Determine the Optimal Shipments to minimize total production and transportation

Question

Determine the Optimal Shipments to minimize total production and transportation costs for the apparel company.

Please Solve this Linear Model Using Excel.

A sports apparel company has received an order for a college basketball team's national championship T-shirt. The company can purchase the T-shirts from textile factories in Mexico Puerto Rico, and Haiti. The shirts are shipped from the factories to companies in the United States that silk-screen the shirts before they are shipped to distribution centers. Following are the production and transportation costs (S/shirt) from the T-shirt factories to the silk-screen companies to the distribution centers, plus the supply of T-shirts at the factories and demand for the shirts at the distribution centers Silk-Screen Company T-shirt Factory 4. Miami|5. Atlanta 6. Houston Supply (1,000s) 1. Mexico $3 18 2. Puerto Rico 3 15 3. Hait 4 4 23 Distribution Center Silk-Screen Company 7. New York 8. St. Louis|9. Los Angeles 4. Miami $5 $7 $9 5. Atlanta 10 6. Houston Demand (1,000s 20 12 20 Determine the optimal shipments to minimize total production and transportation costs for the apparel company

Explanation / Answer

ANSWER:

There are three factories located at three different places( say represented by 'i' taking the values 1, 2, and 3) and three silk screen companies located at three different places( say represented by 'j' taking the values 4, 5, and 6) and finally three different demand centres( say represented by 'k' taking the values 7, 8, and 9). In other words the places are represented by their respective numbers as given in the question.

Decision variables are defined as Xijk representing the number ( positive integer ) that are produced at ith factory and silk screen at jth silk screen company and delivered at kth demand center.

The Objective function is to minimize the total production and transportation costs. The cost co-efficients denoted as Cijk corresponding to decision variables are calculated as follows: Cijk = Cij + Cjk

In simple terms Objective function is to Minimize Sigma (over i, j, k ) Cijk * Xijk

Constraints are on account of supply and demand. For supply i is kept constant and for demand k is kept constant as follows:

Sigma ( i is fixed and j varying) Xijk <= (less than equal to ) Si for i = 1,2,3 S1=18, S2=15, S3=23

Sigma ( k is fixed and j varying) Xijk >= ( greater than equal to ) Dk for k= 7, 8, 9 D7=20, D8=12, D9=20

For screen companies total quantity into must match with the total quantity supplied that is for fixed j Sigma (i varying ) Xijk = Sigma (k varying) Xijk

Total demand (52) is less than total supply (56)

Variable Coefficient X147 8 X148 10 X149 7 X157 9 X158 11 X159 11 X167 5 X168 7 X169 7 X247 7 X248 9 X249 6 X257 8 X258 10 X259   10 X267 7 X268 9 X269 9 X347 6 X348 8 X349 5 X357 7 X358 9 X359 9 X367 6 X368 8 369 8

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