You work as an operations consultant for a textile company. Your client has a well-established distribution system in the US market. The company has hundreds of stores and four distribution centers. You will work with your client to determine the best way to distribute goods from distribution centers to retail stores.

The following information will guide you in analyzing the facility location decisions.

The company has 300 stores in 14 different states. See the locations in the Excel file.

The company has 4 distribution centers: Valdosta GA, Salisbury NC, Ft. Worth TX and Salt Lake City UT.

For each distribution center, information on estimated monthly capacity (in terms of number of truckloads of goods that can be shipped), and the distance between the distribution center locations and store locations

are provided in the Excel file.

Your client tells you that the transportation cost will be $1.5 per mile per truckload.

Following Zara’s strategy of “fast fashion” or fast-response supply chain, your client makes two deliveries to each store every week. Each delivery involves only one truckload. That is, a total of two truckloads of goods are shipped from the distribution centers to each retail store each week. In your analysis, assume that a month is only four weeks; this brings monthly store demand to 2x4=8 truckloads per month.

You will analyze the information provided, build spreadsheet models, and use optimization to make a recommendation to your client on which distribution centers should supply which retail stores.

NOTE: SHOW ALL YOUR WORK.

PART-I: Transportation Problem

You first analyze the cost of logistics network. For this problem, you will formulate a transportation problem and determine the monthly shipment plan that minimizes the total transportation costs. Your client is not familiar with optimization models. You must clearly explain your approach to computing the minimum transportation. To do so,

(2 points) Clearly define your decision variables in English. How many decision variables do you have?

NOTE: In your Solver model, the unit of shipment should be number of truckloads.

(2 points) What is the objective function in this problem? Explain how you would compute the objective function (in English). The mathematical expression for the objective function is very long, so do NOT explicitly state the mathematical formula.

(4 points) Clearly explain the constraints (in English). What type of constraints do you have? Do NOT provide the mathematical formula for the constraints.

(2 point) What are the sign and type restrictions in optimization problems? Are there any sign or type restrictions in this problem? Clearly explain why or why not.

Prepare a spreadsheet to determine the optimal production plan using Excel Solver.

(2 points) Provide a screenshot of your spreadsheet model. The screenshot should show all the cells involved in formulating the optimization problem. NOTE: Do not copy and paste your spreadsheet to your answer report as an Excel object. Instead, capture a screenshot in Excel and paste this as an image.

(2 points) Provide a screenshot of your spreadsheet with the formula view. You can switch from normal view to formula view in Excel by clicking the CTRL and ~ keys on your keyboard simultaneously.

(2 points) Prepare the Solver model by logging all the information to the Solver Parameters window in Excel. Provide a screenshot of the Solver Parameters window in Excel.

NOTE: Ignore all type restrictions in this problem. Do not enforce your decision variables to be integer while using Excel Solver. If you formulate the problem correctly, the optimal solution provided by the Solver will be in whole numbers even without the type restrictions.

(2 points) Finally, solve the optimization problem using the Simplex LP method in Excel Solver. What is the optimal solution, i.e. the optimal objective function value and the optimal values of decision variables?

PART-II Expand the Outbound Logistics Capacity

(3 points) Your client asked you to determine the capacity utilization level of the existing distribution system. Compute the capacity utilization levels for each distribution center and for the entire distribution network based on your optimal transportation plan in Part-I. Complete the table below to report the capacity utilization levels.

Distribution center(s)	Available capacity (no. of truckloads of shipment)	Capacity used (no. of truckloads of shipment)	Capacity Utilization
Valdosta, GA	450
Salisbury, NC	580
Ft. Worth, TX	440
Salt Lake City, UT	700
All distribution centers

Table-Q6-f. Capacity utilization when the optimal transportation plan is used

(6 points) Your client thinks that they are operating at a very high capacity utilization rate, which does not give them any flexibility to effectively respond to unexpected hikes in demand. The client asked you to determine whether the distribution network capacity should be expanded by 100 units (where capacity is measured in terms of number of truckloads of shipment). Expansion, if approved, will occur at only one location. You have to determine which distribution center, if expanded by 100 units, will lead to largest savings in transportation costs. Complete the table below to report your findings.

Location of capacity expansion	Optimal transportation cost after expansion	Reduction in optimal transportation cost	Percentage change in optimal transportation cost due to expansion
Valdosta, GA
Salisbury, NC
Ft. Worth, TX
Salt Lake City, UT

Table-Q6-g. Transportation costs of different capacity expansion plans

Which distribution center, if expanded, leads to the highest reduction in optimal transportation costs?

NOTE: Optimal transportation cost is obtained by solving a transportation problem using Excel Solver. You already computed this in Part-I for the existing network capacity. To complete Table-Q6-g, you have to formulate and solve four optimization problems. In each optimization problem, only one of the distribution centers has 100 additional units of capacity available.

(3 points) What additional information should be taken into account before making a final decision about the capacity expansion decision? Clearly explain the type of information needed and how that information could be obtained.

Numbers:

Distance From/To (in miles)
Planned store locations	Valdosta, GA	Salisbury, NC	Ft. Worth, TX	Salt Lake City, UT		Number of Stores in each State	State
Texas	1109	1286	193	1090		35	Texas
Florida	261	638	1202	2372		25	Florida
Arizona	1918	1981	986	644		22	Arizona
North Carolina	495	105	1216	2126		18	North Carolina
Louisiana	619	833	361	1590		14	Louisiana
Georgia	111	326	973	2031		16	Georgia
Alabama	280	482	642	1840		18	Alabama
Missouri	894	840	593	1293		17	Missouri
Oklahoma	1040	1107	182	1236		19	Oklahoma
Tennessee	454	450	715	1691		16	Tennessee
Arkansas	685	747	397	1491		12	Arkansas
California	2546	2551	1533	814		35	California
South Carolina	288	153	1020	2080		23	South Carolina
New York	977	586	1579	2172		30	New York
Monthly capacity	550	700	450	800