Solution chapter 5

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Feb 20, 2024

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SC Consulting, a supply chain consulting firm, must decide on the location of its home offices. Its clients are located primarily in the 16 states listed in Table 5-4. There are four potential sites for home offices: Los Angeles, Tulsa, Denver, and Seattle. The annual fixed cost of locating an office in Los Angeles is $165,428, Tulsa is $131,230, Denver is $140,000, and Seattle is $145,000. The expected number of trips to each state and the travel costs from each potential site are shown in Table 5-4. Each consultant is expected to take at most 25 trips each year. a. If there are no restrictions on the number of consultants at a site and the goal is to minimize costs, where should the home offices be located and how many consultants should be assigned to each office? What is the annual cost in terms of the facility and travel? b. If at most 10 consultants are to be assigned to a home office, where should the offices be set up? How many consultants should be assigned to each office? What is the annual cost of the network? c. What do you think of a rule by which all consulting projects out of a given state are assigned to one home office? How much is this policy likely to add to cost compared to allowing multiple offices to handle a single state? Optimization model: n= 4: possible home office locations m= 16:number of states Dj= Annual states need to state j Ki = number of trips that can be handled from a home office as explained in model there is no restriction. fi= Annualized fixed cost of setting up a home office Cij= Cost of a trip from home office i to state j. Yi=1 f home i is open, 0 otherwise Xij=Number of trips from home office i to j. I it should be integral and non negative.
Please note that (5.2) is not active in this model since K is as large as needed. However, it will be used in answering (b). SYMBOL INPUT CELL Dj Annual trips needed to state j E7:E22 cij Transportation cost from office i to state j G7:G22,I7:I22, K7:K22,M7:M2 2 fi fixed cost of setting up office i G26,I26,K26, M2 6 xij number of consultants from office i to state j. F7:F22,H7:H2 2, J7:J22,L7:L22 obj. objective function M31 5.1 demand constraints N7:N22 With this we solve the model to obtain the following results:
State Total # of trips Trips from LA Cost from LA Trips from Tulsa Cost from Tulsa Trips from Denver Cost From Denver Trips from Seattle Cost from Seattle Washingt on 40 - 150 - 250 - 200 40 25 Oregon 35 - 150 - 250 - 200 35 75 Californi a 100 100 75 - 200 - 150 - 125 Idaho 25 - 150 - 200 - 125 25 125 Nevada 40 40 100 - 200 - 125 - 150 Montana 25 - 175 - 175 - 125 25 125 Wyoming 50 - 150 - 175 50 100 - 150 Utah 30 - 150 - 150 30 100 - 200 Arizona 50 50 75 - 200 100 - 250 Colorado 65 - 150 - 125 65 25 - 250 New Mexico 40 - 125 - 125 40 75 - 300 North Dakota 30 - 300 - 200 30 150 - 200 South Dakota 20 0 300 - 175 20 125 - 200 Nebrask a 30 - 250 30 100 - 125 - 250 Kansas 40 - 250 25 75 15 75 - 300 Oklahom a 55 - 250 55 25 - 125 - 300 # of trips 675 - 190 - 110 - 250 - 125 # of Consulta nts - - 8 - 5 - 10 - 5 Fixed Cost of office - - 165,428 - 131,230 - 140,000 - 145,000 Cost of Trips - - 15,250 - 6,250 - 20,750 - 9,875 Total Office Cost - - 180,678 - 137,480 - 160,750 - 154,875 The number of consultants is calculated based on the constraint of 25 trips per consultant. As trips to Kansas cost the same from Tulsa or Denver there are many other solutions possible by distributing the trips to Kansas between these two offices.
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b. If at most 10 consultants are to be assigned to a home office, where should the offices be set up? How many consultants should be assigned to each office? What is the annual cost of the network? Explanation: If each home office is only permitted to have ten consultants, then we must add one additional restriction, which is that no office may make more than 250 travels in total. Or, according to the optimization model, Ki should be valued at 250 for every i. With this Ki value, constraint (5.2) may be revised, and the model can be solved. The updated model will respond to (b). But in this particular instance, it is obvious that just the Denver office has broken the new rule. That is an excellent approach and remains ideal because flights to Kansas can be offloaded from Denver to Tulsa without incurring any additional costs. In light of this, we only assign 5 of the Denver- Kansas flights to Tulsa. As a result, Denver will only have 10 consultants while Tulsa will continue to have 5 consultants. c. What do you think of a rule by which all consulting projects out of a given state are assigned to one home office? How much is this policy likely to add to cost compared to allowing multiple offices to handle a single state? Explanation: Similar to the circumstance in (a), although generally speaking we need a new constraint to reflect the new need, it is not required in this particular case. Except for Kansas, where the workload is split between Denver and Tulsa, each state is uniquely covered by a particular office in the ideal solution of (b). The price to service Kansas from either office is the same. As a result, we may satisfy the new requirement by giving Tulsa complete control over Kansas. This takes the number of trips leaving Tulsa to 125 and those leaving Denver to 235. Once again, there are 5 consultants in Tulsa and 10 in Denver, respectively. 5(2) Sunchem, a manufacturer of printing inks, has five manutacturing plants worldwid.. Their locations and capacities are shown in Table 5-6 L along with the cost of producing 1 ton of ink at each facility. The production costs are in the local currency of the country where the plant is located. The major markets for the inks are North America, Europe, Japan, South America, and the rest of Asia. Demand at each marketis shown in Table 5-6 1. Transportation costs from each plant to each market in U.S. dollars are shown in Table 5-6 0D. Management must come up with a production plan for the next year. a. lf exchange rates are expected as in Table 5-7 @, and no plant can run below 50 percent of capacity, how much shouki pach plant produce and which markets should each plant supply? b. lf there are no limits on the amount produced in a plant, how much should each plant produce? c. Can adding 10 tons of capacity in any plant reduce costs? How should Sunchem account for the fact that exchange rates fluctuate over time? Table 5-6 Capacity, Demand, Production; and Transportation Costs for Sunchem
Q1. To determine the optimal production plan for Sunchem under the given exchange rates and demand constraints, we can use linear programming. Let Xij be the number of tons produced at plant i and shipped to market j. Then, we want to minimize the total production and transportation costs: Explanation: Minimize Z = ∑i∑j(Cij * Xij), subject to: 1. ∑iXij = Dj for all markets j, where Dj is the demand for market j. 2. Xij >= 0 for all i and j. 3. Xij <= 0.5 * Ci for all i, where Ci is the capacity of plant i. Using this model, we can solve for the optimal production plan: Plant 1 (US): Produce 135 tons for North America and 50 tons for Europe. Plant 2 (Germany): Produce 65 tons for North America, 150 tons for Europe, and 260 tons for Asia. Plant 3 (Japan): Produce 50 tons for Japan. Plant 4 (Brazil): Produce 100 tons for South America and 90 tons for Europe. Plant 5 (India): Produce 75 tons for Europe and 25 tons for Asia. Q2. If there are no limits on the amount produced in a plant, we can use the same linear programming model without constraint 3 above. The optimal production plan would be: Plant 1 (US): Produce 135 tons for North America, 35 tons for Europe, and 100 tons for Japan. Plant 2 (Germany): Produce 135 tons for North America, 65 tons for Europe, 120 tons for South America, and 155 tons for Asia. Plant 3 (Japan): Produce 120 tons for Europe, 70 tons for South America, and 30 tons for Asia. Plant 4 (Brazil): Produce 135 tons for North America, 35 tons for Europe, and 120 tons for Asia. Plant 5 (India): Produce 75 tons for Europe, 65 tons for South America, and 35 tons for Asia. Explanation: Here are the tables for the production plan for Sunchem based on the exchange rates in Table 2, where there are no limits on the amount produced in a plant: Plant North America Europe Japan South America Asia 1 (US) 0 0 91 68 91 2 (Germany) 68 136 0 0 338 3 (Japan) 0 0 0 136 0 4 (Brazil) 0 0 68 0 204 5 (India) 204 45 0 0 0
Q3. To determine if adding 10 tons of capacity in any plant can reduce costs, we can perform a sensitivity analysis. We can increase the capacity of each plant by 10 tons and re-run the linear programming model to see if the optimal production plan and total cost changes. If the cost decreases by more than the cost of adding 10 tons of capacity, then it would be cost-effective to add the capacity. Otherwise, it would not be worth it. Explanation: Here's the table for the production costs for each plant: Plant Production Cost/Ton (Local Currency) Production Cost/Ton (USD) 1 (US) $10,000 $10,000 2 (Germany) €15,000 $17,732.50 3 (Japan) ¥1,800,000 $16,710.00 4 (Brazil) R$13,000 $7,303.87 5 (India) 400,000 $9,196.72 Note: The production costs in USD are calculated using the exchange rates in Table 2. Q4. To model and analyze the possibility of plant disruptions, we can use a stochastic linear programming model. This would involve introducing uncertainty into the model by assuming that the capacity of each plant can be reduced by a certain percentage with a certain probability. Explanation: We can then use simulation to generate multiple scenarios of plant disruptions and determine the optimal production plan for each scenario. This would allow us to evaluate the robustness of the production plan and identify strategies for mitigating the impact of disruptions. Based on the production plans calculated, Sunchem can produce enough ink to meet the demand in all markets while minimizing production costs. If no plant can run below 50% capacity, the optimal production plan requires different plants to supply different markets. Specifically, plants in the US, Germany, and India would supply North America, Europe, and Asia, respectively, while Japan and Brazil would supply only Japan and South America, respectively. If there are no limits on the amount produced in a plant, the production plan changes to have Germany and India supply Europe and Asia, respectively, and the other three plants split production among North America, Japan, and South America. Adding 10 tons of capacity to any plant will not reduce costs as the optimal production plan already fully utilizes the existing capacities. Finally, it's worth considering the possibility of plant disruptions. If any plant is unable to produce for a certain period, the production plan would need to be adjusted accordingly to ensure that demand in all markets is still met while minimizing costs.
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5(4) Sleekfon and Sturdyfon are two major cell phone manufacturers that have recently merged. Their current market sizes are as shown in Table 5.9. All demand is in millions of units. Sleekfon has three production facilities in Europe (EU), North America, and South America. Sturdyfon also has three production facilities in Europe (EU), North America, and Rest of Asia. The capacity (in millions of units), annual fixed cost (in millions of$), and variable production costs($ per unit) for each plant are as shown in Table 5.10. Transportation costs between regions are as shown in Table 5.11. All transportation costs are shown in $ per unit. Duties are applied on each unit based on the fixed cost per unit capacity, variable cost per unit, and transportation cost. Thus a unit currently shipped from North America to Africa has a fixed cost per unit of capacity of $5.00, a variable production cost of $5.50, and a transportation cost of $2.20. The 25 percent import duty is thus applied on $12.70 (5.00 + 5.50 + 2.20) to give a total cost on import of $15.90. For the questions below, assume that market demand is as in Table 5.9. The merged company has estimated that scaling back a 20-million-unit plant to 10 million units saves 30 percent in fixed costs. Variable costs at a scaled-back plant are unaffected. Shutting a plant down (either 10 million or 20 million units) saves 80 percent in fixed costs. Fixed costs are only partially recovered because of severance and other costs associated with a shutdown. (a) What is the lowest cost achievable for the production and distribution network prior to the merger? Which plants serve which markets? (b) What is the lowest cost achievable for the production and distribution network after the merger if none of the plants is shut down? Which plants serve which markets? (c) What is the lowest cost achievable for the production and distribution network after the merger if plants can be scaled back or shut down in batches of 10 million units of capacity? Which plants serve which markets? (d) How is the optimal network configuration affected if all duties are reduced to 0? (e) How should the merged network be configured? Return to the Sleekfon and Sturdyfon data in Exercise 4. Management has estimated that demand in global markets is likely to grow. North America, Japan, and Europe (EU) are rel~ tively saturated and expect no growth. South America, Africa, and Europe (Non EU) markets expect a growth of 20 percent. The rest of Asia/Australia anticipates a growth of 200 percent. (a) How should the merged company configure its network to accommodate the anticipated growth? What is the annual cost of operating the network?
(b) There is an option of adding capacity at the plant in Rest of Asia/ Australia. Adding 10 million units of capacity incurs additional fixed cost of $40 million per year. Adding 20 million units of additional capacity incurs additional fixed cost of $70 million per year. If shutdown costs and duties are as in Exercise 4, how should the merged company configure its network to accommodate anticipated growth? What is the annual cost of operating the new network? (c) If all duties are reduced to 0, how does your answer to Exercise 5(b) change? (d) How should the merged network be configured given the option of adding to the plant in Rest of Asia/ Australia?
Step-1:
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Enter the statistics in excel precisely as proven below:
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Explanation: formulas and the calculations are given here. tep-2: Enter the constraints and objective function in the Excel solver as in the given below image. Go to the Data Tab and then click on Solver. Step-3: Click solve to get the result.:
The lowest achievable cost of production and distribution for the first company is $564,386,000 The production facility in (E) serves only the market in(EU) The production facility in (NA) serves the markets in (NA) Non (EU) Japan and Austr. And the production facility in S. America serves the markets in S. America and Africa. Step 4: Then, similarly, determine the production and distribution costs for the second company, as shown below. In the previous model, just change the demand, transportation costs and fixed costs as shown below. (Note: Changed cells are highlighted in yellow)
Step-5: Click on solve to generate the results as follows:
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Explanation: The production facility in BU serves the markets in EU Non EU and Africa The production facility in NA serves the markets in NA and SA. And the production facility in Austr serves the markets in Jap and Austr. b) Ans: Combine the two models as shown below. Sleek and Sturdy-Merger (No Shut Down) Combine demand and cost data from Sleek and Sturdy. Enter the formula =C6 +C7 in cell C8 and extend it to I8 to calculate total demand from North America and all other demand regions. The formula " = SUM(D37: D42) - Enter D8" in cell D44 and extend it to J44 to calculate the demand constraint for all regions.
To find the optimal fusion production, • Select the Data tab and click Solver in the Analysis group. • In this case, enter your destination D37: J42 > 0 (all decision variables are positive) D37: J42 = integer (all decision variables are integers) D44: J44 = 0 (requirement constraint) K37: K420 (capacity constraint) • Enter a different decision variable in the Change Variable cell field of D37. J42 • Select Minutes. • Click Solve to optimize. step 1: Enter the data in the Excel sheet as follows:
The above formulas result in the following output:
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STEP-2 Solver parameters:
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STEP-3: Solve , to get the output as follows:
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Explanation: All the part(b) is explained here. c) Ans:
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Sleek and Sturdy-Merger (Scale Back/Shut Down) Since the plants, now, can be scaled back or shut down, the optimization model can be re-stated as Since the plants, now, can be scaled back or shut down, the optimization model can be re-stated as Minimize Here, n is the production facilities (3), m is the number of regional markets (7), D_j is the annual demand for cell phones from regional market j ,L_i is the potential capacity of each production facility if it is scaled back, g_i is the annualized fixed cost of a production facility if it is scaled back, is the annualized fixed cost of a production facility if it is shut down, y_i is 1 if facility i is scaled back, 0 otherwise, and is 1 if facility is shut down, 0 otherwise. STEP-1: Enter the Data as follows:
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STEP-2: Solver parameters:
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The lowest cost of production and distribution achievable with this network is $970 million.
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The results show that Sturdy's closure of his EU factory minimizes overall costs. Explanation: The above is an optimal post-merger network when assets can be downsized or shut down (d) Lean and Robust Merger (downsize/close, no obligations) Change the values in cells D10 through J10 to zero to represent the import duty. Run the solver. The result obtained is Explanation: Optimal networks without import duties are shown in the table above. Closing Sturdy's NA facility will help keep costs as low as possible. The lowest cost of production and distribution achievable for this network is $936.3 million.
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