Chapter 5: Network Design in the Supply ChainExercise Solutions 1. (a) The objective of this model is to decide optimal locations of home offices, and number of trips from each home office, so as to minimize the overall network cost. The overall network cost is a combination of fixed costs of setting up home offices and the total trip costs. There are two constraint sets in the model. The first constraint set requires that a specified number of trips be completed to each state j and the second constraint set prevents trips from a home office i unless it is open. Also, note that there is no capacity restriction at each of the home offices. While a feasible solution can be achieved by locating a single home office for all trips to all states, it is easy to see that this might not save on trip costs, since trip rates vary between home offices and states. We need to identify better ways to plan trips from different home offices to different states so that the trip costs are at a minimum. Thus, we need an optimization model to handle this. Optimization model: n m Dj Ki fi cij yi xij = 4: possible home office locations. = 16: number of states. = Annual trips needed to state j = number of trips that can be handled from a home office As explained, in this model there is no restriction = Annualized fixed cost of setting up a home office = Cost of a trip from home office i to state j = 1 if home office i is open, 0 otherwise = Number of trips from home office i to state j. It should be integral and non-negative n Min i 1 n m f i yi cij xij i 1 j 1 Subject to n x i 1 ij m x j1 ij D j for j 1,...,m (5.1) K i yi for i 1,...,n (5.2) yi {0,1} for i 1,...n (5.3) 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). 1 L7:L22 M31 N7:N22 2 .xls) CELL E7:E22 G7:G22.SYMBOL INPUT Dj Annual trips needed to state j cij Transportation cost from office i to state j fi xij fixed cost of setting up office i number of consultants from office i to state j.I26.1 demand constraints (Sheet SC consulting in workbook exercise5.I7:I22.M26 F7:F22. objective function 5. obj.H7:H22.K26.1. J7:J22.M7:M22 G26. K7:K22. 250 6.750 154.875 180.480 160.428 131.000 145. 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.875 The number of consultants is calculated based on the constraint of 25 trips per consultant.230 140.000 15.250 20. 3 .750 9.With this we solve the model to obtain the following results: Tota l# of trips Trip s from LA Cost from LA Trip s from Tuls a Washington 40 - 150 - Oregon 35 - 150 California 100 100 Idaho 25 Nevada Trips from Denve r Cost From Denver Trips from Seattl e 250 - 200 40 25 - 250 - 200 35 75 75 - 200 - 150 - 125 - 150 - 200 - 125 25 125 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 Nebraska 30 - 250 30 100 - 125 - 250 Kansas 40 - 250 25 75 15 75 - 300 Oklahoma 55 - 250 55 25 - 125 - 300 # of trips 675 190 - 110 - 250 - 125 State # of Consultants Fixed Cost of office Cost of Trips Total Office Cost Cost from Tulsa Cost from Seattle 8 5 10 5 165.678 137. Optimization model: n m Dj Ki fi cij yi xij = 4: potential sites. All shipment numbers need to be positive integers. = Annualized fixed cost of setting up a potential site. 0 otherwise = Number of air conditioners from site i to regional market j. though in general we need a new constraint to model the new requirement. 4 . This reduces the number of consultants at Denver to 10 while maintaining 5 consultants at Tulsa. should have a value of 250. for all i. We note that in the optimal solution of (b). each state is uniquely served by an office except for Kansas where the load is divided between Denver and Tulsa. then we need to add one more constraint i. The typical scenarios that need to be considered are either having regional manufacturing if the shipping costs are significant or have a centralized facility if the fixed costs show significant economies to scale. b. it is not necessary in this specific case. Or in terms of the optimization model. Each Ki is assigned value 400000. that is a good solution and still optimal. We keep the units shipped from each plant to every region as variable and choose the fixed cost based on the emerging production quantities in each plant location. = Cost of producing and shipping an air conditioner from site i to regional market j = 1 if site i is open. the total number of trips from an office may not exceed 250. Again the number of consultants remains at 5 and 10 in Tulsa and Denver. The cost to serve Kansas is the same from either office. faces the tradeoff between fixed cost (that is lower per item in a larger plant) versus the cost of shipping and manufacturing.000 c. We can revise constraint (5. (c) Just like the situation in (b). As trips to Kansas can be offloaded from Denver to Tulsa without any incremental cost. Hence we can meet the new constraint by making Tulsa fully responsible for Kansas.e.(b) If at most 10 consultants are allowed at each home office.2) with this Ki value and resolve the model. This brings the trips out of Tulsa to 125 and those out of Denver to 235. respectively. DryIce Inc. The new model will answer (b). All shipments to a region should add up to the requirement for 2006 . = Annual units needed of regional market j = maximum possible capacity of potential sites. Ki. The maximum production capacity is 400. 2. it is clear that only the Denver office violates this new condition. The total system cost is then minimized with the following constraints: a. = 4: number of regional markets. we choose fixed cost accordingly. However in this specific case. If actually needed capacity is less than or equal to 200000. Hence we just allocate 5 of the Denver-Kansas trips to Tulsa. 120000.. All calculations are performed at the exchange rates provided.. This is done by utilizing the following constraints: Each plant runs at least at half capacity.F10:F13 H10:H13...xls) We get the following results: The optimal solution suggests setting up 4 regional plants with each serving the needs of its own region.m ij K i yi for i 1.J10:J13 K21 L10:L13 (Sheet DryIce in workbook exercise5...1) SYMBOL INPUT Dj requirement at market j cij Variable cost from plant i to market j fi fixed cost of setting up plant i xij number of consultants from office i to state j. Sum of all shipments from the plant needs to be less than or equal to the capacity in that plant. Chicago and San Diego should each have a 200. 180000. Atlanta.2. the shipments from each plant to every market are assumed to be variable and solved to find the minimum total cost.000 capacity plant with production levels of 110000.1 objective function demand constraints CELL K10:K13 C10:C13. 5. New York.E10:E13 G10:G13.It should be integral and non-negative n n m f y c x Min i 1 i i i 1 j 1 ij ij Subject to n x i 1 ij D j for j 1.. All production volumes are non-negative.. 5 . 3 (a) Sunchem can use the projections to build an optimization model as shown below. In this case.n (5.E7:E8 G7:G8.I7:I8 D10:D13.2) m x j1 (5. respectively. obj. 100000.I10:I13 C7:C8. 502 7.100 1.000.000 4. cij = Cost of shipping one ton of printing ink from plant i to regional market j pi = Cost of producing one ton of printing ink at plant i xij = Tons of printing ink shipped from site i to regional market j..000 120 1.100 800 190 2.000 10.2 5.576.740 418.625.200 270 1.K10.300 200 1.I4:I8.L12 xij printing ink shipped from site i to regional market j obj.000 64...500 7.400 95 300 25 2.I10.Optimization model: n m Dj Ki = 5: five manufacturing plants = 5: number of regional markets.G10.000 1..2) ij 0.400 1. M4:M8 N18 O4:O8 E10. objective function 5.F12.n (5.400 2. K4:K8. Especially for (a) lower limit for capacity is 50%*Ki .000 1.400 1.063.H12.. It should be integral and non-negative n m (c Min i 1 j 1 ij pi ) xij Subject to n x ij D j for j 1.000 1.000 1. America Europe Japan Asia Capacity (ton/yr) Minimum Run Rate 600 1.000 0.000 0.009 16.000 400.700 185 ShipmentGermany Shipment Japan Shipment Brazil Shipment India Shipment Demand(ton/yr) 100 1.750 487.300 160 2.000 Total 1.800.000 0 0 - 7.n (5.200 1.000 0.H4:H8...M10 E10.m (5.000 1...1 5.G4:G8.3.I10.750 426.200 800.000 1. = Annual tons of ink needed for regional market j = Maximum possible capacity of manufacturing plants.461. America S.060. J4:J8.306 1.300 2. Production Cost per Ton Exch Rate Prod Cost per Ton(US$) Production Cost In US$ Tpt Cost in US$ 10.950 6 .200 10 2.1) ij K i for i 1..3 demand constraints capacity constraints 50% capacity constraints E4:E8.3) i 1 m x j1 m x j1 SYMBOL INPUT Dj Annual demand at market j cij shipping cost from plant i to regional market j pi CELL N4:N8 D4:D8.K10.G10.F4:F8.000 0.200 164.xls) The optimal result is summarized in the following table: US N.5 K i for i 1..M10 (Sheet capacity_constraints in workbook exercise5.300 20 900 2.300 190 600 200 1.100 800 80 100 100 475 475 50 25 200 200 80 80 93 $ 238 Mark 25 Yen 100 Real 40 Rs.200 736..500 13.000 15.530 3.023 9.000 60.974.562 7.J12. L4:L8 production cost of at plant i D12. 200 270 2.795.400 120 300 2.219. we see that it is more economical to produce higher volumes in Brazil.000 0.306 1.760 476.000 400.000 0.3).000 1. US N.300 200 1.200 270 1.800.700 185 ShipmentGermany Shipment Japan Shipment Brazil Shipment India Shipment Demand(ton/yr) 115 1.300 2.000 120 1.000 - (0) - 7. Low cost structure plants need to operate at capacity.200 - - 3.009 16.000 0.162.576.100 800 100 475 420 50 200 460 80 - 93 $ 238 Mark 25 Yen 100 Real 40 Rs. America S.260 176. 7 .300 2.000 - 15.000 0.530 3.740 - 13.800.000 1.836.200 1.710.502 7.300 135 2. And in (b). there is merit in shifting some of the production to another plant.400 1.200 736.534.000 1.200 1.200 0 270 1.795.750 489.000 0. (c) From the scenario in (a) we see that two of the plants are producing at full capacity. Production Cost per Ton Exch Rate Prod Cost per Ton(US$) Production Cost In US$ Tpt Cost in US$ 10. America Europe Japan Asia Capacity (ton/yr) Minimum Run Rate 600 1.000 10.740 - 13.306 3. the cost reduces to $7.510 (d) It is clear that fluctuations in exchange rates will change the cost structure of each plant.400 120 300 2.400 1.400 2.This is clearly influenced by the production cost per ton and the local market demand.580.150.200 20 2.502 7.100 1. Sunchem should plan on making excess capacity available at its plants.590.000 15.400 1. This gives us the following results: US N.000 Total 1.000 0.000 1.000 10.000 0.300 190 600 200 1.360.009 16.000 64.260 800.300 200 1.066. If the cost at a plant becomes too high.500 1. America S.300 20 900 2.400 2. The analysis shows that there are gains from shifting a significant portion of production to Brazil and having no production in Japan. there is merit in shifting some of the production from other plants to this plant.760 - 0 0 - 7. (b) If there are no limits on production we can perform the same exercise as in (a) but without the capacity constraints (5.562 7.100 1.000 1.600 - 3.100 800 190 2.000 120 1.023 9.100 800 190 2.023 9.000 69.300 190 600 200 1.600 418. Either of these scenarios requires that the plants have built in excess capacity.700 185 Minimum Run Rate Production Cost per Ton Exch Rate Prod Cost per Ton(US$) Production Cost In US$ Tpt Cost in US$ Total ShipmentGermany Shipment Japan Shipment Brazil Shipment India Shipment Demand(ton/yr) 1.530 3.250 1.510.000 1.400 1.2) and (5.000 4.417.562 7. 10. US and India.300 2.000 0.300 100 900 2. America Europe Japan Asia Capacity (ton/yr) 600 1.100 800 80 100 115 475 475 50 210 210 80 80 93 $ 238 Mark 25 Yen 105 Real 40 Rs.000 1.360 Clearly by having no restrictions on capacity SunChem can reduce costs by $557.000 400. Similarly if a plant’s cost structure becomes more favorable.000 1. Once we add 10 tons/year to Brazil. Dj = Annual market size of regional market j Ki = maximum possible capacity of production facility i cij = Variable cost of producing..1) ij K i for i 1. we will build more advanced models in the subsequent parts of this question.4) CELL B4:H4 C12:C14 B22:H28 D12:D17 C43:I45 D48 J43:J45 C46:I46 The above model gives optimal result as in following table: 8 ... Sleekfon and Sturdyfon operate independently. and so we need to build separate models for each of them. Prior to merger. transporting and duty from facility i to market j Annual fixed cost of facility i fi xij Number of units from facility i to regional market j. obj.1 demand constraints 5. m = 7: number of regional markets. n Min i 1 n m fi cij xij i 1 j 1 Subject to n x i 1 ij D j for j 1.m (5.4 (a) Starting from the basic models in (a).2 capacity constraints (Sheet sleekfon in workbook problem5.. It should be integral and non-negative. transporting and duty from facility i to market j fi = Annual fixed cost of facility i xij = Number of units from facility i to regional market j. Optimization model for Sleekfon: n = 3: Sleekfon production facilities. objective function 5..n (5.. Variable cost cij is calculated as follows: cij = production cost per unit at facility i + transportation cost per unit from facility i to market j + duty*( production cost per unit at facility i + transportation cost per unit from facility i to market j + fixed cost per unit of capacity) SYMBOL INPUT Dj Annual market size of regional market j Ki maximum possible capacity of production facility i cij Variable cost of producing.2) m x j1 Please note that we need to calculate the variable cost cij before we plug it into the optimization model... i.00 2.00 0.00 0.00 0 0.00 1.00 0. the previous model is still applicable. and hence has more choices for production and distribution plans. America Europe (EU) N.00 0.00 0.00 5.00 4.00 8.68 (b) Under conditions of no plant shutdowns.00 0.Quantity Shipped Sleekfon N. America Demand S.00 0.00 0 7.00 10.00 0 0.00 0.00 0 0.00 4.00 12.e. 3 from Sleekfon and 3 from Sturdyfon.00 0. 9 .00 1.00 0.00 0. America Europe (EU) Sturdyfon N.. America S.00 512.00 3.00 0.00 0.00 0.00 0.00 0.00 0.39 And we use the same model but with data from Sturdyfon to get following optimal production and distribution plan for Sturdyfon: Quantity Shipped N. However.00 0 0. America Europe (EU) Europe Japan (Non EU) Rest of Asia/Australia Africa Capacity 0.00 Total Cost for Sleekfon = $ 564.00 0. The optimal result is summarized in the following table.00 0.00 0 0. America Rest of Asia Demand Total cost for Sturdyfon = S. we need to increase the number of facilities to 6.00 0. And the market demand at a region needs revised by combining the demands from the two companies.00 0.00 0.00 1.00 0.00 0.00 7.00 0.00 0. Decision maker has more facilities and greater market share in each region.00 2.00 0 3.00 0.00 3. America Europe (EU) Europe Japan (Non EU) Rest of Asia/Australia Africa Capacity 0.00 7.00 0.00 0.00 20.00 0. 00 0. America Europe (EU) Sturdyfon N.00 0.00 0.N. Sleekfon S America and Sturdyfon Rest of Asia.00 0.00 4. 0 otherwise. Ki =capacity of production facility i Li =capacity of production facility if it is scaled back cij = Variable cost of producing.00 5.00 0. Dj = Annual market size of regional market j.00 0.00 5.00 20.00 0. America Europe (EU) Sleekfon N. yi = Binary variable indicating whether to scale back facility i.00 0.00 3.00 0. yi = 1 means to scale it back.00 0.00 0. transporting and duty from facility i to market j fi = Annual fixed cost of facility i gi = Annual fixed cost of facility i if it is scaled back hi = Shutdown cost of facility i xij = Number of units from facility i to regional market j.00 0.00 14.00 0. (1-yi –zi) would be the binary variable indicating whether the facility is unaffected. Optimization model for Sleekfon: n = 6: Sleekfon and Sturdyfon production facilities. Accordingly we need more variables to reflect this new complexity. the index i doesn’t include these two facilities.82 (c) This model is more advanced since it allows facilities to be scaled down or shutdown. America Rest of Asia Demand S.00 0.00 0.00 0.00 0.00 0. America S.00 0.00 0. It should be integral and non-negative.00 0.00 4. Since two facilities.00 Total Cost for Merged Network = 1066.00 0. zi =1 means to shutdown it.00 0.00 0. zi = Binary variable indicating whether to shutdown facility i.00 5.00 0.00 0.00 0.00 0.00 0.00 16.00 0. can not be scaled back. America Europe (EU) Europe Japan (Non EU) Rest of Asia/Australia Africa Capacity 0.00 6.00 0.00 0.00 0. sum of the Sleekfon and Sturdyfon market share.00 0.00 0. m = 7: number of regional markets.00 11.00 0.00 0. 0 otherwise.00 0.00 5. 10 .00 2.00 0.00 0.00 0. 11 .00 0...00 1..xls.00 0.4) Please note that we need to calculate the variable cost cij before we plug it into the optimization model.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 5.00 1.00 0. America S.00 0. zi are binary for i 1.00 0.. n (5.00 0.00 0.n n m ( f (1 y z ) g y h z ) c x Min i 1 i i i i i i i i 1 j 1 ij ij Subject to n x i 1 ij m x j 1 ij D j for j 1.00 2.00 0. Quantity Shipped Europe (EU) Sleekfon N.00 20.00 0. much lower than the result we got in (b) $1066.00 5..00 0.00 0.00 5. m (5.00 Total Cost for Merged Network = 988.America facility is shutdown.00 0.00 0. fixed cost for a scaled back facility is 70% of the original one..00 2.00 0.82.00 2.00 0.00 0.00 0.00 0.00 0.America market share is 22.00 0 0.00 0.00 0. America S. The lowest cost possible in this model is $988.00 0.00 0. As explained in the problem description.00 0. n (5...00 0.2) 1 yi zi 0 for i 1.00 0.93.00 0. We can achieve this by resetting B7:H7 to zeros in sheet merger (shutdown) in workbook problem5.00 3.00 0.00 0.00 0.America facility. n (5. the Sleekfon N.00 0.00 1. Above model gives optimal solution as summarized in the following table.00 0.00 0.93 For questions (d) and (e).. we need to change the duty to zero and run the optimization model again to get the result.00 0.3) yi . As shown in the result.. and there are 40 in terms of production capacity. America Europe (EU) Sturdyfon N.00 0.00 0.00 0.00 0.00 0. and the market is mainly served by Sturdyfon N.00 0. America Rest of Asia Demand N.1) K i (1 yi zi ) Li yi for i 1.00 11..00 0.00 1. Variable cost cij is calculated as following: cij = production cost per unit at facility i + transportation cost per unit from facility i to market j + duty*( production cost per unit at facility i + transportation cost per unit from facility i to market j + fixed cost per unit of capacity) And we also need to prepare fixed cost data for the two new scenarios: shutdown and scale back.00 0.00 1.00 0...00 19.. hence it is wise to shutdown one facility whichever is more expensive..00 1.00 0.00 0.4. America Europe (EU) Europe Japan (Non EU) Rest of Asia/Australia Africa Scale back Shut down Plant Capacity unaffected 0. and it costs 20% of the original annual fixed cost to shutdown it..00 5. The N.00 0. 00 0.00 20. only two constraints are relevant.5 (a) The model we developed in 4.00 0.00 0.d is applicable to this question. America Europe (EU) Europe Japan (Non EU) Rest of Asia/Australia Africa Scale back Shut down Plant Small unaffected addition Large Addition Capacity 0.00 0. and it may be more expensive to ship from other regions than to increase local capacity. adjust the regional imbalance.00 0. i.00 1.00 0.00 2.00 0. it must be noted that as the demand increases steadily.America is not shutdown in this optimal result.00 0.00 0.00 0.77 As shown in the table.00 0.00 0.00 4.00 0.00 0.00 0.00 Total Cost for Merged Network 0.00 4.60 0.00 0.00 0. America Rest of Asia Demand N.00 0.00 0.00 0.00 0. And the new demand structure yields a quite different optimal configuration of the network.00 0. America Europe (EU) Sturdyfon N.00 1.00 0.00 0.00 0. Instead. We only need to update the demand data accordingly. 12 .00 0. even when the total capacity at a certain period is greater than or equal to the total demand.00 0.40 0.00 0. However due to the discount factor.00 0.00 0 0. we might want to increase capacity anyway.00 1.00 0. we want to increase capacities as late as possible.00 9.40 0.00 15. If we treated each period separately. Sturdyfon N.00 0. Quantity Shipped Europe (EU) Sleekfon N. and take benefit of discount effect over periods. Sturdyfon EU facility is shutdown.00 6.00 0.e. satisfy capacity constraints. Considering the multi-period nature of this problem.00 15.00 0. This complexity calls for an optimization model to find an optimal solution which can serve all demands.00 0.00 9.00 0.00 0.60 0.00 0. For questions (b). America S.00 0.00 1.00 0. (c) and (d).00 1.00 1.00 4. On the other hand.00 1.00 0.00 0. we need to update Excel sheet data accordingly and rerun the optimization model.40 0.00 6. we need to add capacities eventually.00 0. This is because a regional market might run short while the total supply is surplus.00 0..00 1141. America S.00 1. the demand and capacity constraints.60 0.00 3. 6 (a) StayFresh faces a multi-period decision problem.00 0.00 0.00 0. 1 means to increase capacity of plant i using option k at time t.I: set of plants and potential plants J: set of regional markets T: set of periods under consideration.1 xijt M i yikt ek for each plant i at each period t x j t x ijt dj for each regional market j 5. 0 otherwise. and capacity incremental options 13 . market. K: set of capacity incremental options d jt : demand of regional market j at period t M i : capacity of plant i at beginning cij : production and transportation cost from plant i to reginal market j e k : capacity increment amount of option k f k : capacity increment cost of option k r : discount factor Yikt : binary variable. shipment amount from plant i to market j at time t Min 10 t t c f y /( 1 r) xij 5cij f k yik5 /( 1 r) ijt ij k ikt t i j i k t 5 i j i k 5. period. And T 5 in this model. X ijt : decision variable. 6~10 year is treated separately.2 i xijt 0 . binary yikt for all plant. However. we need to change data in the Excel sheet accordingly. 0 otherwise X ijt decision variable.2 demand constraint I5:Q8 B22:E25 H22:K25 N22:Q25 T22:W25 Z22:AC25 C31 G22:G25 M22:M25 S22:S25 Y22:Y25 Ae22:Ae25 B26:E26 H26:K26 N26:Q26 T26:W26 Z26:AC26 (Sheet StayFresh in workbook problem5. a new plant in Kolkata is built in the optimal solution anyway.1 capacity constraint 5. Since the cost of fifth year will be added into the total cost six times.000 units. the solution is optimal in this particular model. In the second year. This explains why extra capacity is built into the network earlier than necessary. shipment amount from plant i to market j at time t obj objective function 5. which has lead to high surplus capacity. In the third and fourth years. since it is cheaper to server the local market from Kolkata than to ship from other regions. no new capacity is added. While there is no reason to add capacity earlier than necessary.6. original total capacity was 600. since the plant location is reasonable and the total capacity still exceeds the demand. 14 .xls) In the first year. especially under the consideration of the discount factor.SYMBOL INPUT d jt demand of regional market j at period t CELL B9:H9 Mi capacity of plant i at beginning C12:C14 cij Production and transportation cost from plant i to regional market j B5:F8 ek capacity increment amount of option k D12:D17 fk r capacity increment cost of option k C43:I45 discount factor D48 Yikt binary variable. new capacity is added consecutively. For questions (b) and (c). it is strategically correct to spend as little as possible in the fifth year.000 units more than the total demand. which was 60. Note that this additional capacity is needed for the fifth year. 1 means to increase capacity of plant i using option k at time t. n = 2 potential sites. However.2) y3 y4 1 add at most one site (5. and N.xls) 15 . Dj = Annual units needed of regional market j Ki = maximum possible capacity of potential sites.3) y3 . and California might yield higher before-tax profit than a network of Kentucky.1) K i yi for i 1. 0 otherwise xij = Number of products from site i to regional market j. fi = Annualized fixed cost of setting up a potential site.1 5. 5.7.4) SYMBOL INPUT Dj Annual market size of regional market j Ki maximum possible capacity of production facility i cij Variable cost of producing.Carolina since California has a lower variable cost serving West regional market.. the after-tax profit might be worse. n (5.. Pennsylvania. Hence it is not hard to see that Blue Computers needs a new plant. however both have high variable costs to serve the West regional market. California is a better choice than N. transporting and duty from facility i to market j Annual fixed cost of facility i fi xij Number of units from facility i to regional market j. cij = Cost of producing and shipping a computer r from site i to regional market j yi = 1 if site i is open..2 5. Pennsylvania.Carolina. y4 are binary (5.. m = 4: number of regional markets.Carolina has extra tax benefit.. From this point of view. obj. Even if a network of Kentucky. On the other hand. N. m (5. It should be integral and non-negative n n m f y c x Min i 1 i i i 1 j 1 ij ij Subject to n x i 1 ij m x j 1 ij D j for j 1. which can serve the West regional market at a lower cost...4 objective function demand constraints capacity constraints see explanation in next paragraph CELL B9:F9 H5:H8 B5:F8 G5:G8 B17:F20 I21 B21:F21 H17:H20 (Sheet Blue in workbook problem5. West regional market has 2 nd highest demand.7 (a) Blue Computers has two plants in Kentucky and Pennsylvania.. 763. we can do better in practice.01E-10 South West 0 450 0 0 -1. m = 4: number of regional markets.500 Total Profit = $ (0) 513.000 $ 516.000 $ 490.00 900 1 0. The result below shows the optimal solution when California is picked up. Shipment Kentucky Pennsylvania N. and change the direction of optimization from minimizing to maximizing. Prior to merger. It should be integral and non-negative. Dj = Annual market size of regional market j Ki = maximum possible capacity of production facility i cij = Variable cost of producing.565 $ $ $ 0 $ 396.500 - $ 1.00 0 1.000 Total Cost = 1501500 (b) We only need to change the objective function from minimize cost to maximize profit.000.500.000 $ 530.000 $ 672. Hot&Cold and CaldoFreddo operate independently.00 1500. It is easy to see that lowest cost doesn’t mean maximum after tax profit. The following table shows the result. Since at most one site can be open.3E-06 Cost $ 255.040.Carolina. Shipment Northeast Kentucky Pennsylvania N.5E-06 5. we can run the optimization three times for three scenarios respectively: none open. and we need to build separate models for each of them. or only N.000 $ 1. we will build more advanced models in the subsequent parts of this question.00 -2.100.500 $ 1.5E-06 South 400 0 0 50 -1.1E-06 Open (1) / Capacity Shut (0) Constraint 0 1 400.500 $ 759. Optimization model for Hot&Cold: n = 3: Hot&Cold production facilities. It is much faster to solve these three scenarios separately given that EXCEL solver cannot achieve a converging result with constraint (5. Carolina California Demand constraint Northeast 0 1050 0 0 -2.1E-06 West Open (1) / Capacity Shut (0) Constraint 0 1 0.555 $ 1.680 8 (a) Starting from the basic models in (a). And we compare the three results and choose the best one.00 900 1 400.000 $ 1.00 -2.65E-06 Southeast Midwest 0 600 150 2. 16 .65E-06 Southeast Midwest 600 0 0 0 -1.00 0 1 0.3E-06 Total Cost = Cost Revenue Profit After tax profit $ 328.500 $ 200.560 $ 459. only California. On the Excel sheet.00 0.4).4) is simple in its mathematical notation.5E-06 0 450 0 150 -1.000 1184000 $ (0) $ 703.00 0 0.43E-09 0 0 450 0 -1.Even though constraint (5. transporting and duty from facility i to market j fi = Annual fixed cost of facility i ti =Tax rate at facility i xij = Number of units from facility i to regional market j. all we need to do is to set the target cell from I21 to L21.00 0 1 0. Carolina California Demand constraint 0 0 0 1050 -2. Italy Demand Capacity Annual Cost Quantity Shipped (million units) 15 0 0 15 5 0 0 30 0 20 0 0 0 $ 6.400 17 .0 0 Annual Cost 0 6150 35 2475 0 4650 Total Cost 13275 And we use the same model but with data from CaldoFreddo to get following optimal production and distribution plan for CaldoFreddo: Shipment CaldoFreddo U.225 Total Cost $ 10.0 0 Capacity 35..2) m x j1 And replace above objective function to the following one to maximize after tax profit: n Max m n n m (1 t ) px f c x i i 1 j 1 ij i 1 i i 1 j 1 ij ij SYMBOL INPUT Dj Annual market size of regional market j Ki maximum possible capacity of production facility i cij Variable cost of producing.xls) The above model gives optimal result as in following table: Shipment Hot&Cold France Germany Finland Demand North East 0.0 0 West 15.0 0.0 20...K.8.1 demand constraints 5. transporting and duty from facility i to market j Annual fixed cost of facility i fi xij CELL C8:F8 G5:G7 C5:F7 H5:H7 C20:F22 H24 C23:F23 G20:G22 of units from facility i to regional market j...0 10.m (5.0 0..2 capacity constraints (Sheet Hot&Cold in workbook problem5.n n m f c x Min i 1 i i 1 j 1 ij ij Subject to n x i 1 ij D j for j 1.0 0. obj.0 5.175 25 $ 4..0 20.n (5. objective function 5.1) ij K i for i 1.0 0 South 0..0 0. sum of the : Hot&Cold and caldoFreddo market share. and hence has more choices for production and distribution plans.. which should be the sum of market shares. Shipment Merged Company France Germany Finland U.(b) If none of the plants is shut down. Decision maker has more facilities and greater market share in each region.. n (5.. transporting and duty from facility i to market j fi = Annual fixed cost of facility i xij = Number of units from facility i to regional market j. m K i (1 zi ) zi are binary for i 1.. Italy Demand Capacity Open (1) / Shut (0) Quantity Shipped (million units) 0 30 15 0 0 0 0 25 0 10 0 0 0 0 50 5 0 0 50 0 1.9E-09 1 Annual Cost 45 $ 1. And we need to update the market demand Dj. zi =1 means to shutdown it.700 1 0 $ 5. Dj = Annual market size of regional market j..500 1 20 $ 3.1) for i 1. However. Ki =capacity of production facility i cij = Variable cost of producing.450 $ 22..000 (c) This model is more advanced since it allows facilities to be shutdown. the previous model is still applicable... The optimal result is summarized in the following table.. with 3 from Hot&cold and 2 from CaldoFreddo.500 1 0 $ 6. m = 4: number of regional markets.0E+00 -1. zi = Binary variable indicating whether to shutdown facility i. n (5.K.3E-10 -9. Accordingly we need more variables to reflect this new complexity..850 1 7.9492E-09 $ 4..2) (5.8E-09 0. It should be integral and non-negative. Optimization model for Sleekfon: n = 5: Hot&Cold and caldoFreddo production facilities. we need to update the number of facilities to 5. 0 otherwise.3) 18 . n n m f (1 z ) c x Min i 1 i i i 1 j 1 ij ij Subject to n x i 1 ij m x j 1 ij D j for j 1.. SYMBOL INPUT Dj Annual market size of regional market j Ki maximum possible capacity of production facility i cij Variable cost of producing.0 0.0 Annual Cost 1 45 $ 1.0 0.K. Following table shows the optimal configuration.500 1 10 $ 4.800 1 0 $ 4.400 $ 22.500 1 10 $ 5.2 capacity constraints (Sheet Merged in workbook problem5.000 Total Cost 19 . transporting and duty from facility i to market j Annual fixed cost of facility i fi xij Zi CELL C8:F8 C12:C14 C5:F7 H5:H7 C19:F23 G19:G23 I25 C24:F24 H19:H23 Number of units from facility i to regional market j. objective function 5.1 demand constraints 5.0 0. Italy Capacity Open (1) / Shut (0) Quantity Shipped (million units) 0 40 5 0 0 0 0 35 0 0 0 0 0 0 50 5 0 0 50 0 0. Shipment Merged Company Demand France Germany Finland U.xls) It turned out that all sites are open so as to achieve best objective value.8. open or shutdown facility i obj.800 1 0 $ 5.