Question

In this homework, you will implement a Python script that solves the quadratic assignment problem using CPLEX. The quadratic assignment prob- lem is an integer problem defined as follows: as with the assignment problem, we consider the problem as related to two sets of objects S and T. S and T both have the same number of members, which are indexed from 1 to N. The problem is to assign each member of S to exactly one member of T to achieve some objective. There are two sorts of conditions we must fulfill: each member of S must be assigned to exactly one member of T, and each member of T must have exactly one member of S assigned to it. Binary variables x_(ij) can be introduced with the following interpretations: x_(ij)={(1 if i( a member of S) is assigned to j( a member of T),),(0 otherwise. ):} Conditions are imposed by the following two types of constraints: sum_(j=1)^N x_(ij)=1,i=1,dots,N, sum_(i=1)^N x_(ij)=1,j=1,dots,N. The objective is more complex than with the assignment problem. We have cost coefficients c_(ijkl), which have the following interpretations. c_(ijkl) is the cost incurred by assigning i (a member of S ) to j (a member of T ) at the same time as assigning k (a member of S ) to l (a member of T ). This cost will clearly be incurred only if x_(ij)=1 and x_(kl)=1, that is, if the product x_(ij)x_(kl)=1. The objective becomes a quadratic expression in binary variables: minimizez=sum_(i=1)^N sum_(j=1)^N sum_(k=1)^N sum_(l=1)^N c_(ijkl)x_(ij)x_(kl)It is very common for the coefficients c_(ijkl) to be derived from the product of other coefficients t_(ik) and d_(jl) so that c_(ijkl)=t_(ik)d_(jl). Firstly, we consider S to be a set of N factories and T to be a set of N cities. The problem is to locate one factory in each city and to minimize total communication costs between factories. The communication costs depend on (i) the frequency of communication between each pair of factories, and (ii) the distances between the two cities where each pair of factories is located. Clearly, some factories will have little to do with each other and can be located far apart at little cost. On the other hand, some factories may need to communicate a lot. The cost of communication will depend on the distance apart. In this application, we can interpret the coefficients t_(ik) and d_(jl) as follows: t_(ik) is the frequency of communication between factories i and k;d_(jl) is the cost per unit of communication between cities j and l (clearly, this will be related to the distance between j and l ). Obviously, the cost of communication between the factories i and k, located in cities j and l, will be given by c_(ijkl)=t_(ik)d_(jl). The total cost is therefore represented by the objective function. This problem will be represented using two .txt files, namely, frequencies . txt and distances.txt. The first file contains the frequencies (i.e., t_(ik) ) in N lines with N values. It contains the following lines for an example problem: frequencies.txt 25,63,34,52 63,39,21,57 34,21,10,66 525766,63 The second file contains the distances (i.e., d_(jl) ) in N lines with N values. It contains th It is very common for the coefficients ci to be derived from the product of other coefficients ti and dj so that ci = tid. Firstly, we consider S to be a set of N factories and T to be a set of N cities. The problem is to locate one factory in each city and to minimize total communication costs between factories. The communication costs depend on i the frequency of communication between each pair of factories, and (ii the distances between the two cities where each pair of factories is located Clearly, some factories will have little to do with each other and can be located far apart at little cost. On the other hand, some factories may need to communicate a lot. The cost of communication will depend on the distance apart. In this application, we can interpret the coefficients ti and dj as follows: ti is the frequency of communication between factories i and k; d, is the cost per unit of communication between cities j and I (clearly this will be related to the distance between j and l). Obviously, the cost of communication between the factories i and k, located in cities j and l will be given by cij = tidjr. The total cost is therefore represented by the objective function. This problem will be represented using two .txt files, namely, frequencies .txt and distances.txt. The first file contains the frequencies (i.e., ti) in N lines with N values. It contains the following lines for an example problem: frequencies.txt 2563 3452 63 39 21 57 34 21 10 66 5257 66 63 The second file contains the distances i.e., d in N lines with N values. It contains the following lines for an example problem: distances.txt 44 26 68 66 26 52 51 62 68 51 96 60 66 62 60 83 The optimum solution of the example problem is as follows: x=0 x=1 x=0 x=0 =0 =0 x3=0 x2=1 x=0 x=0 x3=1 43=0 x4=1 x4=0 x=0 4=0. Implement your algorithm to solve the quadratic assignment problem in a single interactive Python notebook using Azure Lab Services. Your notebook should include at least the following function definition that takes the file paths of two input files as parameters and returns the solution found. def quadratic_assignment_problem(frequencies_file, distances_file): # your implementation starts below # your implementation ends above return(x_star)

Name: in this homework you will implement a python script that solves the quadratic assignment problem using cplex the quadratic assignment prob lem is an integer problem defined as follows as wit 93085
Uploaded: 2022-11-05T06:23:20-08:00
Duration: 2 min 14 s
Channel: Sri K
Description: in this homework you will implement a python script that solves the quadratic assignment problem using cplex the quadratic assignment prob lem is an integer problem defined as follows as wit 93085

          In this homework, you will implement a Python script that solves the
quadratic assignment problem using CPLEX. The quadratic assignment prob-
lem is an integer problem defined as follows: as with the assignment problem,
we consider the problem as related to two sets of objects S and T. S and
T both have the same number of members, which are indexed from 1 to N.
The problem is to assign each member of S to exactly one member of T to
achieve some objective. There are two sorts of conditions we must fulfill:
each member of S must be assigned to exactly one member of T, and
each member of T must have exactly one member of S assigned to it.
Binary variables x_(ij) can be introduced with the following interpretations:
x_(ij)={(1 if i( a member of S) is assigned to j( a member of T),),(0 otherwise. ):}
Conditions are imposed by the following two types of constraints:
sum_(j=1)^N x_(ij)=1,i=1,dots,N,
sum_(i=1)^N x_(ij)=1,j=1,dots,N.
The objective is more complex than with the assignment problem. We have
cost coefficients c_(ijkl), which have the following interpretations. c_(ijkl) is the
cost incurred by assigning i (a member of S ) to j (a member of T ) at the
same time as assigning k (a member of S ) to l (a member of T ). This cost
will clearly be incurred only if x_(ij)=1 and x_(kl)=1, that is, if the product
x_(ij)x_(kl)=1. The objective becomes a quadratic expression in binary variables:
minimizez=sum_(i=1)^N sum_(j=1)^N sum_(k=1)^N sum_(l=1)^N c_(ijkl)x_(ij)x_(kl)It is very common for the coefficients c_(ijkl) to be derived from the product of
other coefficients t_(ik) and d_(jl) so that c_(ijkl)=t_(ik)d_(jl).
Firstly, we consider S to be a set of N factories and T to be a set of N
cities. The problem is to locate one factory in each city and to minimize total
communication costs between factories. The communication costs depend on
(i) the frequency of communication between each pair of factories, and (ii)
the distances between the two cities where each pair of factories is located.
Clearly, some factories will have little to do with each other and can be
located far apart at little cost. On the other hand, some factories may
need to communicate a lot. The cost of communication will depend on the
distance apart. In this application, we can interpret the coefficients t_(ik) and
d_(jl) as follows: t_(ik) is the frequency of communication between factories i and
k;d_(jl) is the cost per unit of communication between cities j and l (clearly,
this will be related to the distance between j and l ). Obviously, the cost
of communication between the factories i and k, located in cities j and l,
will be given by c_(ijkl)=t_(ik)d_(jl). The total cost is therefore represented by the
objective function.
This problem will be represented using two .txt files, namely, frequencies . txt
and distances.txt. The first file contains the frequencies (i.e., t_(ik) ) in N
lines with N values. It contains the following lines for an example problem:
frequencies.txt
25,63,34,52
63,39,21,57
34,21,10,66
525766,63
The second file contains the distances (i.e., d_(jl) ) in N lines with N values.
It contains th
It is very common for the coefficients ci to be derived from the product of other coefficients ti and dj so that ci = tid. Firstly, we consider S to be a set of N factories and T to be a set of N cities. The problem is to locate one factory in each city and to minimize total communication costs between factories. The communication costs depend on i the frequency of communication between each pair of factories, and (ii the distances between the two cities where each pair of factories is located Clearly, some factories will have little to do with each other and can be located far apart at little cost. On the other hand, some factories may need to communicate a lot. The cost of communication will depend on the distance apart. In this application, we can interpret the coefficients ti and dj as follows: ti is the frequency of communication between factories i and k; d, is the cost per unit of communication between cities j and I (clearly this will be related to the distance between j and l). Obviously, the cost of communication between the factories i and k, located in cities j and l will be given by cij = tidjr. The total cost is therefore represented by the objective function.
This problem will be represented using two .txt files, namely, frequencies .txt and distances.txt. The first file contains the frequencies (i.e., ti) in N lines with N values. It contains the following lines for an example problem: frequencies.txt
2563 3452 63 39 21 57 34 21 10 66 5257 66 63
The second file contains the distances i.e., d in N lines with N values. It contains the following lines for an example problem: distances.txt
44 26 68 66 26 52 51 62 68 51 96 60 66 62 60 83
The optimum solution of the example problem is as follows:
x=0 x=1 x=0 x=0
=0 =0 x3=0 x2=1
x=0 x=0 x3=1 43=0
x4=1 x4=0 x=0 4=0.
Implement your algorithm to solve the quadratic assignment problem in a single interactive Python notebook using Azure Lab Services. Your notebook should include at least the following function definition that takes the file paths of two input files as parameters and returns the solution found.
def quadratic_assignment_problem(frequencies_file, distances_file): # your implementation starts below
# your implementation ends above return(x_star)

in this homework you will implement a python script that solves the quadratic assignment problem using cplex the quadratic assignment prob lem is an integer problem defined as follows as wit 93085

Added by Soledad M.

We constantly seem to be pricing ourselves out of some markets and not charging enough in others. Our pricing policy is pretty simple: we mark up our full manufacturing cost by 50%. That means a computer that costs us $2,000 to manufacture will sell for $3,000. Until now I thought this was a workable approach, but now I’m not so sure. Steve Works, CEO, Cortland Manufacturing, Inc. (CMI) Steve’s Controller, Sally Nomer, had just told him that she believed the computers might be priced inappropriately. Steve continued: When I was at Leland [School of Management] I focused most of my attention on operations and marketing, convinced that those were the keys to my successful career. Cost accounting was boring and not relevant. But now my accountant is telling me that I need to think about a new way to assign manufacturing overhead to products, and I don’t even know where to start! I never dreamed that some day my career would come to this. I wish I’d paid more attention to those classes so I could understand what is going on here. I’m lost. BACKGROUND CMI manufactures several different models of computers, distributed to retail outlets throughout the 50 states. The company is proud of the user-friendly computers it produces, in particular their graphics capability. CMI claims the superb graphics are what distinguish its products from competitors. CMI purchases raw materials in components and subassemblies made to its specifications from a very small group of highly reliable suppliers. It uses a single facility to house both manufacturing facility and administrative and sales offices. The factory workers operate three kinds of machines. Inspecting machines check the raw materials and test components and subassemblies to assure they are working to specifications. Soldering machines solder various components as necessary. Finally assembly machines put all the components and subassemblies together into finished computers. The processes can vary by computer model. Depreciation, maintenance and repairs on the three types of machines account for about 40% of CMI’s overhead cost. The remainder of the overhead is made up of labor involved in receiving and handling the raw materials, adjusting and setting up machines for each new batch of computers, and inspecting and packing finished computers for shipping. Additional costs include insurance and depreciation on handling equipment, supplies, and utilities. The actual manufacturing effort (soldering, inspecting and assembly) is primarily automated, so CMI uses very little direct labor. The company also operates on a lean production model, so almost no inventories exist at the end of any period. Most of the computers are sold in large orders to national electronic chains. However, the Cortland 2000 is not such a machine. It represents a recent effort by CMI to enter the scientific computing market. The quantity manufactured and sold of this machine is expected to be much lower than other models even when it gains its hoped-for market share. Fewer customers exist for this more sophisticated, powerful, high-priced machine. Nonetheless, Mr. Works has believed from the conception of this product that, when all manufacturing costs were considered, the Cortland 2000 would contribute a reasonable amount to CMI’s selling and administrative costs and profit. The name recognition it brought in its elite community should enhance sales of the more generally used computers. FINANCIAL INFORMATION CMI budgeted direct labor costs for 2010 at $60,000,000. Based on expected sales, the company estimated that raw material purchased and used would be $300,000,000. Manufacturing overhead was budgeted at $196,000,000. It is currently allocated to production on the basis of machine hours (MH). As mentioned above, computers are priced at full production cost plus a mark-up of 50%. EXHIBIT 1 shows the expected direct manufacturing costs for two of the company’s computers. The Cortland 1000 is a very popular computer with a large production and sales volume. By contrast, the Cortland 2000, described above, is a state-of-the-art scientific computer with several special features. In particular, the Cortland 2000 a. Uses a new processing chip imported from Sweden. b. Has special patented random access memory (RAM) that gives it extremely high input/output speed. c. Is manufactured in very small batches to assure uniform quality from one computer to the next, to satisfy the users of the machines who have very high expectations for the performance of the machines they purchase. Mr. Works’ concerns arose when Ms. Nomer told him that she thought the company’s traditional overhead allocation system was providing misleading cost information about the different types of products. She developed an analysis of the 2010 manufacturing overhead costs, shown in EXHIBIT 2. She classified the overhead costs described above into activities. She also gathered data, shown in EXHIBIT 3, for the 2010 production of the Cortland 1000 and the Cortland 2000. Mr. Works commented on the data: I don’t know quite what to make of all this. Clearly I need some further explanation and analysis. I guess what I really need is some sense of what is the true manufacturing cost of each computer. I thought I knew that, but I didn’t really. Even though we cannot at this point change prices for 2010, we at least need to know if we’re covering full production cost on the Cortland 2000 and have something left to contribute to the company’s selling, general and administrative expenses. I thought this was so, but now … well, I’m not so sure. We’d better figure this out before we begin our budget cycle for 2011. Maybe I need to go back to school! EXHIBIT 1 Direct Manufacturing Costs for ONE Computer Cortland 1000 Cortland 2000 Direct Material $1,000 $2,500 Direct Labor $200 $400 EXHIBIT 2 Analysis of 2010 Budgeted Manufacturing Overhead Costs Activity Cost driver Budgeted activity (for entire company) Budgeted cost Receive/handle raw material Orders received 200 orders $90,000,000 Adjust/set up machines Number of setups 2,000 setups $12,000,000 Inspect, pack, ship computers Batches 500 batches $60,000,000 Inspect raw materials Inspection hours 200,000 insp. hr. $10,000,000 Solder parts Soldering hours 200,000 sdr. hr. $12,000,000 Assemble computers Assembly hours 100,000 assm. hr. $12,000,000 Total overhead $196,000,000 EXHIBIT 3 Production Data for Cortland 1000 and Cortland 2,000 Computers Item Measure Cortland 1000 Cortland 2000 Budgeted production Number of computers 20,000 5,000 Received order size* Size of order 10,000 500 Batch size Number of computers 5,000 100 Machine setups Number of s/u per batch 5 6 Inspecting time (Raw mat.) Hours / computer 1 2 Soldering time Hours / computer 3 1 Assembly time Hours / computer 1 1 * “Received order size” is the number of computers one order of raw materials will build. Thus, for example, (see above) the company receives 2 orders of raw materials for the Cortland 1,000. Calculation: 20,000 computers per year, divided by 10,000 computers each order will build = 2 orders per year. REQUIRED: Don’t try to begin answering questions until you have studied the case information carefully. Careless reading will result in careless errors! Just take a few extra minutes to understand it all. 1. Calculate the predetermined overhead rate the company is currently using. (Where should you look for the machine hours?) 2. Using the information from the exhibits and Q1, calculate the manufacturing cost of ONE Cortland 1000 and ONE Cortland 2000. 3. Calculate the selling price, under the current system, of ONE Cortland 1000 and ONE Cortland 2000. 4. Next, you are going to begin to analyze the ABC system. In order to do this, however, your study of Exhibits 2 and 3 should have told you that you are going to have to convert the information the company currently measures into the information you need for the ABC analysis. First, how many raw material orders did the company receive for each model? 5. How many batches of each model did the company pack and ship? 6. How many machine setups does Cortland do for each model? 7. Calculate a rate for each activity in Exhibit 2. 8. Use the rates calculated in Q7 to apply overhead to each product line: Cortland 1000 and 2000. 9. Calculate the overhead per computer for a Cortland 1000 and a Cortland 2000. 10. Calculate the manufacturing cost of ONE Cortland 1000 and ONE Cortland 2000 using ABC overhead allocation. 11. Show how the company would determine a selling price for each model under the ABC system. 12. Should Mr. Works adopt an ABC system for internal analysis? Your answer should address the question very specifically to this case – no generalities about ABC from the book. It should contain two points in favor of ABC, and two drawbacks that he should be aware of. 13. Mr. Works is also concerned about future product development. They have several new models under development at this time. If an ABC system is used, list two characteristics of a new model that will make it more costly to manufacture.

Sri K.

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