International Journal of Mechanical Engineering and Applications
Volume 5, Issue 1, February 2017, Pages: 28-40

Supply Chain Network Design Optimization Model for
Multi-period Multi-product Under Uncertainty

M. S. Al-Ashhab1, 2

1Design & Production Engineering Dept., Faculty of Engineering, Ain-Shams University, Cairo, Egypt

2Dept. of Mechanical Engineering, College of Engineering and Islamic Architecture, Umm Al-Qura University, Makkah, Kingdom Saudi Arabia

M. S. Al-Ashhab. Supply Chain Network Design Optimization Model for Multi-period Multi-product Under Uncertainty. International Journal of Mechanical Engineering and Applications. Vol. 5, No. 1, 2017, pp. 28-40. doi: 10.11648/j.ijmea.20170501.14

Received: September 4, 2016; Accepted: September 13, 2016; Published: February 17, 2017

Abstract: This research is a development of a stochastic mixed integer linear programming (SMILP) model considering stochastic customer demand, to tackle the multi-product SCND problems. It also considers multi-period, multi-echelons, products inventories, considering locations capacities and associated cost elements. The model represents both location and allocation decisions of the supply chain which maximize the total expected profit. The effect of demand mean on the total expected profit and the effect of the number of scenarios on the CPU time are studied. The results have shown the effect of customers’ demands for each product in each period on the quantities of material delivered from each supplier to each factory, the quantities of products delivered from each factory and factory store to each distributor, the inventory of each product in each factory and distributor, the quantities of each type of product delivered from each distributor to each customer in each period. The model has been verified through a detailed example.

Keywords: Supply Chain Network Design (SCND), Stochastic Mixed Integer Linear Programming (SMILP), Location, Allocation, Modeling, Multi-products, Multi-echelon and Multi-periods

Contents

1. Introduction

Supply chain (SC) generally include two main inter-related processes of (1) production planning and inventory control that deals with production, storage, and the relation between them, and (2) logistics and distribution that determine how to transport products to customers (Ballou, R. H., [3]).

Haq, A. N. et al., proposed an integrated production-inventory-distribution model incorporating many realistic conditions to determine optimal production and distribution as well as inventory level, where a mixed-integer-linear programming (MILP) was developed to minimize the total cost of the system [5].

Maqsood, I., et al. proposed an interval-parameter fuzzy two-stage stochastic programming (IFTSP) method for the planning of water-resources-management systems under uncertainty [6].

Santoso, T., et al. proposed a stochastic programming model and solution algorithm for solving supply chain network design problems of a realistic scale. their solution methodology integrates a recently proposed sampling strategy, the sample average approximation (SAA) scheme, with an accelerated benders decomposition algorithm to quickly compute high-quality solutions to large-scale stochastic supply chain design problems with a huge (potentially infinite) number of scenarios [8].

El-Sayed, M., et al. developed a multi-period multi-echelon forward–reverse logistics network design under risk model maximizing the total expected profit. The proposed network structure considers first customer zones in which the demands are stochastic and second customer zones in which the demand is assumed to be deterministic [4].

Wang, F., et al. studied a supply chain network design problem with environmental concerns. They proposed a multi-objective optimization model that captures the trade-off between the total cost and the environment influence [10].

Pishvaee, M. S., & Razmi, J. proposed an interactive fuzzy solution approach to solve a multi-objective fuzzy mathematical programming model of an environmental SCND with objectives of minimizing total cost and minimization of the total environmental impact [7].

Badri, H. et al. developed a new multi-commodity SCND model with different time resolutions for strategic and tactical decisions. In addition, a mathematical technique based on the Lagrangian relaxation method was developed to solve the problem [2].

Wu, J., & Li, J. studied dynamic factory location and supply chain planning through minimizing the costs of factory location, path selection and transportation of coal under demand uncertainty [11].

Xia R. & Matsukawa H. investigated a supplier-retailer supply chain that experiences disruptions in supplier during the planning horizon. While determining what suppliers, parts, processes, and transportation modes to select at each stage in the supply chain, options disruption must be considered [13].

Adabi, F., & Omrani, H. formulated a mixed integer programming model considering two objective functions where the first one maximizes the efficiency of the supply chain and the second one minimizes the cost of facility layout as well as the production of different products [1].

Serdar E. T. & Al-Ashhab M. S. mathematically modeled an SCN in a mixed integer linear programming (MILP) form considering deterministic demand maximizing the total profit [9].

This research is a development of a stochastic mixed integer linear programming (SMILP) model considering stochastic customer demand, to tackle the multi-product SCND problems. It also considers multi-period, multi-echelons, products inventories, considering locations capacities and associated cost elements. The model represents both location and allocation decisions of the supply chain which maximize the total expected profit. The nature of the logistic decisions encompasses procurement of raw materials from suppliers, production of finished product at factories, distribution of finished product to customers via distributors, and the storage of end product at factories and distributors. The proposed scheme of supply chain consists of three echelons (three suppliers, three factories, and three distributors) to serve four customers as shown in Figure 1.

Figure 1. Supply chain network scheme.

2. Model Assumptions and Limitations

The problem is formulated using multi-stage stochastic mixed integer linear programming (SMILP) and it is solved using Xpress-SP software which uses Mosel language [12].

The following assumptions are considered:

a.  Customers’ demands are stochastic and known for all product in all periods.

b.  The model is multi-product, where actions and flow of materials take place for multi-product.

c.   Product weight affected material, transportation, holding

d.  Costs parameters (fixed costs, material costs, manufacturing costs, non-utilized capacity costs, shortage costs, transportation costs, and inventory holding costs) are known for each location, each product at each period.

e.   The shortage cost depends on the shortage quantity for each product and time.

f.   The manufacturing cost depends on the manufacturing hours for each product and manufacturing cost per hours

The stochastic demand of the customers is normally distributed with mean µ and standard deviation σ, it is discretized into N points.

3. Model Formulation

The model involves the following sets, parameters, and variables:

Sets:

S, F, D, and D: potential number of suppliers, factories, distributors, and first customers,

T: number of periods, indexed by t.

P: number of products, indexed by p.

Parameters:

Fi: fixed cost of opening location i,

DEMANDcpt: demand of customer c from product p in period t,

µct: demand mean of customer c in period t,

σct: demand standard deviation of customer c in period t,

Ppct: unit price of product p at customer c in period t,

Wp: weight of product p,

MHp: manufacturing hours for product p,

Dsf: distance between supplier s and factory f,

Dfd: distance between factory f and distributor d,

Ddc: distance between distributor d and customer c,

CAPst: capacity of supplier s in period t (kg),

CAPMft: capacity of factory f Raw Material Store in period t,

CAPHft: capacity in manufacturing hours of factory f in period t,

CAPFSft: storing capacity of factory f in period t,

CAPdt: capacity of distributor d in period t (kg),

MatCost: material cost per unit supplied by supplier s in period t,

MCft: manufacturing cost per hour for factory f in period t,

MHp: manufacturing hours for product p,

NUCCf: non-utilized manufacturing capacity cost per hour of factory f,

SCPUp: shortage cost per unit per period of product p,

HFp: holding cost per unit per period at factory f store (kg) of product p,

HDp: holding cost per unit per period at distributor d store (kg) of product p,

Bs: batch size for supplier s,

Bfp& Bdp: batch size for factory f for product p and distributor d for product p,

TCperkm: transportation cost per unit per kilometer,

M: big number,

S: small number.

Decision Variables:

Li: binary variable equal to 1 if a location i is opened and equal to 0 otherwise,

Liij: binary variable equal to 1 if a transportation link is activated between location i location j,

Qijpt: flow of batches from location i to location j of product p in period t,

Ifpt: flow of batches from factory f to its store of product p in period t,

Ifdpt: flow of batches from the store of factory f to distributor d of product p in period,

Rfpt: residual inventory of the period t at the store of factory f for product p,

Rdpt: residual inventory of the period t at distributor d for product p.

3.1. Objective Function

The objective of the model is to maximize the total expected profit of the supply chain network.

Total expected profit = Total expected income – Total expected cost

3.1.1. Total Expected Income

(1)

3.1.2. Total Expected Cost

Total expected cost = fixed costs + material costs + manufacturing costs + non-utilized capacity costs + shortage costs + transportation costs + inventory holding costs.

i.        Fixed Costs

(2)

ii.       Material Cost

(3)

iii.     Manufacturing Costs

(4)

iv.     Non-Utilized Capacity Cost (for factories)

(5)

v.      Shortage Cost (for distributor)

(6)

vi.     Transportation Costs

(7)

vii.   Inventory Holding Costs

(8)

3.2. Constraints

3.2.1. Balance Constraints

(9)

(10)

(11)

(12)

3.2.2. Capacity Constraints

(13)

(14)

(15)

(16)

(17)

4. Effect of Demand Mean and Number of Scenarios

The relationship between demand mean and total expected profit has been studied at different values of scenarios of 1, 8, 27 and 64. Figures 2-6 show that the general shape of the relation between demand means and total expected profit is almost the same for a different number of scenarios. In general, the increase in demand mean increases the total expected profit as shown in Figure 6. The total expected profit is linearly proportional to the total demand. At transient ranges, it decreases slightly due to the shortage cost as it is not profitable to open an extra location. At certain demand mean, it is profitable to open another location to fulfill the extra demand. The same behavior continues with the increase in demand mean until the total demand exceeds the maximum permissible capacity of the network and it is not possible to open extra locations.

Figure 2. The relationship between demand mean and the total expected profit for 1 scenario.

Figure 3. The Relationship between demand mean and the total expected profit for 8 scenarios.

Figure 4. The Relationship between demand mean and the total expected profit for 27 scenarios.

Figure 5. The Relationship between demand mean and the total expected profit for 64 scenarios.

Figure 6. The Relationship between demand mean and the total expected profit for all scenarios.

The effect of changing number of scenarios on the total expected profit has been studied for different values of demand means. The changing percentage is calculated by dividing the resulted total expected profit of the given number of scenarios by the resulted total expected profit of the 1-scenario case. Figure 7 depicts the effect of the number of scenarios on the total expected profit. As it is noticed in Figure 8 the most changes happened for demand means of 200, 400 and 500 respectively which are located in the transient ranges since the increase of the total expected profit in the high demand scenarios is not equal to the decrease of it in the low demand scenarios.

Figure 7. Effect of no. of scenarios on the total expected profit.

Figure 8. Maximum change percentage in total expected profit vs. demand mean.

5. Computational Results

In this section, we describe numerical experiment using the proposed model for solving a supply chain network design problem.

5.1. Model Inputs

The model has been verified through the following example where the input parameters are considered as shown in Table 1.

Table 1. Verification model parameters.

 Parameter Value Parameter Value Number of suppliers 3 Material Cost per unit weight 10 Number of factories 3 Manufacturing Cost per hour 10 Number of Distributors 3 Manufacturing hours for product (1) 1 Number of Customers 4 Manufacturing hours for product (2) 2 Number of products 3 Manufacturing hours for product (3) 3 Fixed costs for supplier & distributor 20000 Transportation cost per kilometer per unit 0.001 Fixed costs for factory 50000 Factory holding cost 2 Weight of Product (1) Kg 1 Distributor holding cost 2 Weight of Product (2) Kg 2 Capacity of each supplier in each period 6000 Weight of Product (3) Kg 3 Supplier batch size 10 Price of Product (1) 100 Factory Batch size for product p 5 Price of Product (2) 150 Distributor Batch size for product p 1 Price of Product (3) 200 Capacity of each Factory Raw Material Store in each period 5000 Customers’ Demands standard deviation 10 Factory capacity in hours 5000 Customers’ Demands mean of all products for all customers in all periods 100-600 Capacity of each Factory Store in each period 2000 Non-utilized cost per hour per period 10 Capacity of each Distributor Store in each period 6000 Shortage cost of product (1) per period 5 Shortage cost of product (3) per period 15 Shortage cost of product (2) per period 10

In this case, demand means and standard deviations for all customers in all periods for all products are assumed to be the same to simplify discussion, demand mean is assumed to be 200 units and demand standard deviation is assumed to be 10 units

5.2. Model Outputs

In this section, the model outputs are presented. One of the outputs is the probabilities of scenarios generated by the model and they are as shown in Table 2.

Table 2. Scenarios probabilities.

 Scenario Probability Scenario Probability Scenario Probability Scenario Probability Scenario Probability 1 4.0094E-12 26 2.89954E-09 51 1.94308E-08 76 2.89954E-09 101 4.0102E-12 2 2.89954E-09 27 2.0969E-06 52 1.40521E-05 77 2.0969E-06 102 2.90012E-09 3 1.94308E-08 28 1.40521E-05 53 9.41675E-05 78 1.40521E-05 103 1.94347E-08 4 2.89954E-09 29 2.0969E-06 54 1.40521E-05 79 2.0969E-06 104 2.90012E-09 5 4.0102E-12 30 2.90012E-09 55 1.94347E-08 80 2.90012E-09 105 4.01101E-12 6 2.89954E-09 31 2.0969E-06 56 1.40521E-05 81 2.0969E-06 106 2.90012E-09 7 2.0969E-06 32 0.00151645 57 0.0101622 82 0.00151645 107 2.09733E-06 8 1.40521E-05 33 0.0101622 58 0.0681006 83 0.0101622 108 1.40549E-05 9 2.0969E-06 34 0.00151645 59 0.0101622 84 0.00151645 109 2.09733E-06 10 2.90012E-09 35 2.09733E-06 60 1.40549E-05 85 2.09733E-06 110 2.90071E-09 11 1.94308E-08 36 1.40521E-05 61 9.41675E-05 86 1.40521E-05 111 1.94347E-08 12 1.40521E-05 37 0.0101622 62 0.0681006 87 0.0101622 112 1.40549E-05 13 9.41675E-05 38 0.0681006 63 0.456365 88 0.0681006 113 9.41865E-05 14 1.40521E-05 39 0.0101622 64 0.0681006 89 0.0101622 114 1.40549E-05 15 1.94347E-08 40 1.40549E-05 65 9.41865E-05 90 1.40549E-05 115 1.94386E-08 16 2.89954E-09 41 2.0969E-06 66 1.40521E-05 91 2.0969E-06 116 2.90012E-09 17 2.0969E-06 42 0.00151645 67 0.0101622 92 0.00151645 117 2.09733E-06 18 1.40521E-05 43 0.0101622 68 0.0681006 93 0.0101622 118 1.40549E-05 19 2.0969E-06 44 0.00151645 69 0.0101622 94 0.00151645 119 2.09733E-06 20 2.90012E-09 45 2.09733E-06 70 1.40549E-05 95 2.09733E-06 120 2.90071E-09 21 4.0102E-12 46 2.90012E-09 71 1.94347E-08 96 2.90012E-09 121 4.01101E-12 22 2.90012E-09 47 2.09733E-06 72 1.40549E-05 97 2.09733E-06 122 2.90071E-09 23 1.94347E-08 48 1.40549E-05 73 9.41865E-05 98 1.40549E-05 123 1.94386E-08 24 2.90012E-09 49 2.09733E-06 74 1.40549E-05 99 2.09733E-06 124 2.90071E-09 25 4.01101E-12 50 2.90071E-09 75 1.94386E-08 100 2.90071E-09 125 4.01182E-12

The model also generates the demand for each scenario according to the given distribution which is assumed to be normal in this research. Figure 9 depicts the generated scenario tree.

Figure 9. Scenario tree and demands.

The resulted optimal supply chain network obtained from the model is shown in Figure 10 where it is decided to open the second supplier, factory, and distributor to serve the four customers.

Figure 10. The Resulted Optimal Network Design.

The resulted total expected profit is 691870. Table 3 presents the number of material batches transferred from the supplier to the factory in all scenarios. Table 4 presents the number of batches transferred from the factory to the distributor from product 1 in all scenarios while the numbers of batches transferred from product 2 in all scenarios are presented Table 5, and Table 6 presents the number of batches transferred from product 3 in all scenarios.

Table 3. Number of material batches transferred from supplier to factory in all scenarios.

 Scen. Qsft Scen. Qsft Scen. Qsft Scen. Qsft Scen. Qsft 222 223 224 222 223 224 222 223 224 222 223 224 222 223 224 1 380 347 3 26 440 383 204 51 497 384 328 76 500 438 98 101 500 459 45 2 380 347 199 27 440 383 386 52 497 384 385 77 500 438 385 102 500 459 149 3 380 347 482 28 440 383 443 53 497 384 442 78 500 438 442 103 500 459 480 4 380 347 194 29 440 383 500 54 497 384 499 79 500 438 500 104 500 459 336 5 380 347 0 30 440 383 322 55 497 384 500 80 500 438 191 105 500 459 0 6 380 443 221 31 440 440 329 56 497 443 326 81 500 496 327 106 500 500 138 7 380 443 386 32 440 440 386 57 497 443 383 82 500 496 384 107 500 500 440 8 380 443 443 33 440 440 443 58 497 443 440 83 500 496 443 108 500 500 497 9 380 443 500 34 440 440 500 59 497 443 500 84 500 496 500 109 500 500 500 10 380 443 370 35 440 440 500 60 497 443 500 85 500 496 499 110 500 500 164 11 380 500 327 36 440 500 326 61 497 500 326 86 500 499 381 111 500 500 440 12 380 500 386 37 440 500 383 62 497 500 383 87 500 499 441 112 500 500 497 13 380 500 443 38 440 500 440 63 497 500 443 88 500 499 498 113 500 500 500 14 380 500 500 39 440 500 500 64 497 500 500 89 500 499 500 114 500 500 500 15 380 500 500 40 440 500 500 65 497 500 500 90 500 499 500 115 500 500 500 16 380 500 111 41 440 500 383 66 497 500 383 91 500 500 440 116 500 500 198 17 380 500 443 42 440 500 440 67 497 500 443 92 500 500 497 117 500 500 500 18 380 500 500 43 440 500 500 68 497 500 500 93 500 500 500 118 500 500 500 19 380 500 500 44 440 500 500 69 497 500 500 94 500 500 500 119 500 500 500 20 380 500 372 45 440 500 500 70 497 500 500 95 500 500 500 120 500 500 320 21 380 500 0 46 440 500 142 71 497 500 443 96 500 500 119 121 500 500 31 22 380 500 253 47 440 500 500 72 497 500 500 97 500 500 500 122 500 500 407 23 380 500 500 48 440 500 500 73 497 500 500 98 500 500 500 123 500 500 500 24 380 500 254 49 440 500 500 74 497 500 500 99 500 500 500 124 500 500 406 25 380 500 0 50 440 500 336 75 497 500 500 100 500 500 323 125 500 500 0

Table 4. Number of batches of product 1 transferred from factory to distributor in all scenarios.

 Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. ..12 ..13 ..14 ..12 ..13 ..14 ..12 ..13 ..14 ..12 ..13 ..14 ..12 ..13 ..14 1 123 120 0 26 141 122 90 51 178 104 53 76 179 121 29 101 196 122 0 2 123 120 35 27 141 122 138 52 178 104 72 77 179 121 141 102 196 122 36 3 123 120 160 28 141 122 157 53 178 104 91 78 179 121 160 103 196 122 160 4 123 120 44 29 141 122 176 54 178 104 110 79 179 121 178 104 196 122 45 5 123 120 0 30 141 122 76 55 178 104 123 80 179 121 50 105 196 122 0 6 123 140 2 31 141 141 119 56 178 123 121 81 179 141 121 106 196 142 30 7 123 140 112 32 141 141 138 57 178 123 140 82 179 141 140 107 196 142 142 8 123 140 131 33 141 141 157 58 178 123 159 83 179 141 160 108 196 142 161 9 123 140 150 34 141 141 176 59 178 123 179 84 179 141 179 109 196 142 180 10 123 140 22 35 141 141 195 60 178 123 198 85 179 141 197 110 196 142 49 11 123 159 120 36 141 160 120 61 178 143 120 86 179 159 122 111 196 160 124 12 123 159 140 37 141 160 139 62 178 143 139 87 179 159 142 112 196 160 143 13 123 159 159 38 141 160 158 63 178 143 159 88 179 159 161 113 196 160 162 14 123 159 178 39 141 160 178 64 178 143 178 89 179 159 180 114 196 160 181 15 123 159 197 40 141 160 197 65 178 143 197 90 179 159 197 115 196 160 200 16 123 176 31 41 141 178 121 66 178 160 122 91 179 178 123 116 196 180 30 17 123 176 142 42 141 178 140 67 178 160 142 92 179 178 142 117 196 180 142 18 123 176 161 43 141 178 160 68 178 160 161 93 179 178 161 118 196 180 161 19 123 176 181 44 141 178 179 69 178 160 180 94 179 178 180 119 196 180 180 20 123 176 48 45 141 178 198 70 178 160 199 95 179 178 199 120 196 180 50 21 123 148 0 46 141 145 74 71 178 179 123 96 179 198 31 121 196 199 0 22 123 148 35 47 141 145 193 72 178 179 142 97 179 198 141 122 196 199 34 23 123 148 208 48 141 145 212 73 178 179 161 98 179 198 160 123 196 199 160 24 123 148 0 49 141 145 231 74 178 179 180 99 179 198 179 124 196 199 0 25 123 148 0 50 141 145 94 75 178 179 199 100 179 198 47 125 196 199 0

Table 5. Number of batches of product 2 transferred from factory to distributor.

 Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. ..22 ..23 ..24 ..22 ..23 ..24 ..22 ..23 ..24 ..22 ..23 ..24 ..22 ..23 ..24 1 122 110 0 26 141 122 69 51 162 121 120 76 178 122 1 101 198 122 0 2 122 110 0 27 141 122 107 52 162 121 139 77 178 122 112 102 198 122 29 3 122 110 171 28 141 122 126 53 162 121 158 78 178 122 131 103 198 122 160 4 122 110 49 29 141 122 145 54 162 121 177 79 178 122 150 104 198 122 45 5 122 110 0 30 141 122 65 55 162 121 197 80 178 122 28 105 198 122 0 6 122 141 46 31 141 141 106 56 162 140 120 81 178 142 121 106 198 141 0 7 122 141 126 32 141 141 125 57 162 140 139 82 178 142 140 107 198 141 141 8 122 141 145 33 141 141 144 58 162 140 158 83 178 142 159 108 198 141 160 9 122 141 164 34 141 141 163 59 162 140 178 84 178 142 178 109 198 141 179 10 122 141 98 35 141 141 182 60 162 140 197 85 178 142 198 110 198 141 135 11 122 161 120 36 141 162 119 61 162 160 119 86 178 160 122 111 198 159 123 12 122 161 139 37 141 162 138 62 162 160 138 87 178 160 142 112 198 159 142 13 122 161 158 38 141 162 157 63 162 160 158 88 178 160 161 113 198 159 161 14 122 161 177 39 141 162 177 64 162 160 177 89 178 160 179 114 198 159 180 15 122 161 196 40 141 162 196 65 162 160 196 90 178 160 199 115 198 159 199 16 122 175 34 41 141 177 123 66 162 174 124 91 178 180 122 116 198 179 30 17 122 175 144 42 141 177 142 67 162 174 144 92 178 180 141 117 198 179 141 18 122 175 163 43 141 177 162 68 162 174 163 93 178 180 160 118 198 179 160 19 122 175 183 44 141 177 181 69 162 174 182 94 178 180 179 119 198 179 179 20 122 175 126 45 141 177 200 70 162 174 201 95 178 180 198 120 198 179 97 21 122 171 0 46 141 198 30 71 162 196 122 96 178 197 30 121 198 156 31 22 122 171 129 47 141 198 141 72 162 196 141 97 178 197 143 122 198 156 78 23 122 171 186 48 141 198 160 73 162 196 160 98 178 197 162 123 198 156 201 24 122 171 95 49 141 198 179 74 162 196 179 99 178 197 181 124 198 156 127 25 122 171 0 50 141 198 49 75 162 196 198 100 178 197 52 125 198 156 0

Table 6. Number of batches of product 3 transferred from factory to distributor.

 Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. Scen. Qfdpt22.. ..32 ..33 ..34 ..32 ..33 ..34 ..32 ..33 ..34 ..32 ..33 ..34 ..32 ..33 ..34 1 121 118 2 26 141 111 60 51 164 118 121 76 155 150 55 101 136 184 30 2 121 118 121 27 141 111 140 52 164 118 140 77 155 150 135 102 136 184 68 3 121 118 154 28 141 111 159 53 164 118 159 78 155 150 154 103 136 184 160 4 121 118 82 29 141 111 178 54 164 118 178 79 155 150 174 104 136 184 179 5 121 118 0 30 141 111 146 55 164 118 161 80 155 150 92 105 136 184 0 6 121 136 116 31 141 141 109 56 164 143 97 81 155 188 97 106 136 192 82 7 121 136 136 32 141 141 128 57 164 143 116 82 155 188 116 107 136 192 152 8 121 136 155 33 141 141 147 58 164 143 135 83 155 188 136 108 136 192 171 9 121 136 174 34 141 141 166 59 164 143 155 84 155 188 155 109 136 192 154 10 121 136 174 35 141 141 147 60 164 143 136 85 155 188 135 110 136 192 3 11 121 173 98 36 141 172 98 61 164 179 98 86 155 173 132 111 136 174 170 12 121 173 118 37 141 172 117 62 164 179 117 87 155 173 152 112 136 174 189 13 121 173 137 38 141 172 136 63 164 179 137 88 155 173 171 113 136 174 172 14 121 173 156 39 141 172 156 64 164 179 156 89 155 173 154 114 136 174 153 15 121 173 137 40 141 172 137 65 164 179 137 90 155 173 135 115 136 174 134 16 121 158 41 41 141 156 133 66 164 164 132 91 155 154 171 116 136 154 102 17 121 158 152 42 141 156 152 67 164 164 152 92 155 154 190 117 136 154 192 18 121 158 171 43 141 156 172 68 164 164 171 93 155 154 173 118 136 154 173 19 121 158 151 44 141 156 153 69 164 164 152 94 155 154 154 119 136 154 154 20 121 158 148 45 141 156 134 70 164 164 133 95 155 154 135 120 136 154 132 21 121 170 0 46 141 153 50 71 164 143 173 96 155 136 49 121 136 163 0 22 121 170 71 47 141 153 175 72 164 143 192 97 155 136 191 122 136 163 208 23 121 170 140 48 141 153 156 73 164 143 173 98 155 136 172 123 136 163 146 24 121 170 106 49 141 153 137 74 164 143 154 99 155 136 153 124 136 163 186 25 121 170 0 50 141 153 160 75 164 143 135 100 155 136 165 125 136 163 0

There are a huge amount of data representing the number of batches transferred from the distributor to customers so, the data of only first 25 scenarios are represented by the paper and the full data are represented in appendix A or through this link https://drive.google.com/open?id=0B448W9rNzRcPRk9iWHlPN0dQa1U. Table 7 represents the number of batches (units) transferred from the distributor to the first and second customers for the three products for only 25 scenarios and Table 8 represents the number of batches (units) transferred from the distributor to the third and fourth customers for the three products for only 25 scenarios.

Table 7. Number of batches (units) transferred from distributors to the first and second customers for the three products for only 25 scenarios.

 Scen. Qdcpt21.. Qdcpt21.. Qdcpt21.. Qdcpt22.. Qdcpt22.. Qdcpt22.. ..12 ..13 ..14 ..22 ..23 ..24 ..32 ..33 ..34 ..12 ..13 ..14 ..22 ..23 ..24 ..32 ..33 ..34 1 152 152 0 151 152 0 149 154 0 152 151 0 152 152 0 152 152 40 2 152 152 0 151 152 0 149 154 169 152 151 175 152 152 0 152 152 176 3 152 152 200 151 152 200 149 154 200 152 151 200 152 152 200 152 152 200 4 152 152 0 151 152 0 149 154 0 152 151 220 152 152 219 152 152 221 5 152 152 0 151 152 0 149 154 0 152 151 0 152 152 0 152 152 0 6 152 176 0 151 176 152 149 178 149 152 176 1 152 176 152 152 176 152 7 152 176 176 151 176 176 149 178 177 152 176 176 152 176 176 152 176 176 8 152 176 199 151 176 200 149 178 200 152 176 200 152 176 200 152 176 200 9 152 176 223 151 176 224 149 178 225 152 176 224 152 176 224 152 176 224 10 152 176 0 151 176 100 149 178 168 152 176 5 152 176 217 152 176 248 11 152 200 152 151 200 153 149 201 152 152 200 152 152 200 152 152 199 152 12 152 200 176 151 200 176 149 201 176 152 200 176 152 200 176 152 199 177 13 152 200 200 151 200 198 149 201 200 152 200 200 152 200 200 152 199 200 14 152 200 224 151 200 222 149 201 224 152 200 224 152 200 224 152 199 224 15 152 200 248 151 200 248 149 201 248 152 200 248 152 200 248 152 199 248 16 152 224 0 151 224 0 149 227 0 152 224 0 152 206 170 152 224 0 17 152 224 176 151 224 174 149 227 176 152 224 176 152 206 194 152 224 176 18 152 224 200 151 224 197 149 227 200 152 224 200 152 206 218 152 224 200 19 152 224 224 151 224 225 149 227 224 152 224 224 152 206 242 152 224 224 20 152 224 0 151 224 248 149 227 248 152 224 240 152 206 134 152 224 248 21 152 248 0 151 248 0 149 169 0 152 248 0 152 248 0 152 248 0 22 152 248 0 151 248 174 149 169 0 152 248 175 152 248 176 152 248 0 23 152 248 200 151 248 200 149 169 282 152 248 200 152 248 200 152 248 200 24 152 248 0 151 248 0 149 169 306 152 248 0 152 248 0 152 248 224 25 152 248 0 151 248 0 149 169 0 152 248 0 152 248 0 152 248 0

Table 8. Number of batches (units) transferred from distributors to the third and fourth customers for the three products for only 25 scenarios.

 Scen. Qdcpt 23.. Qdcpt 23.. Qdcpt 23.. Qdcpt 24.. Qdcpt 24.. Qdcpt 24.. ..12 ..13 ..14 ..22 ..23 ..24 ..32 ..33 ..34 ..12 ..13 ..14 ..22 ..23 ..24 ..32 ..33 ..34 1 152 152 0 152 152 0 152 152 0 152 152 0 152 97 0 152 152 0 2 152 152 0 152 152 0 152 152 174 152 152 0 152 97 0 152 152 116 3 152 152 200 152 152 200 152 152 200 152 152 200 152 97 255 152 152 200 4 152 152 0 152 152 26 152 152 0 152 152 0 152 97 0 152 152 219 5 152 152 0 152 152 0 152 152 0 152 152 0 152 97 0 152 152 30 6 152 176 152 152 175 1 152 176 152 152 176 0 152 176 0 152 175 152 7 152 176 175 152 175 177 152 176 176 152 176 176 152 176 176 152 175 176 8 152 176 199 152 175 200 152 176 200 152 176 200 152 176 200 152 175 200 9 152 176 223 152 175 224 152 176 224 152 176 223 152 176 223 152 175 222 10 152 176 0 152 175 248 152 176 248 152 176 248 152 176 0 152 175 231 11 152 199 148 152 200 152 152 200 152 152 199 152 152 200 151 152 200 149 12 152 199 176 152 200 176 152 200 176 152 199 176 152 200 175 152 200 176 13 152 199 199 152 200 200 152 200 200 152 199 200 152 200 200 152 200 200 14 152 199 223 152 200 224 152 200 224 152 199 223 152 200 223 152 200 223 15 152 199 247 152 200 248 152 200 248 152 199 246 152 200 244 152 200 56 16 152 224 0 152 224 0 152 171 205 152 215 155 152 224 0 152 218 0 17 152 224 176 152 224 176 152 171 229 152 215 182 152 224 176 152 218 179 18 152 224 200 152 224 200 152 171 253 152 215 205 152 224 200 152 218 202 19 152 224 224 152 224 224 152 171 277 152 215 233 152 224 224 152 218 30 20 152 224 0 152 224 0 152 171 244 152 215 0 152 224 248 152 218 0 21 152 4 0 152 114 0 152 248 0 152 247 0 152 248 0 152 235 0 22 152 4 0 152 114 295 152 248 176 152 247 0 152 248 0 152 235 179 23 152 4 444 152 114 334 152 248 200 152 247 196 152 248 196 152 235 18 24 152 4 0 152 114 358 152 248 0 152 247 0 152 248 117 152 235 0 25 152 4 0 152 114 0 152 248 0 152 247 0 152 248 0 152 235 0

The number of batches transferred from factory to its store (Ifpt), the number of batches transferred from this store to the distributor (Ifdpt), and the residual batches in both factory (Rfpt) and distributor (Rdpt) stores for the three products for the first 25 scenarios are shown in Table 9.

Table 9. Number of batches transferred from factory to its store (Ifpt), Transferred from this store to the distributor (Ifdpt) and Residual batches in both factory (Rfpt) and distributor (Rdpt) stores for the three products for only 25 scenarios.

 Scen. Ifpt 2 Ifpt 2 Ifpt 2 Ifdpt 22.. Ifdpt 22.. Ifdpt 22.. Rfpt 2.. Rfpt 2.. Rfpt 2.. Rdpt 2.. Rdpt 2.. Rdpt 2... 12 13 22 23 32 33 13 14 23 24 33 34 12 13 22 23 32 33 12 13 22 23 32 33 1 0 0 0 0 10 0 0 0 0 0 4 6 0 0 0 0 10 6 7 0 3 0 0 0 2 0 0 0 0 10 0 0 0 0 0 4 6 0 0 0 0 10 6 7 0 3 0 0 0 3 0 0 0 0 10 0 0 0 0 0 4 6 0 0 0 0 10 6 7 0 3 0 0 0 4 0 0 0 0 10 0 0 0 0 0 4 6 0 0 0 0 10 6 7 0 3 0 0 0 5 0 0 0 0 10 0 0 0 0 0 4 6 0 0 0 0 10 6 7 0 3 0 0 0 6 0 28 0 14 10 0 0 28 0 14 5 5 0 28 0 14 10 5 7 3 3 5 0 0 7 0 28 0 14 10 0 0 28 0 14 5 5 0 28 0 14 10 5 7 3 3 5 0 0 8 0 28 0 14 10 0 0 28 0 14 5 5 0 28 0 14 10 5 7 3 3 5 0 0 9 0 28 0 14 10 0 0 28 0 14 5 5 0 28 0 14 10 5 7 3 3 5 0 0 10 0 28 0 14 10 0 0 28 0 14 5 5 0 28 0 14 10 5 7 3 3 5 0 0 11 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 10 10 7 4 3 8 0 65 12 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 10 10 7 4 3 8 0 65 13 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 10 10 7 4 3 8 0 65 14 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 10 10 7 4 3 8 0 65 15 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 10 10 7 4 3 8 0 65 16 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 17 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 18 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 19 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 20 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 21 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 22 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 23 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 24 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 25 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 7 0 3 0 0 0 26 1 1 0 33 11 0 0 2 0 33 10 1 1 2 0 33 11 1 1 3 1 4 1 0

5.3. Model Results Analysis

Table 10 represents the mean demand and its required material weight in kilograms and the required manufacturing hours. It can be noticed that the totalexpected required material equals 1200 kg which is smaller than the supplying capacity of one supplier so it is reasonable to open only one supplier. The total expected required manufacturing hours equals 1200 hour which is smaller than the manufacturing capacity of one factory so also, it is reasonable to open only one factory and one distributor to transfer them to customers.

Table 10. Mean demand fulfillment requirements.

 Product Mean demand per period Unit weight (Kg.) Exp. Req. weight (Kg.) Man. Hours Exp. Req. Hours 1 200 1 200 1 200 2 200 2 400 2 400 3 200 3 600 3 600 1200 1200

Considering transportation cost, the model optimally decided to open the second raw of facilities as shown in Figure 10 to reduce the total transportation cost to the four customers.

Verification of the network flow balancing is done for only the middle scenario which has the most probability. Demands of all products for all customers are 200, 200 and 200 in the three periods. Table 11 depicts the quantities of batches and weights of raw material transferred from supplier 2 to factory 2, and the quantities of batches and weights of products transferred from factory 2 to all customers through distributor 2. Balancing is noticed in Table 11 for the transferred weights since the transferred amount from any echelon to other are the same of 14,400 kilograms.

Table 11. No. of batches and weights transferred from the supplier to customers.

 From-To S2-F2 F2-D2 D2C1 D2C2 D2C3 D2C4 Period RM P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3 1 497 178 162 164 200 200 200 200 200 200 200 200 200 200 200 200 2 500 143 160 179 200 200 200 200 200 200 200 200 200 200 200 200 3 443 159 158 137 200 200 200 200 200 200 200 200 200 200 200 200 Weight (Kg.) 14400 2400 4800 7200 600 1200 1800 600 1200 1800 600 1200 1800 600 1200 1800 Total Weights 14400 14400 14400

Figure 11. CPU seconds versus No. of Scenarios up to 64 scenarios.

Figure 12. CPU seconds versus No. of Scenarios up to 125 scenarios.

The effect of changing the number of scenarios on the processing time is studied and it was as shown in Figures 11 and 12 from which it is noticed that increasing the number of scenarios dramatically increases CPU time. Figure 12 shows that the processing time of 125-scenarios is too big comparing to other values. The CPU time of this example is 485,925 seconds, 135 hours or 5.6 days.

6. Conclusion

The proposed model is successful in designing supply chain networks while considering multi-product, multi-period stochastic demand with three echelons (suppliers, facilities, and distributors). It can only be used for single item problems. The model is flexible to solve larger problems; however, it requires more powerful hardware since the CPU time increases exponentially with the increase of the number of scenarios which increases by increasing the number of periods.

The application of the proposed model showed that the total expected profit is directly affected by demand mean for a given capacity of the network.

The proposed design model is capable of supply chain networks while considering inventory at the factory and distribution centers.

The proposed design model takes into account different types of costs like the non-utilized capacity cost for factories, transportation cost between all nodes, and the holding cost of inventory in both factories and distributors and shortage cost to enhance customers’ satisfaction.

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 Contents 1. 2. 3. 3.1. 3.2. 4. 5. 5.1. 5.2. 5.3. 6.
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