Suppose that you have two possible locations to locate warehouses: city A and ci

Question

Suppose that you have two possible locations to locate warehouses: city A and city B. You can locate a warehouse in city A, or city B, or both cities. You have the following information about the possible warehouse locations: . A warehouse in city A has a fixed monthly payment of $4,000 and a capacity of 5000 units. . A warehouse in city B has a fixed monthly payment of 55,000 and a capacity of 6000 units. Once you locate a warehouse or warehouses, you will ship the product from these warehouses to the retailer stores you locate. There are also two possible locations to locate retailer stores: site 1 and site 2. You can locate at most 5 retailer stores at each one of these sites. You have the following information about the possible retailer store sites. . A retailer store in site 1 has a fixed monthly payment of $2,000. The sale potential of a single retailer store in site 1 is 1000 units per month. . A retailer store in site 2 has a fixed monthly payment of $2,500. The sale potential of a single retailer store in site 2 is 1500 units per month. Once you locate retailer stores, you have to ship the total sale potential at each site from the warehouses you have. For instance, if you locate 3 retailer stores in site 1, you need to ship exactty 3000 units to site 1 from the warehouses you have located. You can ship from any warehouse to any site. The costs of shipping one unit from the warehouse locations to the retailer store sites are given in the table below. Each unit is sold for $15 at any retailer store site and the sale potential of each retailer store will be the average number of items sold at the retailer store. As the manager of distribution network, you need to find where to locate the warehouses, how many retailer stores to locate at each potential site, and how to ship from the warehouses to the retailer store sites to satisfy the sale potentials at the sites. In doing so, you want to maximize the total monthly profit, which is equal to revenues from sales minus monthly warehouse and retailer store costs and shipping costs. Note that you cannot ship from a city if there is no warehouse in that city; and, the total amount you ship to a site will be determined by the number of retailer stores you locate on that site. a) (1.5 points) Mathematically formulate a mixed integer linear programming model for the above distribution planning problem by defining decision variables, objective function, and constraints.

Explanation / Answer

Decision Variables:

Location of the warehouses and location & number of retailers are required to be determined as the question alongwith the quantity to be transported from each warehouse to each retailer.

Let us define Xij as the variable representing quantity to be transported from ith warehouse to the jth retailer. "i" takes the value 1 for locating warehouse in city A and "i" takes the value 2 for warehouse in city.

"j" takes the value 1 for locating retailers at site1 and "j" takes the value 2 for locating retailers at site2.

Objective function:

As mentioned in the question the objective is to maximise the total monthly profit. For its caculations we need to calculate per unit profit for transporting unit from warehouses to retailers. Given each unit is sold for $15.

Per unit cost at warehouse in city A is$.80 ( 4000/5000), at warehouse in city B is $.83 ( 5000/6000), per unit cost at site 1 is $2 (2000/1000), per unit cost at site 2 is $1.67 (2500/1500). Therefore total cost(warehouse cost + site cost + transportation cost) matrix and profit (sales price - total cost) matrix are as follows:

The above mentioned profit contributions per unit from each warehouse to each site are objective function coefficients denoted by Cij (per unit profit from each warehouse to each retailer).

Therefore the objective function is to Maximize double summation Cij * Xij

Constraints:

Supply constraint of warehouse at city A is Sigma (over j) X1j = 5000

Supply constraint of warehouse in city B is Sigma (over j) X2j =6000

Demand constraint at site 1 for maximum five retailers is Sigma (over i ) Xi1 <= 5000

Demand constraint at site 2 for maximum five retailers is Sigma (over i ) Xi2 <=7 500

Further Xi1 needs to be in multiples of 1000 that is either 0 or 1000 or 2000 or 3000 or 4000 or 5000

and Xi2 needs to be in multiples of 1500 that is either 0 or 1500 or 3000 or 4500 or 6000 or 7500

The optimal solution in the table is as follows:

To conclude it is recommended to have warehouses at both the places that is in city A as well as in city B.

Have five retailers at site 1 to be served from warehouse in city A.

Have four retailers at site 2 to be served from warehouse in city B.

Maximum profit earned is $75,000

Cost matrix cityA cityB site 1 7.8 10.83 site 2 12.47 8.5 Profit matrix cityA cityB site 1 7.2 4.17 site 2 2.53 6.5

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