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Problem 1

TOC Company produces two products, Y and Z that are processed in four departments, A, B, C and D. Product Y requires three types of materials, M1, M2 and M4. Product Z requires two types of materials, M2 and M3. The company's production process is illustrated in the following graphic.

The requirements for each product are summarized in the table below.

Resource Required per unit of

Product Y Required per unit of

Product Z

Material 1 $100 -

Material 2 $100 $100

Material 3 - $100

Material 4 $15 -

Department A 15 minutes 10 minutes

Department B 15 minutes 30 minutes

Department C 15 minutes 5 minutes

Department D 15 minutes 5 minutes

Each department has 2,400 minutes of available time per week. The Company's operating expenses are $30,000 per week. Based on current demand, the company can sell 100 units of product Y and 50 units of

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product Z per week. Sales prices are $450 for product Y and $500 for product Z. All four materials are available in sufficient quantities. The needed workers are also available. Required:

1. Determine the company's constraint. 2. Determine the throughput per unit for each product. 3. Determine the throughput per minute of the constrained resource for each product. 4. Determine the product mix needed to maximize throughput, i.e., the number of units of Y and Z that

should be produced per week. 5. Determine the maximum net income per week for TOC Company. 6. Suppose the company broke the current constraint by doubling the capacity of that resource. What would

become the new constraint?

Solution

The following approach is useful when there are only two products and there is only one binding constraint in addition to demand. However, a different approach is needed when there are overlapping constraints, i.e., more than one binding constraint. Linear programming is needed to solve the more difficult problems involving multiple products with multiple binding constraints.

1. Determine the company's constraint.

Time requirements to meet demand for each department are calculated as follows:

Department Product Y Product Z Total Time

Required Per Week

A (15 min)(100 units) (10 min)(50 units) 2,000 minutes

B (15 min)(100 units) (30 min)(50 units) 3,000 minutes

C (15 min)(100 units) (5 min)(50 units) 1,750 minutes

D (15 min)(100 units) (5 min)(50 units) 1,750 minutes

Each machine center has only 2,400 minutes of available time per week. B is the constraint because it does not have enough capacity to process 100 units of Y and 50 units of Z per week.

2. Determine the throughput per unit for each product.

Throughput per unit for each product is needed so that we can determine how to use the constraint to maximize throughput. Throughput per unit is as follows:

Product Sales price - Materials Cost Throughput Per Unit

Y $450 - 215 $235

Z $500 - 200 $300

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3. Determine the throughput per minute of the constrained resource for each product.

Product Throughput Per Unit Minutes required in B

Throughput Per Minute

Y $235 ÷ 15 $15.67

Z $300 ÷ 30 $10.00

4. Determine the product mix needed to maximize throughput, i.e., the number of units of Y and Z that should be produced per week.

Maximizing throughput requires producing as much of the product with the highest throughput per minute of the constrained resource as needed to meet demand. So the company should produce 100 units of product Y. This requires (100 units)(15 minutes) = 1,500 minutes of time in the constraint department B and leaves 2,400 - 1,500 = 900 minutes for the production of 30 units of product Z, i.e., 900 minutes ÷ 30 minutes per unit = 30 units of Z.

Graphic Analysis

The following graphic analysis provides a general approach for solving simple product mix problems that is also applicable when there are overlapping constraints.

First, plot the constraints to find the feasible solution space. The B constraint is 15Y + 30Z = 2,400 so Department B could produce 160 Y's (i.e., 2,400/15) or 80 Z's (i.e., 2,400/30), or some combination of Y and Z indicated by the constraint line connecting those two points on the graph. The department B constraint and demand constraints define the feasible solution space indicated by the mustard colored area on the graph.

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Problem 2

Hart Furniture Company produces two products, End Tables and Sofas that are processed in five departments, Saw Lumber, Cut Fabric, Sand, Stain, and Assemble. End tables are produced from raw lumber. Sofas require lumber and fabric. Glue and thread are plentiful and represent a relatively insignificant cost that is included in operating expense. The specific requirements for each product are provided in the table below.

Resource or Activity & (Quantity available per month)

Required per End Table Required per Sofa

Lumber (4,300 board feet) 10 board ft @ $10 = $100 7.5 board ft @ $10 = $75

Fabric (2,500 yards) - 10 yards @ 17.50 = $175

Saw Lumber (280 hours) 30 minutes 20 minutes

Cut Fabric (140 hours) - 20 minutes

Sand (280 hours) 30 minutes 10 minutes

Stain (140 hours) 20 minutes 30 minutes

Assemble (700 hours) 60 minutes 90 minutes

The Company's operating expenses are $75,000 per month. Based on current demand, the company can sell 300 End Tables and 180 Sofas per month. Sales prices are $300 for End Tables and $500 for Sofas. Required:

1. Determine Hart Company's constraint. 2. Determine the throughput per minute of the constrained resource for each product. 3. Determine the product mix needed to maximize throughput, i.e., the number of End Tables and Sofas that

should be produced per month. 4. Determine the maximum net income per month for Hart Company. 5. Suppose Hart Company broke the current constraint resource. What would become the new constraint? 6. Solve the Hart Company product mix problem assuming that only 3,000 board feet of lumber can be

obtained rather than 4,300 board feet.

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Solution

As indicated in the first example, the following approach can be used to solve a simple product mix problem when there is only one binding constraint, i.e., no overlapping constraints. 1. Determine Hart Company's binding constraint. Resource requirements to meet demand for each department are calculated as follows:

Activity & (Quantity available per month)

End Table Sofa Total Amount of Resource Required Per Month

Lumber (4,300 board feet) (10 board ft)(300) (7.5 board ft)(180) 4,350 board feet

Fabric (2,000 yards) - (10 yards)(180) 1,800 yards

Saw (16,800 min) (30 min)(300) (20 min)(180) 12,600 minutes

Cut &Trim (8,400 min) - (20 min)(180) 3,600 minutes

Sand (16,800 min) (30 min)(300) (10 min)(180) 10,800 minutes

Stain (8,400 min) (20 min)(300) (30 min)(180) 11,800 minutes

Assemble (42,000 min) (60 min)(300) (90 min)(180) 34,200 minutes

The Stain activity is the binding constraint because it does not have enough capacity to process 300 End Tables and 180 Sofas per month. 2. Determine the throughput per minute of the constrained resource for each product. First we need to determine the throughput per unit for each product so that we can determine how to use the constraint to maximize throughput. Throughput per unit is as follows:

Product Sales price - Materials Cost Throughput Per Unit

End Tables $300 - 100 $200

Sofas $500 - 250 $250

Then we can determine the throughput per minute of the constrained resource for each product as follows.

Product Throughput Per Unit Minutes required in Stain

Throughput Per Minute

End Tables $200 ÷ 20 $10.00

Sofas $250 ÷ 30 $8.33

3. Determine the product mix needed to maximize throughput, i.e., the number of End Tables and Sofas that should be produced per month. Maximizing throughput requires producing as much of the product with the highest throughput per minute of the constrained resource as needed to meet demand. So the company should produce 300 End Tables. This requires (300 units)(20 minutes) = 6,000 minutes of time in the constraint and leaves 8,400 - 6,000 = 2,400 minutes for the production of 80 Sofas, i.e., 2,400 minutes ÷ 30 minutes per unit = 80.

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Graphic Analysis

The following graphic analysis provides a general approach for solving simple product mix problems that is also applicable when there are overlapping constraints. First, plot the constraints to find the feasible solution space. The staining constraint is 20ET + 30S = 8,400 minutes, so staining could produce 420 ETs (8,400/20) or 280 Sofas (i.e., 8,400/30), or some combination of ETs and Sofas indicated by the constraint line connecting those two points on the graph. The Staining constraint and demand constraints define the feasible solution space indicated by the green area on the graph. The lumber constraint is 10ET + 7.5S = 4,300 so the company could produce 430 ETs (4,300/10) or 573.33 Sofas (4,300/7.5) with the available lumber. Plotting the lumber constraint shows that it is not a binding constraint, i.e., it does not limit the feasible solution space on the graph. Next, check the amount of throughput that could be obtained at each corner point, or plot the objective function 200 ET + 250S and move it up and to the right as far as possible without leaving the feasible solution space. The first objective function is plotted to indicate the slope of the function (200/250 = .8 means that it takes only .8 of a Sofa to produce as much throughput as 1 ET) and shows that 250 ETs produces the same throughput as 200 Sofas, i.e., (250 ETs)($200) = (200 Sofas)($250) = $23,500. Either approach (checking the corner points or using the objective function) reveals that 300 ETs and 80 Sofas is the optimum solution.

4. Determine the maximum net income per month for Hart Company.

Sales: 300 End Tables = (300)($300)

$90,000

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80 Sofas = (80)($500) COGS: 300 End Tables = (300)($100) 80 Sofas = (80)($250) Throughput Less Operating expense Net income

40,000 $30,000 20,000

$130,000 50,000 80,000 75,000 $5,000

Note: An assumption in this illustration is that there are no beginning or ending inventories of work in process or finished goods. See the Pop Company problem for an illustration with beginning and ending inventories. 5. Suppose Hart Company broke the current constraint. What would become the new constraint? Breaking the Stain activity constraint would cause Lumber to become the constraint resource because 4,350 board feet are needed and only 4,300 board feet are available per month. 6. Solve the Hart Company product mix problem assuming that only 3,000 board feet of lumber can be obtained rather than 4,300 board feet. Where there are overlapping constraints as in this case, the solution obtained using the first approach indicated above is not recommended. The graphic approach is more reliable.With only 3,000 board feet of lumber, the company can produce 300 ETs (3,000/10) or 400 Sofas (3,000/7.5), or some combination of the two as indicated by the new lumber constraint line on the graph.

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The feasible solution space is smaller than before and is now defined by lumber and staining as well as product demand as indicated in the graph below.

The solution can be found by examining the potential throughput at each of the corner points 1, 2, 3, and 4.

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Corner Point Throughput

1. 300 ETs and zero Sofas (300)(200) = 60,000

2. 180 ETs and 160 Sofas (180)(200) + (160)(250) = 76,000

3. 150 ETs and 180 Sofas (150)(200) + (180)(250) = 75,000

4. Zero ETs and 180 Sofas (180)(250) = 45,000

Point 2 provides the solution because it provides the greatest amount of throughput. The solution can also be found by using the objective function. If we plot the objective function we can locate the solution by moving it to the outer most point in the feasible solution space as illustrated below. The point indicated by 180 ETs and 160 Sofas is the last point the objective function touches in the feasible solution space as we move it up and to the right. This point indicates the solution to our product mix problem. The objective functions in the graph are iso-throughput lines indicating that any combination of ETs and Sofas on the line produces the same amount of throughput, i.e., $23,500 of throughput for the lower function and $76,000 for the function indicating the solution point.

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The graphic solutions to the Hart Company problem and the TOC problem are both fairly simple. For problems with multiple products and multiple constraints there is no graphic equivalent, but these simple problems provide a conceptual view and introduction to more realistic product mix problems and are useful for introducing both TOC and the linear programming technique.

Problem 3

Buffa Company produces two products, X and Y that are processed in two departments, A and B. The requirements for each product are provided in the table below.

Resource Required per unit of

Product X Required per unit of

Product Y

Department A 2 hours 4 hours

Department B 3 hours 2 hours

Weekly capacity is 80 hours for Department A and 60 hours for Department B. Throughput per unit is $60 for X and $50 for Y.

Required:

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1. Assume that the company can sell as much of X and Y as it can produce. Determine the number of units of X and Y that should be produced per week to maximize throughput.

2. Now assume that Weekly demand is 12 for Product X and 14 for Product Y. What would be the new product mix needed to maximize throughput?

Solution

Question 1: Where there are no demand constraints. The Department A constraint is 2X + 4Y = 80 so the company can produce 40 units of X (80/2) or 20 units of Y (80/4) or some combination of X and Y on the line connecting those two points on the graph. The Department B constraint is 3X + 2Y = 60 so the company can produce 20 units of X (60/3) or 30 units of Y (60/2) or some combination of X and Y on the line connecting those two points on the graph.

We can find the solution by checking the throughput at each of the corner points.

1. 20Y @ $50 = $1,000 throughput. 2. 20X @ $60 = $1,200 throughput. 3. 15Y @ $50 + 10X @ $60 = $1,350 throughput.

The last point represents the solution since it produces the greatest throughput.

Another way to find the solution is to plot the objection function. The objection is to maximize throughput = 60X + 50Y. It takes 1.2Y (60/50) to generate as much throughput as 1X, i.e., (1.2)(50) = 60. The first objective function plotted is 12Y and 10X. Moving the objective function up and to the right, parallel to this function indicates the solution point on the graph at 15 units of Y and 10 units of X.

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Question 2: Where the demand constraints limit the solution space. We can see that the solution space becomes smaller after plotting the demand constraints and the original solution is no longer feasible.

Checking the throughput at the corner points we have: 1. 14Y @ $50 = $700. 2. 14Y @ $50 + 10.67@ $60 = $1,340.20. If we use (14)(2 hours) = 28 hours for Y, then there are 60-28 = 32 hours left for the production of 10.67 X (i.e., 32/3). 3. 12X @ $60 = $720. 4. 12X @ $60 + 12Y@ $50 = $1,320.

Point 2 is the solution.

Using the objective function also shows that the point where Y = 14 and X = 10.67 is the solution point since that is the last point within the solution space that the objection function touches as we move it up and to the right.

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As indicated in previous illustrations, graphic solutions are only useful for solving simple introductory problems. A technique such as linear programming is needed for more realistic product mix problems. Linear programming and many other techniques are illustrated in operations management and quantitative methods textbooks.