Exam 3: Linear Programming: Computer Solution and Sensitivity Analysis

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. -What is the shadow price for assembly?

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. -What is the shadow price for fabrication?

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. -What is the optimal product mix and maximum profit?

A shadow price reflects which of the following in a maximization problem?

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 -A change in the market has increased the profit on the super product by $5. Total profit will increase by ________.

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 optimal number of regular products to produce is ________, and the optimal number of super products to produce is ________, for total profits of ________.

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 -What is the optimal daily profit?

Aunt Anastasia operates a small business: she produces seasonal ceramic objects to sell to tourists. For the spring, she is planning to make baskets, eggs, and rabbits. Based on your discussion with your aunt you construct the following table: $\begin{array} { l c c c c } \text { Product } & \begin{array} { c } \text { Mix/mold } \\\text { requirements (lb) }\end{array} & \begin{array} { c } \text { Kiln } \\\text { (units) }\end{array} & \begin{array} { c } \text { Paint \& Seal } \\\text { (hr) }\end{array} & \begin{array} { c } \text { Profit/product } \\\text { (\$) }\end{array} \\\hline \text { Baskets } & 0.500 & 1 & 0.250 & \$ 2.50 \\\text { Eggs } & 0.333 & 1 & 0.333 & \$ 1.50 \\\text { Rabbits } & 0.250 & 1 & 0.750 & \$ 2.00 \\\text { Capacity } & 20 \text { pounds } & 50 \text { units } & 80 \text { hours } & \\\hline\end{array}$ Your aunt also has committed to make 25 rabbits for a charitable organization. Based on the information in the table, you formulate the problem as a linear program. B = number of baskets produced E = number of eggs produced R = number of rabbits produced MAX 2.5B + 1.5E + 2R s.t. 0.5B + 0.333E + 0.25R ≤ 20 B + E + R ≤ 50 0.25B + 0.333E + 0.75R ≤ 80 R ≥ 25 The Excel solution and the answer and sensitivity report are shown below. The Answer Report: Target Cell (Max) Cell Name Original Value Final Value \ C\ 21 Profit 0 \ 112.5 $\text { Adjustable Cells }$ Cell Name Original Value Final Value \ C\ 18 Baskets 0 25 \ C\ 19 Eggs 0 0 \ C\ 20 Rabbits 0 25 $\text { Constraints }$ Cell Name Cell Value Formula Status Slack Not \G \1 3 Mix/mold 18.75 \G \1 3<= \F \1 3 Binding 1.25 \G \1 4 Kiln 50 \G \1 4<= \F \1 4 Binding 0 Not \G \1 5 Paint and Seal 25 \G \1 5<= \F \1 5 Binding 55 \G \1 6 Demand 25 \G \1 6>= \F \1 6 Binding 0 The Sensitivity Report: $\text { Adjustable Cells }$ Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \ C\ 18 Baskets 25 0 2.5 1+30 0.5 \C \1 9 Eggs 0 -1 1.5 1 1+30 \C \2 0 Rabbits 25 0 2 0.5 1+30 $\text { Constraints }$ Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \ G\ 13 Mix/mold 18.75 0 20 1+30 1.25 \ G\ 14 Kiln 50 2.5 50 2.5 25 \ G\ 15 Paint and Seal 25 0 80 1+30 55 \G \1 6 Demand 25 -0.5 25 25 5 -Aunt Anastasia is planning for next spring, and she is considering making only two products. Based on the results from the linear program, which two products would you recommend that she make?

Max Z = 3x1 + 3x2 Subject to : 10x1 + 4x2 ? 60    25x1 + 50x2 ? 200    x1, x2 ? 0 Determine the sensitivity range for each objective function coefficient.

The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. -What is the optimal weekly profit?

Sensitivity analysis is the analysis of the effect of ________ changes on the ________.

Given the following linear programming problem that minimizes cost: Min Z = 2x + 8y Subject to (1) 8x + 4y ? 64       (2) 2x + 4y ? 32       (3) y ? 2 -Determine the optimum values for x and y.

The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. -If the production manager decides to produce of 400 bottles of light beer and 0 bottles of dark beer, it will result in slack of

Aunt Anastasia operates a small business: she produces seasonal ceramic objects to sell to tourists. For the spring, she is planning to make baskets, eggs, and rabbits. Based on your discussion with your aunt you construct the following table: $\begin{array} { l c c c c } \text { Product } & \begin{array} { c } \text { Mix/mold } \\\text { requirements (lb) }\end{array} & \begin{array} { c } \text { Kiln } \\\text { (units) }\end{array} & \begin{array} { c } \text { Paint \& Seal } \\\text { (hr) }\end{array} & \begin{array} { c } \text { Profit/product } \\\text { (\$) }\end{array} \\\hline \text { Baskets } & 0.500 & 1 & 0.250 & \$ 2.50 \\\text { Eggs } & 0.333 & 1 & 0.333 & \$ 1.50 \\\text { Rabbits } & 0.250 & 1 & 0.750 & \$ 2.00 \\\text { Capacity } & 20 \text { pounds } & 50 \text { units } & 80 \text { hours } & \\\hline\end{array}$ Your aunt also has committed to make 25 rabbits for a charitable organization. Based on the information in the table, you formulate the problem as a linear program. B = number of baskets produced E = number of eggs produced R = number of rabbits produced MAX 2.5B + 1.5E + 2R s.t. 0.5B + 0.333E + 0.25R ≤ 20 B + E + R ≤ 50 0.25B + 0.333E + 0.75R ≤ 80 R ≥ 25 The Excel solution and the answer and sensitivity report are shown below. The Answer Report: Target Cell (Max) Cell Name Original Value Final Value \ C\ 21 Profit 0 \ 112.5 $\text { Adjustable Cells }$ Cell Name Original Value Final Value \ C\ 18 Baskets 0 25 \ C\ 19 Eggs 0 0 \ C\ 20 Rabbits 0 25 $\text { Constraints }$ Cell Name Cell Value Formula Status Slack Not \G \1 3 Mix/mold 18.75 \G \1 3<= \F \1 3 Binding 1.25 \G \1 4 Kiln 50 \G \1 4<= \F \1 4 Binding 0 Not \G \1 5 Paint and Seal 25 \G \1 5<= \F \1 5 Binding 55 \G \1 6 Demand 25 \G \1 6>= \F \1 6 Binding 0 The Sensitivity Report: $\text { Adjustable Cells }$ Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease \ C\ 18 Baskets 25 0 2.5 1+30 0.5 \C \1 9 Eggs 0 -1 1.5 1 1+30 \C \2 0 Rabbits 25 0 2 0.5 1+30 $\text { Constraints }$ Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease \ G\ 13 Mix/mold 18.75 0 20 1+30 1.25 \ G\ 14 Kiln 50 2.5 50 2.5 25 \ G\ 15 Paint and Seal 25 0 80 1+30 55 \G \1 6 Demand 25 -0.5 25 25 5 -Aunt Anastasia can obtain an additional 10 hours of kiln capacity free of charge from a friend. If she did this, how would her profits be affected?

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 MAX 30x+30y subject to 2x+y\leq60 2x+3y\leq120 x\leq45 The graphical solution is shown below. -What is the range for the shadow price for assembly?