Alliance Flour produces commercial all-purpose flour from four plants in the U.S

ID: 398464 • Letter: A

Question

Alliance Flour produces commercial all-purpose flour from four plants in the U.S. and ships them to the two biggest bread producers in the country. However, Alliance Flour doesn’t know how to most effectively ship its flour to satisfy their contracted demand for the next month. Alliance Flour’s four manufacturing sites are Los Angeles, Seattle, Chicago, and Boston with a monthly production supply of 8,000, 9,000, 1,000, and 7,000 respectively. The two bread producers are located in Dallas and Orlando with an upcoming monthly demand of 20,000 and 5,000 respectively which Alliance Flour must satisfy. In addition, Alliance Flour does not directly transport its flour as it contracts a shipping company that requires all of Alliance Flour’s production to be sent a distribution center before it is sent to the bread producers. The shipping company has two distribution centers located in Nashville and Denver and they have told Alliance Flour that they can handle at most 24,000 and 11,000 pounds of flour that month respectively.

By using the shipping company, Alliance Flour has contracted the following $/lb to ship its products between all possible destinations:

Using this information, complete the following tasks/questions:

Mathematically formulate (i.e. type out or write down) a linear optimization model to minimize Alliance Flour’s transportation costs for the month including all decision variables, the objective function, and all constraints.

Decision Variables

Objective Function

Constraints

Distribution Center (DC) Nashville Denver Manufacture site LosAngeles $ 4.07 $ 4.21 Seattle $ 3.64 $ 4.70 Chicago $ 3.30 $ 2.53 Boston $ 3.18 $ 2.79 Bread Producer Dallas Orlando DC Nashville $ 1.40 $ 2.04 Denver $ 1.66 $ 2.44

Explanation / Answer

Decision Variables:

Let Xij = represents pounds of flour to be transported form Manufacture site i to the distribution center j.

Xjk = represents pounds of flour to be transported form distribution center j to Bread producer k.

Where, i = 1, 2, 3, and 4 for Manufacture sites at Los Angeles, Seattle, Chicago, and Boston respectively

j = 5 and 6 for distribution centers at Nashville and Denver respectively

k = 7 and 8 for bread producer at Dallas and Orlando respectively

Objective Function:

Objective is to minimize the total cost of transportation from manufacture sites to Dc to bread producer.

Min Z = $4.07X15 + $4.21X16 + $3.64X25 + $4.70X26 + $3.30X35 + $2.53X36 + $3.18X45 + $2.79X46 + $1.40X57 + $2.04X58 + $1.66X67 + $2.44X68

Subject to:

Supply constraint at the manufacturing sites: Manufacturing sites cannot supply more than available capacity to DC’s

Los Angeles

X15 + X16 <= 8000

Seattle

X25 + X26 <= 9000

Chicago

X35 + X36 <= 1000

Boston

X45 + X46 <= 7000

Transshipment constraints: At DC the pounds received from sites should equal to pounds supplied from DC to producer plants.

Nashville

X15 + X25 + X35 + X45 = X57 + X58

X15 + X25 + X35 + X45 - X57 - X58 = 0

Denver

X16 + X26 + X36 + X46 = X67 + X68

X16 + X26 + X36 + X46 - X67 - X68 = 0

DC capacity Constraint: DC’s cannot receive pounds from sites more than their capacity

Nashville

X15 + X25 + X35 + X45 <= 24000

Denver

X16 + X26 + X36 + X46 <= 11,000

Bread producer demand constraints: Each producer demand should be satisfied

Dallas

X57 + X67 >= 20,000

Orlando

X58 + X68 >= 5,000

Nonnegativity Constraint: All Xij, Xjk >= 0