Question

Universal Technologies, Inc. has identified two qualified vendors with the capability to supply some of its electronic components. For the coming year, Universal has estimated its volume requirements for these components and obtained price-break schedules from each vendor. (These are summarized as “all-units” discounts in the table below.) Universal’s engineers have also estimated each vendor’s maximum capacity for producing these components, based on available information about equipment in use and labor policies in effect. Finally, because of its limited history with Vendor A, Universal has adopted a policy that permits no more than 60% of its total unit purchases on these components to come from Vendor A.

		Vendor A				Vendor B
Product	Requirement	Unit price		Volume required		Unit price	Volume required
1	500	$225	0–250		$224		0–300
		$220	250–500		$214		300–500
2	1000	$124	0–600		$120		0–1000
		$115	600–1000				(no discount)
3	2500	$60	0–1000		$54		0–1500
		$56*	1000–2000		$52		1500–2500
		$51	2000–2500
Total capacity (units)				2500			2000

*For example, if 1400 units are purchased from Vendor A, they cost $56 each, for a total of $78,400.

What is the minimum-cost purchase plan for Universal?

Answer 1

Let A1 & B1 be the number of Product 1 items purchased from Vendor 1 & Vendor 2 respectively.

Let A2 & B2 be the number of Product 2 items purchased from Vendor 1 & Vendor 2 respectively.

Let A3 & B3 be the number of Product 3 items purchased from Vendor 1 & Vendor 2 respectively.

Let X1 & Y1 be the Binary variable determining if the Product 1 purchased from Vendor 1 & Vendor 2 exceeds 250 & 300 respectively.

Let X2 be the Binary variable determining if the Product 2 purchased from Vendor 1 exceeds 600.

Let X3, X4 & Y3 be the Binary variable determining if the Product 1 purchased from Vendor 1 & Vendor 2 exceeds 1000, 2000 & 1500 respectively.

Our Objective is to minimize the cost in which the purchase requirements are met. Thus our objective function is-

Min(225*A1-5*X1*(A1-250) + 224*B1-10*Y1*(B1-300) + 124*A2-9*X2*(A2-600) + 120*B2 + 60*A3-4*X3*(A3-1000)-5*X4*(A3-2000) + 54*B3-2*Y3*(B3-1500)

Constraints-

From Demand Side-

A1+B1 = 500

A2+B2 = 1000

A3+B3 = 2500

From Supply Side-

A1+A2+A3 <= 2800

B1+B2+B3 <= 2400

The Solver Solution is as shown-

The Total Cost considering all discounts = 372450

Total Price without any discount = 500*225+1000*120+1100*60+1400*54 = 374100

Percentage of total cost this increase represents = (374100-372450)/372450 = 0.00443 = 0.443%