Tutorial 3 Solutions

Question 1

Gelido, a frozen food distribution company, is evaluating three potential distribution centre (DC) locations to serve four market regions. The company must decide which facilities to open and how to allocate market demands to minimise the total cost, including fixed facility costs and transportation costs.

Table 1. Facility fixed costs and capacities

Facility	Fixed Cost ($)	Capacity (100kg/year)
Linares	82,252	20,000
Monclova	134,400	20,000
Monterrey	82,252	20,000

Table 2. Market demands

Market Region	Demand (100kg/year)
Bustamante	5,000
Saltillo	7,000
Santa Catarina	9,000
Montemorelos	6,000

Table 3. Facility variable costs (per hundred kg)

Facility	Variable Cost ($/100kg)
Linares	18.5
Monclova	4.1
Monterrey	18.5

Table 4. Distances between facilities and markets (miles)

Facility	Bustamante	Saltillo	Santa Catarina	Montemorelos
Linares	165.0	132.5	92.7	32.4
Monclova	90.8	118.5	139.0	176.7
Monterrey	84.2	51.6	11.9	49.5

1. Formulate the optimisation problem using the given cost and constraint structure.
2. Using the given data to determine optimal facility location strategy if the transportation cost, \(C_{ij}\), between facility \(i\) and market \(j\) is given by \[ C_{ij} = \tilde{c}_{ij} D_j, \quad \text{where} \quad \tilde{c}_{ij} = (0.98 \times 2 \times l_{ij}) / 10 \] where \(l_{ij}\) is the distance (in miles) between facility \(i\) and market \(j\), and \(D_j\) is the demand of market \(j\).

Sets

• \(F = \{\text{Linares, Monclova, Monterrey}\}\): Set of potential facilities, indexed by \(i\).
• \(M = \{\text{Bustamante, Saltillo, Santa Catarina, Montemorelos}\}\): Set of market regions, indexed by \(j\).

Parameters

• \(f_i\): Fixed cost of opening facility \(i\).
• \(K_i\): Capacity of facility \(i\) (in hundred kg/year).
• \(D_j\): Demand of market \(j\) (in hundred kg/year).
• \(v_i\): Variable cost per hundred kg at facility \(i\) (in dollars).
• \(\tilde{c}_{ij}\): Variable transportation cost per hundred kg from facility \(i\) to market \(j\) (in dollars).
• \(C_{ij}\): Total transportation cost from facility \(i\) to market \(j\) (in dollars).
• \(l_{ij}\): Distance from facility \(i\) to market \(j\) (in miles).

Decision Variables

• \(x_{ij}\): The fraction of demand of market \(j\) served by facility \(i\) (between 0 and 1).
• \(y_i\): Binary variable indicating whether facility \(i\) is opened (1) or not (0).

Objective Function

\[ \min \sum_{i \in F} \sum_{j \in M} (v_i + \tilde{c}_{ij})D_j x_{ij} + \sum_{i \in F} f_i y_i \]

Constraints

\[ \begin{align} \sum_{i \in F} x_{ij} &= 1, \quad &&\forall j \in M \\ \sum_{j \in M} D_j x_{ij} &\leq K_i y_i, \quad &&\forall i \in F \\ 0 \leq x_{ij} &\leq 1, \quad &&\forall i \in F, j \in M \\ y_i &\in \{0,1\}, \quad &&\forall i \in F \end{align} \]

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Question 2

A company produces three products (\(P_1, P_2, P_3\)) in two factories (\(F_1, F_2\)) and distributes them to four warehouses (\(W_1, W_2, W_3, W_4\)). The goal is to determine the optimal allocation of products from factories to warehouses in order to minimise the total transportation cost.

Table 1. Transportation costs from each factory to each warehouse ($/unit)

Factory/Warehouse	\(W_1\)	\(W_2\)	\(W_3\)	\(W_4\)
\(F_1\)	5	7	6	8
\(F_2\)	6	5	7	6

Table 2. Demand for each product at each warehouse (units)

Warehouse	Demand for \(P_1\)	Demand for \(P_2\)	Demand for \(P_3\)
\(W_1\)	200	150	100
\(W_2\)	180	120	130
\(W_3\)	220	160	90
\(W_4\)	150	140	110

The company must determine how many units of each product to ship from each factory to each warehouse in order to minimise total transportation costs.

Formulate the optimisation model

• Define the decision variables.
• Construct the objective function.
• Identify any relevant constraints.

Sets

• \(F = \{F_1, F_2\}\): Set of factories, indexed by \(i\).
• \(W = \{W_1, W_2, W_3, W_4\}\): Set of warehouses, indexed by \(j\).
• \(P = \{P_1, P_2, P_3\}\): Set of products, indexed by \(h\).

Decision Variables

• \(x_{ij}^h\): Quantity of product \(h\) shipped from factory \(i\) to warehouse \(j\) (units).

Parameters

• \(c_{ij}\): Transportation cost per unit from factory \(i\) to warehouse \(j\).
• \(D_j^h\): Demand for product \(h\) at warehouse \(j\).

Objective Function

\[ \min \sum_{i \in F} \sum_{j \in W} \sum_{h \in P} c_{ij} x_{ij}^h \]

Constraints

\[ \begin{align} \sum_{i \in F} x_{ij}^h &= D_j^h, && \forall j \in W, h \in P \\ x_{ij}^h &\ge 0, && \forall i \in F, j \in W, h \in P \end{align} \]

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Question 3

An FMCG¹ company distributes two products, Canned Vegetables (P1) and Beverages (P2), from three suppliers through two potential distribution centres (DCs) to three retail stores. Clearly formulate the TEMC optimisation problem. Define decision variables, the objective function, and all constraints.

Table 1. Supplier capacities (units)

Supplier	P1	P2
S1	250	150
S2	200	300
S3	200	100

Table 2. Customer demands (units)

Customer	P1	P2
C1	100	100
C2	150	150
C3	200	100

Table 3. DC capacities and operating costs

DC	Capacity	Fixed Cost ($)	Handling Cost ($/unit)
DC1	600	800	2.0
DC2	750	950	1.5

Table 4. Transportation costs from suppliers to DCs ($ per unit)

Supplier	DC1	DC2
S1	4	5
S2	3	2
S3	5	3

Table 5. Transportation costs from DCs to customers ($ per unit)

DC	C1	C2	C3
DC1	2	3	4
DC2	3	2	3

Sets:

• \(S = \{S1, S2, S3\}\): Set of suppliers, indexed by \(i\).
• \(D = \{DC1, DC2\}\): Set of distribution centres, indexed by \(j\).
• \(C = \{C1, C2, C3\}\): Set of customers, indexed by \(k\).
• \(P = \{P1, P2\}\): Set of products, indexed by \(h\).

Decision Variables:

• \(x_{ij}^h\): Quantity of product \(h\) shipped from supplier \(i\) to DC \(j\).
• \(y_{jk}^h\): Quantity of product \(h\) shipped from DC \(j\) to customer \(k\).
• \(z_j\): Binary variable indicating whether DC \(j\) is opened (1) or not (0).

Objective Function:

\[ \min \sum_{i,j,h} c_{ij}^h x_{ij}^h + \sum_{j,k,h} c_{jk}^h y_{jk}^h + \sum_{j \in D} \left( f_j z_j + g_j\sum_{h,k} y_{jk}^h \right) \]

Constraints

\[ \begin{align} \sum_{j} x_{ij}^{h} &\le P_{i}^{h}, && \forall i,h \\[6pt] \sum_{k,h} y_{jk}^{h} &\le K_j z_j, && \forall j \\[6pt] \sum_{j} y_{jk}^{h} &= D_k^{h}, && \forall k,h \\[6pt] \sum_{i} x_{ij}^{h} &= \sum_{k} y_{jk}^{h}, && \forall j,h \\[6pt] x_{ij}^{h},\, y_{jk}^{h} &\ge 0, \\[6pt] z_j &\in \{0,1\} \end{align} \]

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Question 4

A pharmaceutical company distributes two medicines, Medicine A (MA) and Medicine B (MB), from two factories through two potential warehouses (W1, W2) to four hospitals. Clearly formulate this scenario as a TEMC optimisation problem. Explicitly define the decision variables, objective function, and all necessary constraints.

Table 1. Supplier capacities (units)

Supplier	MA	MB
F1	300	200
F2	200	400

Table 2. Hospital demands (units)

Hospital	MA	MB
H1	100	100
H2	80	120
H3	50	150
H4	100	80

Table 3. Warehouse capacities and operating costs

Warehouse	Capacity	Fixed Cost ($)	Handling Cost ($/unit)
W1	400	670	1.25
W2	300	520	1.50

Table 4. Transportation costs from suppliers to warehouses ($ per unit)

Supplier	W1	W2
F1	4	6
F2	3	5

Table 5. Transportation costs from warehouses to hospitals ($ per unit)

Warehouse	H1	H2	H3	H4
W1	3	4	2	5
W2	5	2	4	3

Sets:

• \(S = \{F1, F2\}\): Set of suppliers, indexed by \(i\).
• \(W = \{W1, W2\}\): Set of warehouses, indexed by \(j\).
• \(H = \{H1, H2, H3, H4\}\): Set of hospitals, indexed by \(k\).
• \(M = \{MA, MB\}\): Set of medicines, indexed by \(h\).

Decision Variables:

• \(x_{ij}^h\): Quantity of medicine \(h\) shipped from supplier \(i\) to warehouse \(j\).
• \(y_{jk}^h\): Quantity of medicine \(h\) shipped from warehouse \(j\) to hospital \(k\).
• \(z_j\): Binary variable indicating whether warehouse \(j\) is opened (1) or not (0).

Objective Function:

\[ \min \sum_{i,j,h} c_{ij}^h x_{ij}^h + \sum_{j,k,h} c_{jk}^h y_{jk}^h + \sum_{j \in W} \left( f_j z_j + g_j\sum_{h,k} y_{jk}^h \right) \]

Constraints

\[ \begin{align} \sum_{j} x_{ij}^{h} &\le P_{i}^{h}, && \forall i,h \\[6pt] \sum_{k,h} y_{jk}^{h} &\le K_j z_j, && \forall j \\[6pt] \sum_{j} y_{jk}^{h} &= D_k^{h}, && \forall k,h \\[6pt] \sum_{i} x_{ij}^{h} &= \sum_{k} y_{jk}^{h}, && \forall j,h \\[6pt] x_{ij}^{h},\, y_{jk}^{h} &\ge 0, \\[6pt] z_j &\in \{0,1\} \end{align} \]

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Question 5

A company operates a two-echelon supply chain where it produces three products (\(P_1, P_2, P_3\)) in two factories (\(F_1, F_2\)), ships them to two regional distribution centers (\(DC_1, DC_2\)), and then distributes them to four customer zones (\(C_1, C_2, C_3, C_4\)). The goal is to determine the optimal allocation of products across the two echelons to minimise total transportation and facility operation costs.

Table 1. Transportation costs from factories to DCs ($/unit)

Factory/DC	\(DC_1\)	\(DC_2\)
\(F_1\)	4	6
\(F_2\)	5	3

Table 2. Transportation costs from DCs to customer zones ($/unit)

DC/Customer	\(C_1\)	\(C_2\)	\(C_3\)	\(C_4\)
\(DC_1\)	3	2	4	5
\(DC_2\)	4	3	5	2

Table 3. Customer demands for each product (units)

Zone	Demand for \(P_1\)	Demand for \(P_2\)	Demand for \(P_3\)
\(C_1\)	180	140	90
\(C_2\)	200	160	110
\(C_3\)	170	130	120
\(C_4\)	190	150	100

Table 4. DC capacities (units)

DC	Capacity (Total Units)
\(DC_1\)	500
\(DC_2\)	450

The company must determine:

• The optimal number of units of each product to be shipped from each factory to DC.
• The optimal number of units of each product to be shipped from each DC to customer zones.
• Whether to operate or close a DC to minimise total logistics costs.

Sets

• \(F = \{F_1, F_2\}\): Set of factories, indexed by \(i\).
• \(D = \{DC_1, DC_2\}\): Set of distribution centers, indexed by \(j\).
• \(C = \{C_1, C_2, C_3, C_4\}\): Set of customer zones, indexed by \(k\).
• \(P = \{P_1, P_2, P_3\}\): Set of products, indexed by \(h\).

Decision Variables

• \(x_{ij}^h\): Quantity of product \(h\) shipped from factory \(i\) to DC \(j\) (units).
• \(y_{jk}^h\): Quantity of product \(h\) shipped from DC \(j\) to customer zone \(k\) (units).
• \(z_j\): Binary variable indicating whether DC \(j\) is operated (1) or closed (0).

Parameters

• \(c_{ij}^h\): Transportation cost per unit of product \(h\) from factory \(i\) to DC \(j\).
• \(c_{jk}^h\): Transportation cost per unit of product \(h\) from DC \(j\) to customer zone \(k\).
• \(D_k^h\): Demand for product \(h\) at customer zone \(k\).
• \(K_j\): Capacity of DC \(j\) (units).
• \(f_j\): Fixed cost of operating DC \(j\).

Objective Function

\[ \min \sum_{i \in F} \sum_{j \in D} \sum_{h \in P} c_{ij}^h x_{ij}^h + \sum_{j \in D} \sum_{k \in C} \sum_{h \in P} c_{jk}^h y_{jk}^h + \sum_{j \in D} f_j z_j \]

Constraints

\[ \begin{align} \sum_{i \in F} x_{ij}^h &= \sum_{k \in C} y_{jk}^h, && \forall j \in D, h \in P \\ \sum_{j \in D} y_{jk}^h &= D_k^h, && \forall k \in C, h \in P \\ \sum_{h \in P} \sum_{k \in C} y_{jk}^h &\leq K_j z_j, && \forall j \in D \\ x_{ij}^h &\le M z_j, && \forall i \in F, j \in D, h \in P \\ y_{jk}^h &\le M z_j, && \forall j \in D, k \in C, h \in P \\ x_{ij}^h &\geq 0, && \forall i \in F, j \in D, h \in P \\ y_{jk}^h &\geq 0, && \forall j \in D, k \in C, h \in P \\ z_j &\in \{0,1\}, && \forall j \in D \end{align} \] where \(M\) is a sufficiently large constant to enforce the logical constraints on \(x_{ij}^h\) and \(y_{jk}^h\) based on the operation of DC \(j\).

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1. FMCG = Fast-moving consumer goods (sold quickly and at a relatively low cost)