Exam 3: Linear Programming: Computer Solution and Sensitivity Analysis

The linear programming problem whose output follows is used to determine how many bottles of red nail polish (x1), blue nail polish (x2), green nail polish (x3), and pink nail polish (x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand for blue, green, and pink nail polish bottles combined is at least 50 bottles. MAX $100 x _ { 1 } + 120 x _ { 2 } + 150 x _ { 3 } + 125 x _ { 4 }$ Subject to 1 . $x _ { 1 } + 2 x _ { 2 } + 2 x _ { 3 } + 2 x _ { 4 } \leq 108$    2. $3 x _ { 1 } + 5 x _ { 2 } + x _ { 4 } \leq 120$    3. $x _ { 1 } + x _ { 3 } \leq 25$    4. $x _ { 2 } + x _ { 3 } + x _ { 4 } \geq 50$     $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \geq 0$ Optimal Solution: Objective Function Value = 7475.000 Variable Value Reduced Costs 8 0 0 5 17 0 33 0 Constraint Slack/ Surplus Dual Prices 1 0 75 2 63 0 3 0 25 4 0 -25 Objective Coefficient Ranges Variable Lower Limit Current Value Upper Limit 87.5 100 none none 120 125 125 150 162 120 125 150 Right Hand Side Ranges Constraint Lower Limit Current Value Upper Limit 1 100 108 123.75 2 57 120 none 3 8 25 58 4 41.5 50 54 -a) To what value can the per bottle profit on red nail polish drop before the solution (product mix) would change? b) By how much can the per bottle profit on green nail polish increase before the solution (product mix) would change?

Mallory Furniture buys two products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. Graphically solve this problem and answer the following questions. -If Mallory Furniture is able to increase the profit per medium shelf to $200, would the company purchase medium shelves? If so, what would be the new product mix and the total profit?

Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor and $30 on each lawn mower, and they sell all they can produce. The time requirements in the machine shop, fabrication, and tractor assembly are given in the table. Machine Shop Fabrication Assembly Tractor 2 hours 2 hours 1 hour Lawn Mower 1 hour 3 hours 0 hour Hrs. Available 60 hours 120 hours 45 hours Formulation: Let x = number of tractors produced per period y = number of lawn mowers produced per period Let x = number of tractors produced per period y = number of lawn mowers produced per period MAX 30x + 30y subject to  2x + y ? 60     2x + 3y ? 120     x ? 45 The graphical solution is shown below. -How many tractors and saws should be produced to maximize profit, and how much profit will they make?

Which of the following could not be a linear programming problem constraint?

Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management. Max R = 14Z + 13Y + 17X subject to: Beef 2Z + 3Y + 4X ≤ 28 Cheese 9Z + 8Y + 11X ≤ 80 Beans 4Z + 4Y + 2X ≤ 68 X,Y,Z ≥ 0 The sensitivity report from the computer model reads as follows: $\text { Variable Cells }$ Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \C \4 1.45 0 14 0.63 5.33 \D \4 Y 8.36 0 13 8 0.56 \E \4 0 -0.818 17 0.818 1+30 $\text { Constraints }$ Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \F \6 Beef 28 0.45 28 2 10.22 \F \7 Cheese 80 1.45 80 46 5.33 \FS 8 Beans 39.27 0 68 1+30 28.73 -How many pounds of beans will Taco Loco have left over if they produce the optimal quantity of products X, Y, and Z?

The production manager for the Whoppy soft drink company is considering the production of two kinds of soft drinks: regular (R) and diet (D). The company operates one 8-hour shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup, is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. The formulation for this problem is given below. MAX Z = $3R + $2D s.t.    2R + 4D ? 480    5R + 3D ? 675 The sensitivity report is given below. Adjustable Cells Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \ \ 6 Regular =90.00 0.00 3 0.33 2 \ \ 6 Diet =75.00 0.00 2 4 0.2 Constraints Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \ \ 3 Production (minutes) 480.00 0.07 480 420 210 text E \ 4 Syrup (gallons) 675.00 0.57 675 525 315 -How many cases of regular and how many cases of diet soft drink should Whoppy produce to maximize daily profit?

Sensitivity analysis determines how a change in a parameter affects the optimal solution.

Consider the following linear program, which maximizes profit for two products--regular (R) and super (S): MAX 50R + 75S s.t.    1.2 R + 1.6 S ? 600 assembly (hours)    0.8 R + 0.5 S ? 300 paint (hours) .   16 R + 0.4 S ? 100 inspection (hours) Sensitivity Report: Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \ \ 7 Regular = 291.67 0.00 50 70 20 \C \ 7 Super = 133.33 0.00 75 50 43.75 Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \ \ 3 Assembly (hr/unit) 563.33 0.0 600 1 mathrm E +30 36.67 mathrm E \ 4 Paint (hr/unit) 300.00 33.33 300 39.29 175 mathrm E \ 5 Inspect (hr/unit) 100.00 145.83 100 12.94 40 -The profit on the super product could increase by ________ without affecting the product mix.

Sensitivity ranges can be computed only for the right-hand sides of constraints.

A change in the value of an objective function coefficient will always change the value of the optimal solution.

If we change the constraint quantity to a value outside the sensitivity range for that constraint quantity, the shadow price will change.

Given the following linear programming problem: Max Z = 15x + 20y S .t.    8x + 5y ? 40    4x + y ? 4 What would be the values of x and y that will maximize revenue?

The marginal value of any scarce resource is the dollar amount one should be willing to pay for one additional unit of that scarce resource.

Consider the following linear program, which maximizes profit for two products--regular (R) and super (S): MAX 50R + 75S s.t.    1.2 R + 1.6 S ? 600 assembly (hours)    0.8 R + 0.5 S ? 300 paint (hours) .   16 R + 0.4 S ? 100 inspection (hours) Sensitivity Report: Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \ \ 7 Regular = 291.67 0.00 50 70 20 \C \ 7 Super = 133.33 0.00 75 50 43.75 Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \ \ 3 Assembly (hr/unit) 563.33 0.0 600 1 mathrm E +30 36.67 mathrm E \ 4 Paint (hr/unit) 300.00 33.33 300 39.29 175 mathrm E \ 5 Inspect (hr/unit) 100.00 145.83 100 12.94 40 -If downtime reduced the available capacity for painting by 40 hours (from 300 to 260 hours), profits would be reduced by ________.

Given the following linear program that maximizes revenue: Max Z = 15x + 20y S )t.    8x + 5y ? 40    4x + y ? 4 What is the maximum revenue at the optimal solution?

________ is the analysis of the effect of parameter changes on the optimal solution.

Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management. Max R = 14Z + 13Y + 17X subject to: Beef   2Z + 3Y + 4X ? 28 Cheese  9Z + 8Y + 11X ? 80 Beans  4Z + 4Y + 2X ? 68 X,Y,Z ? 0 The sensitivity report from the computer model reads as follows: $\text { Variable Cells }$ Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \C \4 1.45 0 14 0.63 5.33 \D \4 Y 8.36 0 13 8 0.56 \E \4 0 -0.818 17 0.818 1+30 $\text { Constraints }$ Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \F \6 Beef 28 0.45 28 2 10.22 \F \7 Cheese 80 1.45 80 46 5.33 \FS 8 Beans 39.27 0 68 1+30 28.73 -The optimal quantity of the three products and resulting revenue for Taco Loco is:

Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management. Max R = 14Z + 13Y + 17X subject to: Beef 2Z + 3Y + 4X ≤ 28 Cheese 9Z + 8Y + 11X ≤ 80 Beans 4Z + 4Y + 2X ≤ 68 X,Y,Z ≥ 0 The sensitivity report from the computer model reads as follows: $\text { Variable Cells }$ Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \C \4 1.45 0 14 0.63 5.33 \D \4 Y 8.36 0 13 8 0.56 \E \4 0 -0.818 17 0.818 1+30 $\text { Constraints }$ Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \F \6 Beef 28 0.45 28 2 10.22 \F \7 Cheese 80 1.45 80 46 5.33 \F \8 Beans 39.27 0 68 1+30 28.73 -If Taco Loco reduces the price of the X product by about 82 cents, then their optimal product mix will contain ________ X.

For a resource constraint, either its slack value must be ________ or its shadow price must be ________.

Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management. Max R = 14Z + 13Y + 17X subject to: Beef 2Z + 3Y + 4X ≤ 28 Cheese 9Z + 8Y + 11X ≤ 80 Beans 4Z + 4Y + 2X ≤ 68 X,Y,Z ≥ 0 The sensitivity report from the computer model reads as follows: $\text { Variable Cells }$ Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \C \4 1.45 0 14 0.63 5.33 \D \4 Y 8.36 0 13 8 0.56 \E \4 0 -0.818 17 0.818 1+30 $\text { Constraints }$ Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \F \6 Beef 28 0.45 28 2 10.22 \F \7 Cheese 80 1.45 80 46 5.33 \FS 8 Beans 39.27 0 68 1+30 28.73 -Taco Loco is unsure whether the amount of beef that their computer thinks is in inventory is correct. What is the range in values for beef inventory that would not affect the optimal product mix?