customers for a good. The decisions to be made are which warehouses to operate and how much to ship from any warehouse to any customer. Let if warehouse i is opened Vi-(o if warehouse i is not opened Amunt to be sent from warehouse i to customer j xij The relevant costs are: fi Fixed operating cost for warehouse i, ifopened (for example, a cost to lease the warehouse) Cij Per-unit operating cost at warehouse i plus the transportation cost for shipping from warehouse i to customer j There are two types of constraints for the model: mer must be filled from i) the demand d; of each custo the warehouses; and i gds can be shipped from a warehouse only if it is opened.

Explanation / Answer

Answer (a) : To formulate the above problem considering n warehouses and m customers we will follow a systematic approach for which steps are given below,

Step 1 :  Identify decision variables

Step 2 : Identify objective function

Step 3 : Identify constraints

Since all the terminology is given in the question we will simply move to formulation of LPP(Linear Programming Problem)

Objective Function : min ( ?cij•xij•yi ) for i=1 to m and j=1 to n ,this is the total cost the comany would incur.

Constraint : ? xij = di (demand constraint)

Answer (b) : Consider an example where there are two warehouses and three customers => n = 2 , m = 3. Putting these values we will get

Objective Function : min ( ?cij•xij•yi ) for i=1 to 3 and j=1 to 2.

Constraint : ? xij = di (demand constraint) for i=1 to 3 and j=1 to 2.

Consider this example,

Eg.1 Two reservoirs are available to supply the water needs of three cities. Each reservoir can supply up to 50 million gallons of water per day. Each city would like to receive 40 million gallons per day. For each million gallons per day of unmet demand, there is a penalty. At city 1, the penalty is $20; at city 2, the penalty is $22; and at city 3, the penalty is $23. The cost of transporting 1 million gallons of water from each reservoir to each city is shown in following table

Here a parallel can be drawn as warehouse = reservoir and customer = city, keeping all the rest same considering yi = 1 for all warehouses to make problem simple.

Shipping Costs for Reservoir City 1 City 2 City 3 Resevoir 1 $ 7 $ 8 $ 10 Reservoir 2 $ 9 $ 7 $ 8

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