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Problem 2 Product Mix: Acme Industries produces four types of men’s ties using three types of material. Your job is to determine how many of each type of tie to make each month. The goal is to maximize profit, profit per tie = selling price -labor cost –material cost. Labor cost is $0.75 per tie for all four types of ties. The material requirements and costs are given below. Material Cost per yard Yards available per month Silk $20 1,000 Polyester $6 2,000 Cotton $9 1,250 Product information Type of tie Type of tie Type of tie Type of tie Silk=s Poly=p Blend1=b Blend2=c Selling Price per tie $6.70 $3.55 $4.31 $4.81 Monthly Minimum units 6,000 10,000 13,000 6,000 Monthly Maximum units 7,000 14,000 16,000 8,500 Material Information in yards Type of tie Type of tie Type of tie Type of tie Silk Polyester Blen1 (50/50) Blend 2 (30/70) Silk 0.125 0 0 0 Polyester 0 0.08 0.05 0.03 Cotton 0 0 0.05 0.07 Formulate the problem as a linear program with an objective function and all constraints. Determine the optimal solution for the linear program using any software you want. Include a copy of the code and output. What are the optimal numbers of ties of each type to maximize profit? Problem 3 Making Change: Given coins of denominations (value) 1 = v1< v2< ... < vn, we wish to make change for an amount A using as few coins as possible. Assume that vi’s and A are integers. Since v1= 1 there will always be a solution. Solve the coin change using integer programming. For each of the following denomination sets and amounts, formulate the problem as an integer program with an objective function and constraints. Determine the optimal solution. What is the minimum number of coins used in each case and how many of each coin is used? Include a copy of your code. a)V = [1, 5, 10, 25] and A = 202. b)V = [1, 3, 7, 12, 27] and A = 293 Problem 4 Consider the following linear program. a) Write the following linear program in slack form. b) Please state what are the basic and non-basic variables in your slack form. Maximize 2x1 –6x3 Subject to x1 +x2 –x3 ≤7 3x1 –x2 ≥8 -x1 + 2x2 + 2x3 ≥0 x1 ≥0 x2 ≥0 x3 ≥0

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Problem 2
First of all, I computed the profit per tie for each type.
E.g. for Silk: selling price – material cost – labor cost = 6.70 – 0.125* 20 – 0.75 = 3.45
Similarly, for Polyester, B1 and B2 and got 2.32, 2.81, and 3.25 respectively

The objective function: Max 3.45S + 2.32P + 2.81B1+ 3.25B
Constraints are given by the 3 inequalities for material limitations and by the extreme values (min and max) for each type of tie.
The output can be seen below (highlighted in yellow for no. of ties to be produced for each type, while the profit is in red)....

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