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:

		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

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

Answer 1

Let X; = Pounds of flour transported form manufacturing site i to the distribution center j. Xit = Pounds of flour to be transported form distribution center j to Bread producer k. Where. Decision variables 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 Minimize cost Z= 4.07 X15 +3.64 X25 +3.30 X35 +3.18 X45 + Objective Function 4.21 X16 +4.70 X26 +2.53 X36 +2.79 X46 + 1.40 X57+2.04 X58+ 1.66 X67 +2.44 X68 subject to X15 + X16 <= 8000 X25 + X26 <= 9000 X35 + X36 <= 1000 X45+X46 <= 7000 X37 + X67 >= 20,000 X38 + X68 >= 5,000 Supply capacity Los Angeles Seattle Chicago Boston Demand Dallas Orlando Handling capacity Nashville Denver In-out equality (flour received = flour shipped) Nashville Denver Non negativity constraint Constraints X15 + X25 + X35 + X45 = 24000 X16+X26+X36 + X46 <= 11,000 X15 + X25 +X35 + X45 = X37 + Xs8 X16 + X26 + X36 + X46 = X67 +X68 x; X = 0

F G H L M N O D Distribution Center | Nashville Denver LosAngeles 4.07 4.21 Seattle 3.64 Chicago 2.53 Boston 3.18 2.79 Bread Producer Dallas Orlando |2.04 2.44 Distribution Center Nashville 1.4 Denver 1.66 47 Manufacture site 3.3 Distribution Center Nashville Denver Bread Producer Dallas Orlando Distribution Center Nashville Denver =SUM(K10-L 10) ==C14 =SUM(K11:L11) ==D14 Manufacture site LosAngeles Seattle Chicago Boston =SUM(C10-D10) <=8000 =SUM(C11:D11) <= 9000 =SUM(C12:D12) <=1000 =SUM(C13-D13) <=7000 =SUM(K10:K11) =SUM(L 10:L11) =SUM(C10:C13) =SUMD10:D13) 20000 5000 24000 11000 Cost =SUMPRODUCT(C3:D6.C10:D13)+SUMPRODUCT(K3-L4.K10:L11)

E F G H I D Distribution Center Nashville Denver LosAngeles 4.07 Seattle 3.64 4.7 Chicago 3.3 2.53 Boston 3.18 2.79 4.21 Manufacture site Distribution Center Nashville Denver =SUM(C10:D10) <=8000 =SUM(C11-D11) <= 9000 1=SUM(C12D12) <= 1000 1=SUM(C13-D13) <=7000 =SUM(C10:C13) =SUMD10:D13) LosAngeles Seattle Chicago Boston Manufacture site <= 24000 11000 Cost =SUMPRODUCT(C3:D6,C10-D13)+SUMPRODUCT(K3-L4,K10-L11)

F G H M N O Bread Producer Dallas Orlando 2.04 2.44 Distribution Center Nashville 1.4 Denver 1.66 Bread Producer Dallas Orlando Distribution Center Nashville Denver =SUM(K10-L10) ==C14 =SUM(K11-L11) ==D14 8000 11 = 9000 12 < 1000 13 <= 7000 =SUM(K10:K11) =SUM(L10-L11) 20000 5000

Solver Parameters Set Objective: $B$18 То: Омах о міn Ovalue of o By Changing Variable Cells: $C$10:$D$13,$K$ 10:$L$11 Add Subject to the Constraints: $C$14:$D$14 <= $C$16:$D$16 $E$10:$E$13 <= $G$10:$G$13 $K$12:$L$12 >= $K$14:$L$14. $M$10:$M$11 = $0$10:$0$11 Change Delete Reset All Load/Save Make Unconstrained variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help Solve Close

E F G H M N O C D Distribution Center Nashville Denver LosAngeles $4.07 $4.21 Seattle $3.64 $4.70 Chicago $3.30 $2.53 Boston $3.18 $2.79 K L Bread Producer Dallas Orlando Nashville $1.40 $2.04 Denver $1.66 $2.44 Distribution Center Manufacture site Bread Producer Dallas Orlando Nashville 12000 5000 Denver 80000 20000 5000 Distribution Center Nashville Denver LosAngeles 80000 8000 8000 Seattle 9000 0 9000 <= 9000 Chicago 0 1000 1000 <=1000 Boston 0 7000 7000 <=7000 17000 8000 Distribution Center 17000 =17000 8000 = 8000 Manufacture site 20000 5000 24000 11000 Cost S 127,660

The minimum cost at which the objective can be achieved is $127660