Question

East Food Company imports food products such as meats, cheese, and pastries to the U.S. from warehouses at ports in Rome, Seville, and Rotterdam. Ships from these ports deliver the products to U.S. ports, i.e. Hampton, Charleston, and Jacksonville, where they are stored in company warehouses before being shipped to distribution centers in Houston, Kansas City, and Minneapolis. The products are then distributed to specialty food stores and sold through catalogs. The shipping costs ($/1,000 lb.) from the European ports to the U.S. ports and the available supplies (1,000 lb.) at the European ports are provided in the following table: European Port U.S. Ports Supply Hampton Charleston Jacksonville Rome $420 $390 $610 55 Seville 510 590 470 78 Rotterdam 450 360 480 37 The transportation costs ($/1,000 lb.) from each U.S. ports of the three distribution centers and the demands (1,000 lb.) at the distribution centers are as follows: U.S. Ports Distribution Center Houston Kansas City Minneapolis Hampton 75 63 81 Charleston 125 110 95 Jacksonville 68 82 95 Demand 60 45 50

a. [2 Marks] Develop a mathematical model that minimizes total transportation costs between the European ports and the warehouses and the distribution center at the U.S.

b. [1 Mark] Solve it using software and show the results. c. [1 Mark] What is the minimum total transportation cost?

d. [1 Mark] Discuss the results in the context of the slack/surplus.

QUESTION # 4. Consider Question 1 above. Suppose that due certain reasons, the available supply from Seville has been decreased to 50 (in 1,000 lb.). a. [2 Marks] Develop a mathematical model that minimizes total transportation costs between the European ports and the warehouses and the distribution center at the U.S., considering the decrease in the available supply from Seville.

b. [1 Mark] Solve it using software and show the results. c. [1 Mark] What is the minimum total transportation cost?

d. [1 Mark] Discuss the results in the context of the slack/surplus

Answer 1

a)

Mathematical model is as follows:

Let Xij be the quantity (in 1000 lbs) to be shipped from European port i to US port j , where i ={1,2,3}, j = {4,5,6}

Xjk be the quantity (in 1000 lbs) to be shipped from US port j to Distribution center k , where j = {4,5,6} and k = {7,8,9}

Minimize 420X14+390X15+610X16+510X24+590X25+470X26+450X34+360X35+480X36+75X47+63X48+81X49+125X57+110X58+95X59+68X67+82X68+95X69

s.t.

X14+X15+X16 <= 55

X24+X25+X26 <= 78

X34+X35+X36 <= 37

X14+X24+X34-X47-X48-X49 = 0

X15+X25+X35-X57-X58-X59 = 0

X16+X26+X36-X67-X68-X69 = 0

X47+X57+X67 = 60

X48+X58+X68 = 45

X49+X59+X69 = 50

Xij, Xjk >= 0

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b)

Solution using LINGO is as follows:

Optimal solution:

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c)

In the Solution Report - Lingo 1, refer Objective value

Minimum total transportation cost = 77,362

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d)

In the Solution Report - Lingo 1, refer Slack or Surplus is as follows:

We see that row 3 (X24+X25+X26 <= 78) has a slack of 15 , which means quantity supplied from Seville port is 15 less than its capacity, i.e. 63 out of 78 units available.

Supply from all other European ports is equal to their capacity.

Demand of all the distribution centers is satisfied.