Question

Part II: Linear Programming Model- Forbelt Corporation has a one-year contract to supply motors for all refrigerators produced by the Ice Age Corporation. Ice Age manufacturers the refrigerators at four locations around the country: Boston, Dallas, Los Angeles, and St. Paul. Plans call for the following number (in thousands) of refrigerators to be produced at each location: Boston 50 Dallas 70 Los Angeles 60 St. Paul 80 Forbelt’s three plants ae capable of producing the motors. The plans and production capacities (in thousands) are as follows: Denver 100 Atlanta 100 Chicago 150 Because of varying production and transportation costs, the profit that Forbelt earns on each lot of 1000 units depends on which plant produces the lot and which destination it was shipped to. Ship to: Produced At: Boston, Dallas, Los Angeles ,St. Paul Denver 7 , 11 . 8 13 Atlanta 20 17 . 12 10 Chicago 8 18 13 16 With profit maximization as a criterion, Forbelt’s management wants to determine how many motors should be produced at each plant and how many motors should be shipped form each plant to each destination. Find the optimal solution.

* I have the solution to the above problem, I need help with calculating profit (ex, when distribution changes)

Answer 1

Below is the Step by Step Profit Maximization table as per VAM method-

As the DEmand and Supply is not equal - this is unbalanced so we introduce a Dummy Demand City to balance it as shown below-

Estimates of the profit per unit
	Shipped to
Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Supply Capacity
Denver	7	11	8	13	0	100
Atlanta	20	17	12	10	0	100
Chicago	8	18	13	16	0	150
Demand	50	70	60	80	90	350

As this is a Profit Maximization requirement we convert it into a relative loss Matrix by Subtracting each element from 20 ( Higest profit ) including itself -

		1	2	3	4	5
	Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Capacity
A	Denver	13	9	12	7	20	100
B	Atlanta	0	3	8	10	20	100
C	Chicago	12	2	7	4	20	150
	Demand	50	70	60	80	90

Here the Smallest number shows the minimum cost ( relatively less loss) thus is a place of maximum profit- So we search for the lowest element and allocate the supply as per the demand in that cell from Capacity and Demand- If the Balance is left in Supply or Demand we leave it as it is and move to other lowest Element cell and allocate the value till we come back and have all the Demand met-

	Ref		1	2	3	4	5
Supplying at B1 ( 0)		Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Capacity	Capacity after Action
	A	Denver						100
	B	Atlanta	50					100	50
	C	Chicago						150
		Demand	50	70	60	80	90
		Demand After Action	0

Supplying at c2 ( 2)		Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Capacity	Capacity after Action
	A	Denver						100
	B	Atlanta	50					100	50
	C	Chicago		70				150	80
		Demand	50	70	60	80	90
		Demand After Action	0	0

Supplying at C4 (4)		Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Capacity	Capacity after Action
	A	Denver						100
	B	Atlanta	50					100	50
	C	Chicago		70		80		150	0
		Demand	50	70	60	80	90
		Demand After Action	0	0		0

	Ref		1	2	3	4	5
Supplying at B3 (8)		Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Capacity	Capacity after Action
	A	Denver						100
	B	Atlanta	50		50			100	0
	C	Chicago		70		80		150	0
		Demand	50	70	60	80	90
		Demand After Action	0	0	10	0

	Ref		1	2	3	4	5
Supplying at A3 (12)		Produced At	Boston	Dallas	Los Angeles	St. Paul	Dummy	Capacity	Capacity after Action
	A	Denver			10			100	90
	B	Atlanta	50		50			100	0
	C	Chicago		70		80		150	0
		Demand	50	70	60	80	90
		Demand After Action	0	0	0	0

A ) Network representation of the model

Denvar ---------------10 units-----------> Los Angeles

Atlanta---------------50 units-----------> Los Angeles

Atlanta---------------50 units-----------> Boston

Chicago ---------------70 units-----------> Dallas

Chicago ---------------80 units-----------> St Paul

B ) Optimum Solution in Thousands ( Number of Units Alloted X Profit Rate ) = (50 X 20) + (70 X 18) + (80 X 16) + (50 X 12) + (10 X 8) = $ 4940