Question

Krappy Kraft Beer Inc. has two breweries in BC, with monthly capacities and production costs as follows:

Brewery	Capacity/month	Cost per six-pack
1	10000 six-packs	$1.60
2	9500 six-packs	$1.80

The product is channeled through 4 different distributors. Bart’s Beer is so popular that all stock that can be brewed is quickly purchased by the distributors, but the distributors can only handle a certain amount of product per month. Also, each distributor is willing to pay a different amount per six-pack to Bart:

Distributor	Capacity/month	Price per six-pack
A	8500	$10.75
B	4500	$9.95
C	5500	$11.00
D	6500	$10.50

Bart’s Beer Inc. has to pay the cost of shipping to the 4 distributors. The shipping cost per six-pack from the breweries to distributors is as follows:

Brewery	A	B	C	D
1	$1.00	$1.15	$1.85	$2.00
2	$0.65	$.75	$1.50	$1.75

Formulate an LP model that will enable Bart’s Beer to maximize its profits each month.

Complete the following:

a) Develop the Transportation Network.

b) Formulate the problem into proper LP format(2marks)

c) Use Solver to determine the optimal solution. State the optimal solution in the context of the business problem.

d) Generate an “answer report” and “sensitivity report” from Solver and answer the following questions:

i. Interpret the meaning of the shadow price for the monthly production capacity limitation of Brewery #2.

ii. Interpret the meaning of the reduced cost for shipping six-packs from Brewery 1 to Distributor D.

Answer 1

Solution:

Given

Monthly capacities and production costs:

Brewery	Capacity/month	Cost per six-pack
1	10000 Six-Packs	$1.60
2	9500 Six-Packs	$1.80

Each distributor is willing to pay a different amount per six-pack to Bart:

Brewery	Capacity/month	Cost per six-pack
A	8500	$10.75
B	4500	$9.95
C	5500	$11.00
D	6500	$10.50

The shipping cost per six-pack from the breweries to distributors:

Brewery	A	B	C	D
1	$1.00	$1.15	$1.85	$2.00
2	$0.65	$.75	$1.50	$1.75

Objective Function (Z) = Profit = Revenue - Production Cost - Transportation Costs

Variables = Production shipped to each distributor

Revenue = Sales to each distributor x Price

Production Cost = Production at Brewery 1 x 1.6 + Production at Brewery 2 x 1.8

Transportation Cost = Product shipped to each location x transportaiton cost

Sales	A	B	C	D
			1	5443	0	3557	1000	10000
			2	3057	4500	1943	0	9500
				8500	4500	5500	1000
			Revenue					207150
			Tpt Cost
			1	1.00	1.15	1.85	2.00	14,023
			2	0.65	0.75	1.50	1.75	8,277
			Prod Cost
			1					16000
			2					17100
			Profit	151750

The optimal solution is as given below:

Sales	A	B	C	D
1	5443	0	3557	1000	10000
2	3057	4500	1943	0	9500
	8500	4500	5500	1000

The interpretation is that Brewery 1 should make 10000 units and ship 5443 to A, 3557 to B and 1000 to D. Similar it goes for Brewery 2 as well

Answer Report:

Cell	Name	Original Value	Final Value
$F$18	Profit A	151750	151750
Cell	Name	Original Value	Final Value	Integer
$F$7	A	5443	5443	Integer
$G$7	B	0	0	Integer
$H$7	C	3557	3557	Integer
$I$7	D	1000	1000	Integer
$F$8	A	3057	3057	Integer
$G$8	B	4500	4500	Integer
$H$8	C	1943	1943	Integer
$I$8	D	0	0	Integer
Cell	Name	Cell Value	Formula	Status	Slack
$F$9	A	8500	$F$9<=$F$3	Binding	0
$G$9	B	4500	$G$9<=$G$3	Binding	0
$H$9	C	5500	$H$9<=$H$3	Binding	0
$I$9	D	1000	$I$9<=$I$3	Not Binding	5500
$J$7		10000	$J$7<=$C$3	Binding	0
$J$8		9500	$J$8<=$C$4	Binding	0
$F$7=Integer
$F$8=Integer
$G$7=Integer
$G$8=Integer
$H$7=Integer
$H$8=Integer
$I$7=Integer
$I$8=Integer

   i) Interpret the meaning of the shadow price for the monthly production capacity limitation of Brewery 2

Ans: Shadow price means that by chancing the capacity by 1 units, how much will the ouput change by.

ii) Interpret the meaning of the reduced cost for shipping six-packs from Brewery 1 to Distributor D.  

Ans: By reducing the cost of shipping from Brewery 1 to D, we can increase the product shipped to D instead of B as D is getting more profit than B.

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Thank you.