Chapter 5: Network Design in the Supply Chain Exercise 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
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.
obj. objective function 5.1 demand constraints (Sheet SC consulting in workbook exercise5.1.xls)
CELL E7:E22 G7:G22,I7:I22, K7:K22,M7:M22 G26,I26,K26,M26 F7:F22,H7:H22, J7:J22,L7:L22 M31 N7:N22
2
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
165,428
131,230
140,000
145,000
15,250
6,250
20,750
9,875
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.
3
(b) If at most 10 consultants are allowed at each home office, then we need to add one more constraint i.e. the total number of trips from an office may not exceed 250. Or in terms of the optimization model, Ki, for all i, should have a value of 250. We can revise constraint (5.2) with this Ki value and resolve the model. The new model will answer (b). However in this specific case, it is clear that only the Denver office violates this new condition. As trips to Kansas can be offloaded from Denver to Tulsa without any incremental cost, that is a good solution and still optimal. Hence we just allocate 5 of the Denver-Kansas trips to Tulsa. This reduces the number of consultants at Denver to 10 while maintaining 5 consultants at Tulsa.
(c) Just like the situation in (b), though in general we need a new constraint to model the new requirement, it is not necessary in this specific case. 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. The cost to serve Kansas is the same from either office. Hence we can meet the new constraint by making Tulsa fully responsible for Kansas. This brings the trips out of Tulsa to 125 and those out of Denver to 235. Again the number of consultants remains at 5 and 10 in Tulsa and Denver, respectively.
2. DryIce Inc. faces the tradeoff between fixed cost (that is lower per item in a larger plant) versus the cost of shipping and manufacturing. 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. 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. The total system cost is then minimized with the following constraints: a. All shipment numbers need to be positive integers. b. The maximum production capacity is 400,000 c. All shipments to a region should add up to the requirement for 2006 . Optimization model: n m Dj Ki fi cij yi xij
= 4: potential sites. = 4: number of regional markets. = Annual units needed of regional market j = maximum possible capacity of potential sites. Each Ki is assigned value 400000. If actually needed capacity is less than or equal to 200000, we choose fixed cost accordingly. = Annualized fixed cost of setting up a potential site. = Cost of producing and shipping an air conditioner from site i to regional market j = 1 if site i is open, 0 otherwise = Number of air conditioners from site i to regional market j.
4
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,...,m
ij
K i yi for i 1,...,n (5.2)
m
x j1
(5.1)
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.
obj. 5.1
objective function demand constraints
CELL K10:K13 C10:C13,E10:E13 G10:G13,I10:I13 C7:C8,E7:E8 G7:G8,I7:I8 D10:D13,F10:F13 H10:H13,J10:J13 K21 L10:L13
(Sheet DryIce in workbook exercise5.2.xls) We get the following results: The optimal solution suggests setting up 4 regional plants with each serving the needs of its own region. New York, Atlanta, Chicago and San Diego should each have a 200,000 capacity plant with production levels of 110000, 180000, 120000, 100000, respectively.
3 (a) Sunchem can use the projections to build an optimization model as shown below. In this case, the shipments from each plant to every market are assumed to be variable and solved to find the minimum total cost. This is done by utilizing the following constraints:
Each plant runs at least at half capacity. Sum of all shipments from the plant needs to be less than or equal to the capacity in that plant. All production volumes are non-negative. All calculations are performed at the exchange rates provided.
5
Optimization model: n m Dj Ki
= 5: five manufacturing plants = 5: number of regional markets. = Annual tons of ink needed for regional market j = Maximum possible capacity of manufacturing plants. Especially for (a) lower limit for capacity is 50%*Ki . 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. 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,...,m
(5.1)
ij
K i for i 1,...,n
(5.2)
ij
0.5 K i for i 1,...,n (5.3)
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,F4:F8,H4:H8, J4:J8, L4:L8
production cost of at plant i
D12,F12,H12,J12,L12
xij
printing ink shipped from site i to regional market j
obj.
objective function
5.1 5.2 5.3
demand constraints capacity constraints 50% capacity constraints
E4:E8,G4:G8,I4:I8, K4:K8, M4:M8 N18 O4:O8 E10,G10,I10,K10,M10 E10,G10,I10,K10,M10
(Sheet capacity_constraints in workbook exercise5.3.xls) The optimal result is summarized in the following table: US N. America S. America Europe Japan Asia Capacity (ton/yr) Minimum Run Rate
600 1,200 1,300 2,000 1,700 185
ShipmentGermany Shipment Japan Shipment Brazil Shipment India Shipment Demand(ton/yr) 100 1,300 160 2,000 1,200 10 2,200 270 1,400 2,100 800 190 2,300 190 600 200 1,400 1,400 1,300 200 1,400 95 300 25 2,100 1,000 120 1,300 20 900 2,100 800 80 100 100 475 475 50 25 200 200 80 80
93 $
238 Mark
25 Yen
100 Real
40 Rs.
Production Cost per Ton Exch Rate Prod Cost per Ton(US$) Production Cost In US$ Tpt Cost in US$
10,000 1.000 10,000 1,000,000 60,000
15,000 0.502 7,530 3,576,750 487,000
1,800,000 0.009 16,740 418,500 7,500
13,000 0.562 7,306 1,461,200 164,000
400,000 0.023 9,200 736,000 64,000
Total
1,060,000
4,063,750
426,000
1,625,200
800,000
0 0 -
7,974,950
6
This is clearly influenced by the production cost per ton and the local market demand. Low cost structure plants need to operate at capacity.
(b) If there are no limits on production we can perform the same exercise as in (a) but without the capacity constraints (5.2) and (5.3). This gives us the following results: US N. America S. America Europe Japan Asia Capacity (ton/yr)
600 1,200 1,300 2,000 1,700 185
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) 1,300 2,000 1,200 270 2,200 270 1,400 2,100 800 190 2,300 190 600 200 1,400 1,400 1,300 200 1,400 120 300 2,100 1,000 120 1,300 100 900 2,100 800 100 475 420 50 200 460 80 -
93 $
238 Mark
25 Yen
100 Real
40 Rs.
10,000 1.000 10,000 -
15,000 0.502 7,530 3,162,600 418,000
1,800,000 0.009 16,740 -
13,000 0.562 7,306 3,360,760 476,000
400,000 0.023 9,200 -
-
3,580,600
-
3,836,760
-
0 0 -
7,417,360
Clearly by having no restrictions on capacity SunChem can reduce costs by $557,590. The analysis shows that there are gains from shifting a significant portion of production to Brazil and having no production in Japan, US and India.
(c) From the scenario in (a) we see that two of the plants are producing at full capacity. And in (b), we see that it is more economical to produce higher volumes in Brazil. Once we add 10 tons/year to Brazil, the cost reduces to $7,795,510. US N. America S. America Europe Japan Asia Capacity (ton/yr) Minimum Run Rate
600 1,200 1,300 2,000 1,700 185
ShipmentGermany Shipment Japan Shipment Brazil Shipment India Shipment Demand(ton/yr) 115 1,300 135 2,000 1,200 20 2,200 0 270 1,400 2,100 800 190 2,300 190 600 200 1,400 1,400 1,300 200 1,400 120 300 2,100 1,000 120 1,300 20 900 2,100 800 80 100 115 475 475 50 210 210 80 80
93 $
238 Mark
25 Yen
105 Real
40 Rs.
Production Cost per Ton Exch Rate Prod Cost per Ton(US$) Production Cost In US$ Tpt Cost in US$
10,000 1.000 10,000 1,150,000 69,000
15,000 0.502 7,530 3,576,750 489,500
1,800,000 0.009 16,740 -
13,000 0.562 7,306 1,534,260 176,000
400,000 0.023 9,200 736,000 64,000
Total
1,219,000
4,066,250
1,710,260
800,000
-
(0) -
7,795,510
(d) It is clear that fluctuations in exchange rates will change the cost structure of each plant. If the cost at a plant becomes too high, there is merit in shifting some of the production to another plant. Similarly if a plant’s cost structure becomes more favorable, there is merit in shifting some of the production from other plants to this plant. Either of these scenarios requires that the plants have built in excess capacity. Sunchem should plan on making excess capacity available at its plants.
7
4 (a) Starting from the basic models in (a), we will build more advanced models in the subsequent parts of this question. Prior to merger, Sleekfon and Sturdyfon operate independently, and so we need to build separate models for each of them. Optimization model for Sleekfon: n = 3: Sleekfon production facilities. m = 7: number of regional markets. Dj = Annual market size of regional market j Ki = maximum possible capacity of production facility i cij = Variable cost of producing, 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. It should be integral and non-negative. 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
(5.1)
ij
K i for i 1,...,n
(5.2)
m
x j1
Please note that we need to calculate the variable cost cij before we plug it into the optimization model. 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, 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. objective function 5.1 demand constraints 5.2 capacity constraints (Sheet sleekfon in workbook problem5.4)
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
Quantity Shipped
Sleekfon
N. America
Europe (EU) N. America S. America Demand
S. America
Europe (EU)
Europe Japan (Non EU)
Rest of Asia/Australia
Africa
Capacity
0.00
0.00
20.00
0.00
0.00
0.00
0.00
0.00
10.00
0.00
0.00
3.00
2.00
2.00
0.00
3.00
0.00
4.00
0.00
0.00
0.00
0.00
1.00
5.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Total Cost for Sleekfon =
$ 564.39
And we use the same model but with data from Sturdyfon to get following optimal production and distribution plan for Sturdyfon: Quantity Shipped
N. America
Europe (EU) Sturdyfon N. America Rest of Asia Demand Total cost for Sturdyfon =
S. America
Europe (EU)
Europe Japan (Non EU)
Rest of Asia/Australia
Africa
Capacity
0.00
0.00
4.00
8.00
0.00
0.00
1.00
7.00
12.00
1.00
0.00
0.00
0.00
0.00
0.00
7.00
0.00 0
0.00 0
0.00 0
0.00 0
7.00 0
3.00 0
0.00 0
0.00
512.68
(b) Under conditions of no plant shutdowns, the previous model is still applicable. However, we need to increase the number of facilities to 6, i.e., 3 from Sleekfon and 3 from Sturdyfon. And the market demand at a region needs revised by combining the demands from the two companies. Decision maker has more facilities and greater market share in each region, and hence has more choices for production and distribution plans. The optimal result is summarized in the following table.
9
N. America Europe (EU) Sleekfon N. America S. America Europe (EU) Sturdyfon N. America Rest of Asia Demand
S. America
Europe (EU)
Europe Japan (Non EU)
Rest of Asia/Australia
Africa
Capacity
0.00
0.00
4.00
11.00
0.00
0.00
2.00
3.00
16.00
0.00
0.00
0.00
4.00
0.00
0.00
0.00
0.00
5.00
0.00
0.00
0.00
0.00
0.00
5.00
0.00
0.00
20.00
0.00
0.00
0.00
0.00
0.00
6.00
0.00
0.00
0.00
0.00
0.00
0.00
14.00
0.00
0.00
0.00
0.00
5.00
5.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Total Cost for Merged Network =
1066.82
(c) This model is more advanced since it allows facilities to be scaled down or shutdown. Accordingly we need more variables to reflect this new complexity. Optimization model for Sleekfon: n = 6: Sleekfon and Sturdyfon production facilities. m = 7: number of regional markets. Dj = Annual market size of regional market j, sum of the Sleekfon and Sturdyfon market share. Ki =capacity of production facility i Li =capacity of production facility if it is scaled back cij = Variable cost of producing, 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. It should be integral and non-negative. yi = Binary variable indicating whether to scale back facility i. yi = 1 means to scale it back, 0 otherwise. Since two facilities, Sleekfon S America and Sturdyfon Rest of Asia, can not be scaled back, the index i doesn’t include these two facilities. zi = Binary variable indicating whether to shutdown facility i. zi =1 means to shutdown it, 0 otherwise. (1-yi –zi) would be the binary variable indicating whether the facility is unaffected.
10
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,..., m
(5.1)
K i (1 yi zi ) Li yi for i 1,..., n
(5.2)
1 yi zi 0
for i 1,..., n
(5.3)
yi , zi are binary for i 1,..., n
(5.4)
Please note that we need to calculate the variable cost cij before we plug it into the optimization model. 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. As explained in the problem description, fixed cost for a scaled back facility is 70% of the original one; and it costs 20% of the original annual fixed cost to shutdown it. Above model gives optimal solution as summarized in the following table. The lowest cost possible in this model is $988.93, much lower than the result we got in (b) $1066.82. As shown in the result, the Sleekfon N.America facility is shutdown, and the market is mainly served by Sturdyfon N.America facility. The N.America market share is 22, and there are 40 in terms of production capacity, hence it is wise to shutdown one facility whichever is more expensive. Quantity Shipped
Europe (EU) Sleekfon N. America S. America Europe (EU) Sturdyfon N. America Rest of Asia Demand
N. America
S. America
Europe (EU)
Europe Japan (Non EU)
Rest of Asia/Australia
Africa
Scale back Shut down Plant Capacity unaffected
0.00
0.00
5.00
11.00
0.00
0.00
2.00
0.00
0.00
1.00
2.00
20.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
2.00
5.00
0.00
0.00
3.00
0.00
0.00
0
0.00
1.00
0.00
0.00
0.00
19.00
0.00
1.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
5.00 0.00
5.00 0.00
0.00 0.00
0.00
0.00
1.00
0.00
Total Cost for Merged Network =
988.93
For questions (d) and (e), we need to change the duty to zero and run the optimization model again to get the result. We can achieve this by resetting B7:H7 to zeros in sheet merger (shutdown) in workbook problem5.4.xls.
11
5 (a) The model we developed in 4.d is applicable to this question. We only need to update the demand data accordingly. And the new demand structure yields a quite different optimal configuration of the network. Quantity Shipped
Europe (EU) Sleekfon N. America S. America Europe (EU) Sturdyfon N. America Rest of Asia Demand
N. America
S. 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
0.00
20.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
15.60
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
4.40
0.00
6.00
0.00
0.00
0.00
0.00
0.00
0
0.00
1.00
4.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00
6.40
0.00
4.00
9.60
0.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00 0.00
0.00 0.00
0.00 0.00
3.60 0.00
9.00 0.00
15.00 0.00
2.40 0.00
0.00
0.00
1.00
Total Cost for Merged Network
0.00
1.00 1.00
0.00
1141.77
As shown in the table, Sturdyfon N.America is not shutdown in this optimal result. Instead, Sturdyfon EU facility is shutdown. For questions (b), (c) and (d), we need to update Excel sheet data accordingly and rerun the optimization model.
6 (a) StayFresh faces a multi-period decision problem. If we treated each period separately, only two constraints are relevant, i.e., the demand and capacity constraints. Considering the multi-period nature of this problem, it must be noted that as the demand increases steadily, we need to add capacities eventually. However due to the discount factor, we want to increase capacities as late as possible. On the other hand, even when the total capacity at a certain period is greater than or equal to the total demand, we might want to increase capacity anyway. This is because a regional market might run short while the total supply is surplus, and it may be more expensive to ship from other regions than to increase local capacity. This complexity calls for an optimization model to find an optimal solution which can serve all demands, satisfy capacity constraints, adjust the regional imbalance, and take benefit of discount effect over periods.
12
I: set of plants and potential plants J: set of regional markets T: set of periods under consideration. 6~10 year is treated separately. And T 5 in this model. 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. 1 means to increase capacity of plant i using option k at time t; 0 otherwise. X ijt : decision 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 5.1 xijt M i yikt ek for each plant i at each period t
x j
t
x
ijt
dj
for each regional market j
5.2
i
xijt 0 , binary yikt
for all plant, market, period, and capacity incremental options
13
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. 1 means to increase capacity of plant i using option k at time t; 0 otherwise
X ijt
decision variable, shipment amount from plant i to market j at time t
obj
objective function
5.1
capacity constraint
5.2
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 problem5.6.xls) In the first year, original total capacity was 600,000 units, which was 60,000 units more than the total demand. However, a new plant in Kolkata is built in the optimal solution anyway, since it is cheaper to server the local market from Kolkata than to ship from other regions. In the second year, no new capacity is added, since the plant location is reasonable and the total capacity still exceeds the demand. In the third and fourth years, new capacity is added consecutively, which has lead to high surplus capacity. Note that this additional capacity is needed for the fifth year. While there is no reason to add capacity earlier than necessary, especially under the consideration of the discount factor, the solution is optimal in this particular model. Since the cost of fifth year will be added into the total cost six times, it is strategically correct to spend as little as possible in the fifth year. This explains why extra capacity is built into the network earlier than necessary. For questions (b) and (c), we need to change data in the Excel sheet accordingly.
14
7 (a) Blue Computers has two plants in Kentucky and Pennsylvania, however both have high variable costs to serve the West regional market. On the other hand, West regional market has 2 nd highest demand. Hence it is not hard to see that Blue Computers needs a new plant, which can serve the West regional market at a lower cost. From this point of view, California is a better choice than N.Carolina since California has a lower variable cost serving West regional market. However, N.Carolina has extra tax benefit. Even if a network of Kentucky, Pennsylvania, and California might yield higher before-tax profit than a network of Kentucky, Pennsylvania, and N.Carolina, the after-tax profit might be worse. n = 2 potential sites. m = 4: number of regional markets. Dj = Annual units needed of regional market j Ki = maximum possible capacity of potential sites. fi = Annualized fixed cost of setting up a potential site. cij = Cost of producing and shipping a computer r from site i to regional market j yi = 1 if site i is open, 0 otherwise xij = Number of products from site i to regional market j. 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,..., m
(5.1)
K i yi for i 1,..., n
(5.2)
y3 y4 1 add at most one site (5.3) y3 , y4 are binary
(5.4)
SYMBOL
INPUT
Dj
Annual market size of regional market j
Ki
maximum possible capacity of production facility i
cij
Variable cost of producing, 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. 5.1 5.2 5.4
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 problem5.7.xls)
15
Even though constraint (5.4) is simple in its mathematical notation, we can do better in practice. Since at most one site can be open, we can run the optimization three times for three scenarios respectively: none open, only California, or only N.Carolina. And we compare the three results and choose the best one. It is much faster to solve these three scenarios separately given that EXCEL solver cannot achieve a converging result with constraint (5.4). The result below shows the optimal solution when California is picked up. Shipment
Northeast
Kentucky Pennsylvania N. Carolina California Demand constraint
0 0 0 1050 -2.65E-06
Southeast Midwest 0 600 150 2.43E-09 0 0 450 0 -1.5E-06 5.01E-10
South
West
0 450 0 0 -1.1E-06
Open (1) / Capacity Shut (0) Constraint 0 1 400.00 900 1 0.00 0 1.00 1500.00 0 1 0.00 -2.3E-06
Cost $ 255,000 $ 516,500 $ 200,000 $ 530,000
Total Cost =
1501500
(b) We only need to change the objective function from minimize cost to maximize profit. On the Excel sheet, all we need to do is to set the target cell from I21 to L21, and change the direction of optimization from minimizing to maximizing. The following table shows the result. It is easy to see that lowest cost doesn’t mean maximum after tax profit. Shipment Kentucky Pennsylvania N. Carolina California Demand constraint
Northeast 0 1050 0 0 -2.65E-06
Southeast Midwest 600 0 0 0 -1.5E-06
0 450 0 150 -1.5E-06
South 400 0 0 50 -1.1E-06
West
Open (1) / Capacity Shut (0) Constraint 0 1 0.00 0 1 0.00 0 0.00 0.00 900 1 400.00 -2.3E-06 Total Cost =
Cost
Revenue
Profit
After tax profit
$ 328,000
$ 1,000,000
$
672,000
$
490,560
$ 459,500
$ 1,500,000
$ 1,040,500
$
759,565
$
$
$
0
$ 396,500
-
$ 1,100,000
1184000
$
(0) $ 703,500
Total Profit =
$
(0) 513,555
$ 1,763,680
8 (a) Starting from the basic models in (a), we will build more advanced models in the subsequent parts of this question. Prior to merger, Hot&Cold and CaldoFreddo operate independently, and we need to build separate models for each of them. Optimization model for Hot&Cold: n = 3: Hot&Cold production facilities. m = 4: number of regional markets. Dj = Annual market size of regional market j Ki = maximum possible capacity of production facility i cij = Variable cost of producing, 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. It should be integral and non-negative.
16
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,...,m
(5.1)
ij
K i for i 1,...,n
(5.2)
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, 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.
obj. objective function 5.1 demand constraints 5.2 capacity constraints (Sheet Hot&Cold in workbook problem5.8.xls) The above model gives optimal result as in following table:
Shipment Hot&Cold
France Germany Finland Demand
North
East
0.0 10.0 20.0 0
South 0.0 0.0 20.0 0
West 15.0 5.0 0.0 0
Capacity 35.0 0.0 0.0 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.K. Italy Demand
Capacity
Annual Cost
Quantity Shipped (million units)
15 0 0
15 5 0
0 30 0
20 0 0
0 $
6,175
25 $
4,225
Total Cost $
10,400
17
(b) If none of the plants is shut down, the previous model is still applicable. However, we need to update the number of facilities to 5, with 3 from Hot&cold and 2 from CaldoFreddo. And we need to update the market demand Dj, which should be the sum of market shares. Decision maker has more facilities and greater market share in each region, and hence has more choices for production and distribution plans. The optimal result is summarized in the following table. Shipment
Merged Company
France Germany Finland U.K. 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
1.3E-10
-9.8E-09
0.0E+00
-1.9E-09
1
Annual Cost 45 $
1,500
1
20 $
3,850
1
7.9492E-09 $
4,700
1
0 $
5,500
1
0 $
6,450
$
22,000
(c) This model is more advanced since it allows facilities to be shutdown. Accordingly we need more variables to reflect this new complexity. Optimization model for Sleekfon: n = 5: Hot&Cold and caldoFreddo production facilities. m = 4: number of regional markets. Dj = Annual market size of regional market j, sum of the : Hot&Cold and caldoFreddo market share. Ki =capacity of production facility i cij = Variable cost of producing, 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. It should be integral and non-negative. zi = Binary variable indicating whether to shutdown facility i. zi =1 means to shutdown it, 0 otherwise. 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,..., m K i (1 zi )
zi are binary for i 1,..., n
(5.1) for i 1,..., n
(5.2) (5.3)
18
SYMBOL
INPUT
Dj
Annual market size of regional market j
Ki
maximum possible capacity of production facility i
cij
Variable cost of producing, 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. open or shutdown facility i
obj. objective function 5.1 demand constraints 5.2 capacity constraints (Sheet Merged in workbook problem5.8.xls)
It turned out that all sites are open so as to achieve best objective value. Following table shows the optimal configuration. Shipment
Merged Company
Demand
France Germany Finland U.K. 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
0.0
0.0
0.0
0.0
Annual Cost
1
45 $
1,500
1
10 $
4,800
1
0 $
4,800
1
0 $
5,500
1
10 $
5,400
$
22,000
Total Cost
19
