# 0.7 Linear programing: the simplex method: homework  (Page 2/2)

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## Chapter review

Solve the following linear programming problems using the simplex method.

Maximize $z={5x}_{1}+{3x}_{2}$

subject to $\begin{array}{ccccc}{x}_{1}& +& {x}_{2}& \le & \text{12}\\ {2x}_{1}& +& {x}_{2}& \le & \text{16}\end{array}$

${x}_{1}\ge 0;{x}_{2}\ge 0$

${x}_{1}=4$ , ${x}_{2}=8$ , ${y}_{1}=0$ , ${y}_{2}=0$ , $z=\text{44}$

Maximize $z={5x}_{1}+{8x}_{2}$

subject to $\begin{array}{ccccc}{x}_{1}& +& {2x}_{2}& \le & \text{30}\\ {3x}_{1}& +& {x}_{2}& \le & \text{30}\end{array}$

${x}_{1}\ge 0$ ; ${x}_{2}\ge 0$

${x}_{1}=6$ , ${x}_{2}=\text{12}$ , ${y}_{1}=0$ , ${y}_{2}=0$ , $z=\text{126}$

Maximize $z={2x}_{1}+{3x}_{2}+{x}_{3}$

subject to $\begin{array}{ccccccc}{4x}_{1}& +& {2x}_{2}& +& {5x}_{3}& \le & \text{32}\\ {2x}_{1}& +& {4x}_{2}& +& {3x}_{3}& \le & \text{28}\end{array}$

${x}_{1},{x}_{2},{x}_{3}\ge 0$

${x}_{1}=6$ , ${x}_{2}=4$ , ${x}_{3}=0$ , ${y}_{1}=0$ , ${y}_{2}=0$ , $z=\text{24}$

Maximize $z={x}_{1}+{6x}_{2}+{8x}_{3}$

subject to $\begin{array}{ccccc}{x}_{1}& +& {2x}_{2}& \le & \text{1200}\\ {2x}_{2}& +& {x}_{3}& \le & \text{1800}\\ {4x}_{1}& +& {x}_{3}& \le & \text{3600}\end{array}$

${x}_{1},{x}_{2},{x}_{3}\ge 0$

${x}_{1}=\text{450}$ , ${x}_{2}=0$ , ${x}_{3}=\text{1800}$ , ${y}_{1}=\text{750}$ , ${y}_{2}=0$ , ${y}_{3}=0$ , $z=\text{14},\text{850}$

Maximize $z={6x}_{1}+{8x}_{2}+{5x}_{3}$

subject to $\begin{array}{ccccccc}{4x}_{1}& +& {x}_{2}& +& {x}_{3}& \le & \text{1800}\\ {2x}_{1}& +& {2x}_{2}& +& {x}_{3}& \le & \text{2000}\\ {4x}_{1}& +& {2x}_{2}& +& {x}_{3}& \le & \text{3200}\end{array}$

${x}_{1},{x}_{2},{x}_{3}\ge 0$

${x}_{1}=0$ , ${x}_{2}=\text{200}$ , ${x}_{3}=\text{1600}$ , ${y}_{1}=0$ , ${y}_{2}=0$ , ${y}_{3}=\text{1200}$ , $z=\text{9600}$

Minimize $z=\text{12}{x}_{1}+\text{10}{x}_{2}$

subject to $\begin{array}{ccccc}{x}_{1}& +& {x}_{2}& \ge & 6\\ {2x}_{1}& +& {x}_{2}& \ge & 8\end{array}$

${x}_{1}\ge 0$ ; ${x}_{2}\ge 0$

${x}_{1}=2$ , ${x}_{2}=4$ , $z=\text{64}$

Minimize $z={4x}_{1}+{6x}_{2}+{7x}_{3}$

subject to $\begin{array}{ccccccc}{x}_{1}& +& {x}_{2}& +& {2x}_{3}& \ge & \text{20}\\ {x}_{1}& +& {2x}_{2}& +& {x}_{3}& \ge & \text{30}\end{array}$

${x}_{1},{x}_{2},{x}_{3}\ge 0$

${x}_{1}=\text{10}$ , ${x}_{2}=\text{10}$ , ${x}_{3}=0$ , $z=\text{100}$

Minimize $z=\text{40}{x}_{1}+\text{48}{x}_{2}+\text{30}{x}_{3}$

subject to $\begin{array}{ccccccc}{2x}_{1}& +& {2x}_{2}& +& {x}_{3}& \ge & \text{25}\\ {x}_{1}& +& {3x}_{2}& +& 2{x}_{3}& \ge & \text{30}\end{array}$

${x}_{1},{x}_{2},{x}_{3}\ge 0$

${x}_{1}=\text{15}/4$ , ${x}_{2}=\text{35}/4$ , ${x}_{3}=0$ , $z=\text{570}$

A department store sells three different types of televisions: small, medium, and large. The store can sell up to 200 sets a month. The small, medium, and large televisions require, respectively, 3, 6, and 6 cubic feet of storage space, and a maximum of 1,020 cubic feet of storage space is available. The three types, small, medium, and large, take up, respectively, 2, 2, and 4 sales hours of labor, and a maximum of 600 hours of labor is available. If the profit made from each of these types is $40,$80, and $100, respectively, how many of each type of television should be sold to maximize profit, and what is the maximum profit? ${x}_{1}=0$ , ${x}_{2}=\text{40}$ , ${x}_{3}=\text{130}$ , ${y}_{1}=\text{30}$ , ${y}_{2}=0$ , ${y}_{3}=0$ , $z=\text{16},\text{200}$ A factory manufactures three products, A, B, and C. Each product requires the use of two machines, Machine I and Machine II. The total hours available, respectively, on Machine I and Machine II per month are 180 and 300. The time requirements and profit per unit for each product are listed below.  A B C Machine I 1 2 2 Machine II 2 2 4 Profit 20 30 40 How many units of each product should be manufactured to maximize profit, and what is the maximum profit? ${x}_{1}=0$ , ${x}_{2}=\text{30}$ , ${x}_{3}=\text{60}$ , ${y}_{1}=0$ , ${y}_{2}=0$ , $z=\text{3300}$ A company produces three products, A, B, and C, at its two factories, Factory I and Factory II. Daily production of each factory for each product is listed below.  Factory I Factory II Product A 10 20 Product B 20 20 Product C 20 10 The company must produce at least 1000 units of product A, 1600 units of B, and 700 units of C. 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