MMIS 671:

Question 1. Optimization Models [15 Points]

A company provides 4 types of products to a client A, B, C, and D. In-house production costs are estimated to be $50, $72, $93, and $115 per unit for products A, B, C, and D respectively. Each product requires three types of raw materials RM1, RM2, and RM3 for in-house production:

• 1 unit of RM1, 1 unit of RM2, and 1 unit of RM3 per unit of product A;
• 2 units of RM1, 1 unit of RM2, and 1 unit of RM3 per unit of product B;
• 1 unit of RM1, 2 units of RM2, and 1 unit of RM3 per unit of product C; and
• 1 unit of RM1, 1 unit of RM2, and 2 units of RM3 per unit of product D.

The cost of raw materials RM1, RM2, and RM3 are $2, $3, and $5 per unit, respectively. Over the upcoming month the company has 6,500 units of RM1, 6,000 units of RM2, and 5,500 units of RM3 available. The company is contractually obligated to provide the client with 4,000 units of product A, 3,000 units of product B, 2,000 units of product C, and 1,000 units of product D in the upcoming period.

Limited resource availability prevents the company from meeting the entire demand for all products through in-house production alone. The company has two other options: it can obtain products from either a small-supplier or a large-supplier; no resources are used for products obtained from suppliers. The small-supplier can supply at most 1,000 units of each product; there is no limit to the quantities of products that the large-supplier can supply.

The small-supplier charges $55, $78, $103, and $130 per unit of products A, B, C, and D, respectively. The large-supplier charges $63, $88, $118, and $150 per unit of products A, B, C, and D, respectively.

For your convenience, the information presented above is summarized in the table below:

	Product	A	B	C	D	Available	Cost/unit
Production requires	RM1	1	2	1	1	6,500	$2
	RM2	1	1	2	1	6,000	$3
	RM3	1	1	1	2	5,500	$5
Demand		4,000	3,000	2,000	1,000
Capacity of small supplier		1,000	1,000	1,000	1,000
Cost per unit	production	$50	$72	$93	$115
	small supplier	$55	$78	$103	$130
	large supplier	$63	$88	$118	$150

1. What is the minimum cost attainable under an optimal plan?

The minimum cost attainable is $ .

2. How many products of each type should be produced in-house, obtained from small-supplier, and obtained from large-supplier under this optimal plan?

Number of items under optimal solution

Number of products	A	B	C	D
In-house production
Obtained from small-supplier
Obtained from large-supplier

ii How many units of RM1, RM2, and RM3 are used up under this optimal plan?

Resources used under optimal solution

Resource	Units used	Available
RM1		6,500
RM2		6,000
RM3		5,500

b) At most how much should the company be willing to pay to increase the availability of each raw material by 1 unit (over its current availability)?

Resource	Maximum cost per additional unit
RM1
RM2
RM3

c)

1. The company has located an alternate supplier (New-Supplier) for product C who can supply at most 200 units of product C in the upcoming month. Price per unit of C is subject to negotiations. What is the maximum amount that the company should be willing to pay New-Supplier per unit of product C? Justify your answer by explaining your approach. [3 points]

The maximum amount that the company should be willing to pay New-Supplier per unit of product C is . (round to nearest cent).

Reasoning: