SOLVE & SHOW YOUR WORK STEP BY STEP ON EXCEL

Klein Chemicals, Inc., produces a special oil-based material that is currently in short supply. Four of Klein's customers have already placed orders that together exceed the combined capacity of Klein's two plants. Klein's management faces the problem of deciding how many units it should supply to each customer. Because the four customers are in different industries, different prices can be charged because of the various industry pricing structures. However, slightly different production costs at the two plants and varying transportation costs between the plants and customers make a "sell to the highest bidder" strategy unacceptable. After considering price, production costs, and transportation costs, Klein established the following profit per unit for each plant-customer alternative.

Plant	Customer
	D1 1	D2 2	D3 3	D4 4
Clifton Springs 1	$32	$34	$32	$40
Danville 2	$34	$30	$28	$38

The plant capacities and customer orders are as follows.

Plant	Capacity (units)
Clifton Springs	5,000
Danville	3,000

Distributor Orders (units)
D1	2,000
D2	5,000
D3	3,000
D4	2,000

(a)

How many units should each plant produce for each customer to maximize profits?

Optimal Solution

	Units	Cost
Clifton SpringsD1	_______	$ _______
Clifton SpringsD2	_______	$ _______
Clifton SpringsD3	_______	$ _______
Clifton SpringsD4	_______	$ _______
DanvilleD1	_______	$ _______
DanvilleD2	_______	$ _______
DanvilleD3	_______	$ _______
DanvilleD4	_______	$ _______
Total Cost	_______	$ _______

(b)

Which customer demands will not be met?

Distributor 1 will have a shortfall of _______ units.

Distributor 2 will have a shortfall of _______ units.

Distributor 3 will have a shortfall of _______ units.

Distributor 4 will have a shortfall of_______ units.

(c)

This answer has not been graded yet.

(ii)

linear programming formulation

Let x_ij = number of units i shipped to client j, using the indices from the given table. (It may be necessary to combine plants or distributors in a single node in order to solve this problem. Use index number 5 for this type of node. Enter "DNE" in any unused answer blanks.)

Max ___________________

s.t.

Orders from Clifton Springs ___________________

Orders from Danville ___________________

Orders from/for Dummy Node ___________________

Orders for D₁ ___________________

Orders for D₂ ___________________

Orders for D₃ ___________________

Orders for D₄___________________

x_ij 0 for all i, j.