Topic 7: Perishable Product Supply Chains

Transcription

1 Topic 7: Perishable Product Supply Chains John F. Smith Memorial Professor Director Virtual Center for Supernetworks Isenberg School of Management University of Massachusetts Amherst Advances in Variational Inequalities, Networks, and Game Theory, Spring 2018 c Anna Nagurney 2018

2 The first part of this lecture is based primarily on the paper: Yu, M., Nagurney, A., Competitive food supply chain networks with application to fresh produce. European Journal of Operational Research 224(2), , where a full list of references can be found, along with some additional results.

3 Outline Background and Motivation Literature Review The Fresh Produce Supply Chain Network Oligopoly Model Case Study Relationship of the Model to Others in the Literature Summary

4 Motivation The fundamental difference between food supply chains and other supply chains is the continuous and significant change in the quality of food products throughout the entire supply chain until the points of final consumption.

5 Globalization of Food Supply Chains Consumers expectation of year-around availability of fresh food products has encouraged the globalization of food markets.

6 Globalization of Food Supply Chains Consumers expectation of year-around availability of fresh food products has encouraged the globalization of food markets. The consumption of fresh vegetables has increased at a much faster pace than the demand for traditional crops such as wheat and other grains (USDA (2011)).

7 Globalization of Food Supply Chains Consumers expectation of year-around availability of fresh food products has encouraged the globalization of food markets. The consumption of fresh vegetables has increased at a much faster pace than the demand for traditional crops such as wheat and other grains (USDA (2011)). In the US alone, consumers now spend over 1.6 trillion dollars annually on food (Plunkett Research (2011)). The United States is ranked number one as both importer and exporter in the international trade of horticultural commodities (Cook (2002)).

8 Globalization of Food Supply Chains Consumers expectation of year-around availability of fresh food products has encouraged the globalization of food markets. The consumption of fresh vegetables has increased at a much faster pace than the demand for traditional crops such as wheat and other grains (USDA (2011)). In the US alone, consumers now spend over 1.6 trillion dollars annually on food (Plunkett Research (2011)). The United States is ranked number one as both importer and exporter in the international trade of horticultural commodities (Cook (2002)). The growing global competition, coupled with the associated greater distances between food production and consumption locations, creates new challenges for food supply chain management.

9 Food Waste/Loss It is estimated that approximately one third of the global food production is wasted or lost annually (Gustavsson et al. (2011)). In any country, 20% 60% of the total amount of agricultural fresh products has been wasted or lost (Widodo et al. (2006)). In developed countries, the overall average losses of fruits and vegetables during post-production supply chain activities are approximately 12% of the initial production. The corresponding losses in developing regions are even severer.

10 Product Differentiation Given the thin profit margins in the food industries, product differentiation strategies are increasingly used in food markets (Lowe and Preckel (2004), Lusk and Hudson (2004), and Ahumada and Villalobos (2009)) with product freshness considered one of the differentiating factors (Kärkkäinen (2003) and Lütke Entrup et al. (2005)).

11 Product Differentiation Given the thin profit margins in the food industries, product differentiation strategies are increasingly used in food markets (Lowe and Preckel (2004), Lusk and Hudson (2004), and Ahumada and Villalobos (2009)) with product freshness considered one of the differentiating factors (Kärkkäinen (2003) and Lütke Entrup et al. (2005)). One successful example is fresh-cut produce, including bagged salads, washed baby carrots, and fresh-cut melons (Cook (2002)). Retailers, such as Globus, a German retailer, are also now realizing that food freshness can be a competitive advantage (Lütke Entrup et al. (2005)).

12 Relevant Literature Nahmias (1982, 2011) and Silver, Pyke, and Peterson (1998); Glen (1987) and Lowe and Preckel (2004); Lucas and Chhajed (2004); Lütke Entrup (2005); Akkerman, Farahani, and Grunow (2010); Ahumada and Villalobos (2009) Zhang, Habenicht, and Spieß (2003), Widodo et al. (2006), Monteiro (2007), Blackburn and Scudder (2009), Ahumada and Villalobos (2011), Rong, Akkerman, and Grunow (2011), Kopanos, Puigjaner, and Georgiadis (2012), and Liu and Nagurney (2012) Nagurney and Aronson (1989), Masoumi, Yu, and Nagurney (2012), Nagurney, Masoumi, and Yu (2012), Nagurney and Masoumi (2012), and Nagurney and Nagurney (2011)

13 The Fresh Produce Supply Chain Model This model focuses on fresh produce items, such as vegetables and fruits. I They require simple or limited processing. I The life cycle can be measured in days.

14 The Fresh Produce Supply Chain Topology M 1 1 C 1 1,1 C 1 1,2 D 1 1,1 D 1 1,2 1 I 2 2M 1 n 1 M I 2 M 1 2M I n M I 2 2C 1 nc ,1 C I 2 1,1 2C I nc I......,1 2 2C 1 n C 1,2 C I 2 1,2 2C I nc I,2 2 2D 1 nd ,1 D1,1 I 2 2D I nd I......,1 2 2D 1 n 1 D,2 D1,2 I 2 2D I n 2 D I,2 R 1 R nr 2 The I food firms compete noncooperatively in an oligopolistic manner. The products may be differentiated, due to product freshness and food safety concerns.

15 Food Production M 1 1 C 1 1,1 C 1 1,2 D 1 1,1 D 1 1,2 1 I 2 2M 1 n 1 M I 2 M 1 2M I n M I 2 2C 1 nc ,1 C I 2 1,1 2C I nc I......,1 2 2C 1 n C 1,2 C I 2 1,2 2C I nc I,2 2 2D 1 nd ,1 D1,1 I 2 2D I Soil agitation nd I......,1 Sowing 2 2D 1 n D 1,2 D1,2 I 2 2D I Pest control n 2 D I,2 Nutrient R 1 R nr 2 Water management SCH-MGMT 825 Management Harvesting Science Seminar

16 Food Processing M 1 1 C 1 1,1 C 1 1,2 D 1 1,1 D 1 1,2 1 I 2 2M 1 n 1 M I 2 M 1 2M I n M I 2 2C 1 nc ,1 C I 2 1,1 2C I nc I......,1 2 2C 1 n C 1,2 C I 2 1,2 2C I nc I,2 2 2D 1 nd ,1 D1,1 I 2 2D I nd I......,1 Cleaning Sorting 2 2D 1 n 1 D,2 D1,2 I 2 2D I n 2 D I,2 Labeling Packaging R 1 R nr 2

17 Food Storage M 1 1 C 1 1,1 C 1 1,2 D 1 1,1 D 1 1,2 1 I 2 2M 1 n 1 M I 2 M 1 2M I n M I 2 2C 1 nc ,1 C I 2 1,1 2C I nc I......,1 2 2C 1 n C 1,2 C I 2 1,2 2C I nc I,2 2 2D 1 nd ,1 D1,1 I 2 2D I nd I......,1 2 2D 1 n 1 D,2 D1,2 I 2 2D I n 2 D I,2 R 1 R nr 2

18 Food Shipment/Distribution M 1 1 C 1 1,1 C 1 1,2 D 1 1,1 D 1 1,2 1 I 2 2M 1 n 1 M I 2 M 1 2M I n M I 2 2C 1 nc ,1 C I 2 1,1 2C I nc I......,1 2 2C 1 n C 1,2 C I 2 1,2 2C I nc I,2 2 2D 1 nd ,1 D1,1 I 2 2D I nd I......,1 2 2D 1 n 1 D,2 D1,2 I 2 2D I n 2 D I,2 R 1 R nr 2

19 How to Handle Food Deterioration Most of fresh produce items reach their peak quality at the time of production, and then deteriorate substantially over time. The decay rate varies significantly With different temperatures, and Under other environmental conditions.

20 How to Handle Food Deterioration Most of fresh produce items reach their peak quality at the time of production, and then deteriorate substantially over time. The decay rate varies significantly With different temperatures, and Under other environmental conditions. The food products deteriorate over time even under optimal conditions.

21 How to Handle Food Deterioration Microbiological decay is one of the major causes of the food quality degradation, especially for the fresh produce. Therefore, food deterioration usually follows the first-order reactions with exponential time decay. The decrease in quantity represents the number of units of decayed products (e.g. vegetables and fruits). The degradation in quality emphasizes that all the products deteriorate at the same rate simultaneously (e.g. meat, dairy, and bakery products).

22 How to Handle Food Deterioration The model adopts exponential time decay so as to capture the discarding of spoiled products associated with the post-production supply chain activities. Each unit has a probability of e λt to survive another t units of time, where λ is the decay rate, which is given and fixed. Let N 0 denote the quantity at the beginning of the time interval (link). Hence, the expected quantity surviving at the end of the time interval (specific link), denoted by N(t), can be expressed as: N(t) = N 0 e λt. (1)

23 How to Handle Food Deterioration Let α a denote the throughput factor associate with every link a in the supply chain network, which lies in the range of (0, 1].

24 How to Handle Food Deterioration Let α a denote the throughput factor associate with every link a in the supply chain network, which lies in the range of (0, 1]. For a production link: α a = 1, (2a)

25 How to Handle Food Deterioration Let α a denote the throughput factor associate with every link a in the supply chain network, which lies in the range of (0, 1]. For a production link: α a = 1, For a post-production link: (2a) α a = e λata, (2b) where λ a and t a are the decay rate and the time duration associated with the link a, respectively, which are given and fixed. In rare cases, food deterioration follows the zero order reactions with linear decay. Then, α a = 1 λ a t a for a post-production link.

26 How to Handle Food Deterioration f a Let f a denote the (initial) flow of product on link a; and f a denote the final flow on link a. f a

27 How to Handle Food Deterioration Let f a denote the (initial) flow of product on link a; and f a denote the final flow on link a. f a f a f a = α a f a, a L. (3) The Number of Units of the Spoiled Fresh Produce on Link a f a f a = (1 α a )f a, a L. (4)

28 How to Handle Food Deterioration Total Discarding Cost Functions ẑ a = ẑ a (f a ), a L, (5) which is assumed to be convex and continuously differentiable.

29 How to Handle Food Deterioration Total Discarding Cost Functions ẑ a = ẑ a (f a ), a L, (5) which is assumed to be convex and continuously differentiable. It is imperative to remove the spoiled fresh food products from the supply chain network. For instance, fungi are the common post-production diseases of fresh fruits and vegetables, which can colonize the fruits and vegetables rapidly. The model mainly focuses on the disposal of the decayed food products at the processing, storage, and distribution stages.

30 How to Handle Food Deterioration Multiplier α ap δ ap α b, if {a < a} p Ø, b {a α ap <a} p δ ap, if {a < a} p = Ø, (6) where {a < a} p denotes the set of the links preceding link a in path p, and Ø denotes the null set. Relationship between Link Flows, f a, and Path Flows, x p f a = I n R i=1 k=1 p Pk i x p α ap, a L. (7)

31 How to Handle Food Deterioration Path Multiplier µ p µ p a p α a, p P i k; i = 1,..., I ; k = 1,..., n R. (8)

32 How to Handle Food Deterioration Path Multiplier µ p µ p a p α a, p P i k; i = 1,..., I ; k = 1,..., n R. (8) Relationship between Path Flows, x p, and demands, d ik x p µ p = d ik, i = 1,..., I ; k = 1,..., n R. (9) p P i k d ik can capture production differentiation, due to food safety and health concerns.

33 Demand Price Functions ρ ik = ρ ik (d), i = 1,..., I ; k = 1,..., n R, (10) which captures the demand-side competition. These demand price functions are assumed to be continuous, continuously differentiable, and monotone decreasing. Total Operational Cost Functions ĉ a = ĉ a (f ), a L, (11) where f is the vector of all the link flows. Such cost functions can capture the supply-side competition. The total cost on each link is assumed to be convex and continuously differentiable.

34 The Profit Function of Firm i U i = n R k=1 ρ ik (d)d ik ) (ĉ a (f ) + ẑ a (f a ). (12) a L i In this oligopoly competition problem, the strategic variables are the path flows. X i : the vector of path flows associated with firm i; i = 1,..., I. X : the vector of all the firm strategies, that is, X {{X i } i = 1,..., I }.

35 Supply Chain Network Cournot-Nash Equilibrium A path flow pattern X K = I i=1 K i is said to constitute a supply chain network Cournot-Nash equilibrium if for each firm i; i = 1,..., I : U i (X i, ˆX i ) U i (X i, ˆX i ), X i K i, (13) where ˆX i (X1,..., Xi 1, X i+1,..., X I ) and K i {X i X i R n P i + }. An equilibrium is established if NO firm can unilaterally improve its profit, given other firms decisions.

36 Variational Inequality Formulation Variational Inequality (Path Flows) Determine x K 1 such that: [ I n R Ĉ p (x ) + Ẑ p (x ) ˆρ ik (x )µ p x p x p n R l=1 i=1 k=1 p Pk i ˆρ il (x ) x p µ p xp [x p xp] 0, x K 1, (14) p P i l where K 1 {x x R n P + }.

37 Variational Inequality Formulation Variational Inequality (Link Flows) Determine (f, d ) K 2, such that: [ ] I ĉ b (f ) + ẑ a(fa ) f a f a i=1 a L i b L i [ I n R n R + ρ ik (d ) i=1 k=1 l=1 [f a f a ] ] ρ il (d ) dil [d ik d d ik] 0, ik (f, d) K 2, (15) where K 2 {(f, d) x 0, and (7) and (9) hold}.

38 Existence There exists at least one solution to variational inequality (14) (equivalently, to (15)), since there exists a b > 0, such that variational inequality F (X b ), X X b 0, X K b, (16) admits a solution in K b with x b b. (17)

39 Uniqueness With existence, variational inequality (16) and, hence, variational inequality (16) admits at least one solution. Moreover, if the function F (X ) of variational inequality (15) is strictly monotone on K K 2, that is, (F (X 1 ) F (X 2 )), X 1 X 2 > 0, X 1, X 2 K, X 1 X 2, (18) then the solution to variational inequality (15) is unique, that is, the equilibrium link flow pattern and the equilibrium demand pattern are unique.

40 Algorithm Euler Method Recall that an iteration of the Euler method, which is induced by the general iterative scheme of Dupuis and Nagurney (1993), is: X τ+1 = P K (X τ a τ F (X τ )), In the Euler method, the sequence {a τ } must satisfy: τ=0 a τ =, a τ > 0, a τ 0, as τ. Closed Form Expression for Fresh Produce Path Flows n R xp τ+1 = max{0, xp τ +a τ (ˆρ ik (x τ )µ p + l=1 ˆρ il (x τ ) x p q P i l µ q x τ q Ĉ p (x τ ) x p Ẑ p (x τ ) x p )}, p P i k; i = 1,..., I ; k = 1,..., n R. (19)

41 Case Study Most of cantaloupes consumed in the United States are originally produced in California, Mexico, and in some countries in Central America. Typically, cantaloupes can be stored for days at 2.2 to 5 C (36 to 41 F). It has been noticed that the decay of cantaloupes may result from such post-production disease, depending on the season, the region, and the handling technologies utilized between production and consumption.

42 Supply Chain Topology 1 2 M 1 1 M C1,1 1 9 C1,2 1 D 1 1,1 D 1 1,2 1 M M C 1, C 1, D2,1 1 D ,1 D2, D 1 2,2 D ,2 D ,2 R 1 R 2 2 Firm 1 is located in California. Firm 2 is located in Central America. All the distribution centers and the demand markets are located in the United States.

43 Case 1 Data and Equilibrium Solution Consumers at the demand markets are essentially indifferent between cantaloupes of Firm 1 and Firm 2. Consumers at demand market R 2 are willing to pay relatively more as compared to those at demand market R 1. The Demand Price Functions ρ 11 =.0001d d 21 +4, ρ 12 =.0001d d 22 +6; ρ 21 =.0001d d 11 +4, ρ 22 =.0001d d

44 Link a λ a t a α a ĉ a(f ) ẑ a(f a) fa f f f f f f f f f f f f f f f f f f Table: Data and Equilibrium Solution for Case 1

45 Link a λ a t a α a ĉ a(f ) ẑ a(f a) fa f f f f f f f f f f f f f f f f f f f f f f Table: Data and Equilibrium Solution for Case 1 (continued) There is no shipment from distribution centers D 1 2 and D2 2 to demand market R 1. The volume of product flows on distribution link 22 (or link 26) is higher than that of distribution link 20 (or link 24), which indicates that it is more cost-effective to provide fresh fruits from the nearby distribution centers.

46 The Equilibrium Demands d 11 = 7.86, d 12 = , d 21 = 27.19, and d 22 = The Equilibrium Prices ρ 11 = 4.00, ρ 12 = 5.97, ρ 21 = 4.00, and ρ 22 = The Profits of Two Firms U 1 = and U 2 = Since consumers do not differentiate the cantaloupes produced by these two firms, the prices of these two firms cantaloupes at each demand market are identical.

47 The Equilibrium Demands d 11 = 7.86, d 12 = , d 21 = 27.19, and d 22 = The Equilibrium Prices ρ 11 = 4.00, ρ 12 = 5.97, ρ 21 = 4.00, and ρ 22 = The Profits of Two Firms U 1 = and U 2 = Due to the difference in consumers willingness to pay, the price at demand market R 1 is relatively lower than the price at demand market R 2.

48 The Equilibrium Demands d 11 = 7.86, d 12 = , d 21 = 27.19, and d 22 = The Equilibrium Prices ρ 11 = 4.00, ρ 12 = 5.97, ρ 21 = 4.00, and ρ 22 = The Profits of Two Firms U 1 = and U 2 = As a result of its lower operational costs, Firm 2 dominates both of these two demand markets, leading to a substantially higher profit.

49 Case 2 Data and Equilibrium Solution The CDC reported a multi-state cantaloupe-associated outbreak. Due to food safety and health concerns, the regular consumers of cantaloupes switched to other fresh fruits. The Demand Price Functions ρ 11 =.001d d , ρ 12 =.001d d ; ρ 21 =.001d d , ρ 22 =.001d d

50 Link a λ a t a α a ĉ a(f ) ẑ a(f a) fa f f f f f f f f f f f f f f f f f f Table: Data and Equilibrium Solution for Case 2 The longer time durations associated with shipment links 13 and 14 are caused by more imported food inspections by the U.S. Food and Drug Administration.

51 Link a λ a t a α a ĉ a(f ) ẑ a(f a) fa f f f f f f f f f f f f f f f f f f f f f f Table: Data and Equilibrium Solution for Case 2 (continued) The distribution links: 20, 21, 24, and 25, have zero product flows, since the extremely low demand price cannot cover the costs associated with long-distance distribution.

52 The Equilibrium Demands d 11 = 4.51, d 12 = 3.24, d 21 = 5.96, and d 22 = The Equilibrium Prices ρ 11 = 0.49, ρ 12 = 0.49, ρ 21 = 0.49, and ρ 22 = The Profits of Two Firms U 1 = 1.16 and U 2 = The demand for cantaloupes is battered by the cantaloupe-associated outbreak, with significant decreases in demand prices at demand markets R 1 and R 2. Both Firm 1 and Firm 2, in turn, experience dramatic declines in their profits.

53 Case 3 Data and Equilibrium Solution Firm 1 would like to regain consumers confidence in its own product after the cantaloupe-associated outbreak. Firm 1 had its label of cantaloupes redesigned. The label incorporates the guarantee of food safety. The label also causes additional expenditures associated with its processing activities. The Demand Price Functions ρ 11 =.001d d , ρ 12 =.0003d d 22 +3; ρ 21 =.001d d , ρ 22 =.001d d

54 Link a λ a t a α a ĉ a(f ) ẑ a(f a) fa f f f f f f f f f f f f f f f f f f Table: Data and Equilibrium Solution for Case 3

55 Link a λ a t a α a ĉ a(f ) ẑ a(f a) fa f f f f f f f f f f f f f f f f f f f f f f Table: Data and Equilibrium Solution for Case 3 (continued)

56 The Equilibrium Demands d 11 = 17.52, d 12 = 46.46, d 21 = 5.81, and d 22 = The Equilibrium Prices ρ 11 = 2.48, ρ 12 = 2.99, ρ 21 = 0.48, and ρ 22 = The Profits of Two Firms U 1 = and U 2 = Consumers differentiate cantaloupes due to food safety and health concerns. With the newly designed label, Firm 1 has managed to encourage the consumption of its cantaloupes at both of these two demand markets.

57 The Equilibrium Demands d 11 = 17.52, d 12 = 46.46, d 21 = 5.81, and d 22 = The Equilibrium Prices ρ 11 = 2.48, ρ 12 = 2.99, ρ 21 = 0.48, and ρ 22 = The Profits of Two Firms U 1 = and U 2 = Practicing product differentiation may be an effective strategy for a food firm to maintain its profit at an acceptable level. Considering the cantaloupe-associated outbreak, it is certainly not easy to reclaim the same profit level as in Case 1.

58 The Equilibrium Demands d 11 = 17.52, d 12 = 46.46, d 21 = 5.81, and d 22 = The Equilibrium Prices ρ 11 = 2.48, ρ 12 = 2.99, ρ 21 = 0.48, and ρ 22 = The Profits of Two Firms U 1 = and U 2 = The demand for Firm 1 s product at demand market R 1 in Case 3 is even higher than that of Case 1, which is probably caused by the remarkable decrease in the price as well as the introduced guarantee of food safety.

59 A Multidisciplinary Perspective for Perishable Product Supply Chains In our research on perishable and time-sensitive product supply chains, we utilize results from physics, chemistry, biology, and medicine in order to capture the perishability of various products over time from food to healthcare products such as blood, medical nucleotides, and pharmaceuticals.

60 A variety of perishable product supply chain models, computational procedures, and applications can be found in our book:

61 Supply Chain Networks Optimization Models

62 Blood Supply Chains for the Red Cross A. Nagurney, A. H. Masoumi, and M. Yu, Supply Chain Network Operations Management of a Blood Banking System with Cost and Risk Minimization, Computational Management Science 9(2) (2012), pp

63 Blood Supply Chains for the Red Cross The American Red Cross is the major supplier of blood products to hospitals and medical centers satisfying about 45% of the demand for blood components nationally.

64 Blood Supply Chains for the Red Cross The shelf life of platelets is 5 days and of red blood cells is 42. Over 39,000 donations are needed everyday in the US, and the blood supply was frequently reported to be just 2 days away from running out (American Red Cross (2010)). Some hospitals have delayed surgeries due to blood shortages on 120 days in a year (Whitaker et al. (2007)). The national estimate for the number of units blood products outdated by blood centers and hospitals was 1,276,000 out of 15,688,000 units. As of February 1, 2016, the American Red Cross was facing an emergency need for blood and platelet donors because of severe winter weather in January.

65 Supply Chain Network Topology for a Regionalized Blood Bank 1 ARC Regional Division Blood Collection C 1 C 2 C 3 C nc Blood Collection Sites Shipment of Collected Blood B 1 B nb Blood Centers Testing & Processing P 1 P np Component Labs Storage S 1 S ns Storage Facilities Shipment D 1 D 2 D nd Distribution Centers Distribution R 1 R 2 R 3 R nr Demand Points

66 Blood Supply Chains for the Red Cross We developed a supply chain network optimization model for the management of the procurement, testing and processing, and distribution of a perishable product that of human blood. Novel features of the model include: It captures perishability of this life-saving product through the use of arc multipliers; It contains discarding costs associated with waste/disposal; It handles uncertainty associated with demand points; It assesses costs associated with shortages/surpluses at the demand points, and It quantifies the supply-side risk associated with procurement.

67 Medical Nuclear Supply Chains We developed a medical nuclear supply chain network design model which captures the decay of the radioisotope molybdenum; see Medical Nuclear Supply Chain Design: A Tractable Network Model and Computational Approach, A. Nagurney and L. S. Nagurney, International Journal of Production Economics 140(2) (2012), pp

68 Medical Nuclear Supply Chains Medical nuclear supply chains are essential supply chains in healthcare and provide the conduits for products used in nuclear medical imaging, which is routinely utilized by physicians for diagnostic analysis for both cancer and cardiac problems. Such supply chains have unique features and characteristics due to the products time-sensitivity, along with their hazardous nature. Salient Features: complexity economic aspects underlying physics of radioactive decay importance of considering both waste management and risk management.

69 Medical Nuclear Supply Chains Over 100,000 hospitals in the world use radioisotopes (World Nuclear Association (2011)). Technetium, 99m Tc, which is a decay product of Molybdenum-99, 99 Mo, is the most commonly used medical radioisotope, used in more than 80% of the radioisotope injections, with more than 30 million procedures worldwide each year. The half-life of Molybdenum-99 is 66 hours. Each day, 41,000 nuclear medical procedures are performed in the United States using Technetium-99m.

70 Medical Nuclear Supply Chains A radioactive isotope is bound to a pharmaceutical that is injected into the patient and travels to the site or organ of interest in order to construct an image for medical diagnostic purposes.

71 Medical Nuclear Supply Chains For over two decades, all of the Molybdenum necessary for US-based nuclear medical diagnostic procedures has come from foreign sources.

72 Medical Nuclear Supply Chains 99 Mo Supply Chain Challenges: The majority of the reactors are between 40 and 50 years old. Several of the reactors currently used are due to be retired by the end of this decade (Seeverens (2010) and OECD Nuclear Energy Agency (2010a)).

73 Medical Nuclear Supply Chains 99 Mo Supply Chain Challenges: The majority of the reactors are between 40 and 50 years old. Several of the reactors currently used are due to be retired by the end of this decade (Seeverens (2010) and OECD Nuclear Energy Agency (2010a)). Limitations in processing capabilities make the world critically vulnerable to Molybdenum supply chain disruptions.

74 Medical Nuclear Supply Chains 99 Mo Supply Chain Challenges: The majority of the reactors are between 40 and 50 years old. Several of the reactors currently used are due to be retired by the end of this decade (Seeverens (2010) and OECD Nuclear Energy Agency (2010a)). Limitations in processing capabilities make the world critically vulnerable to Molybdenum supply chain disruptions. The number of generator manufacturers is under a dozen (OECD Nuclear Energy Agency (2010b)).

75 Medical Nuclear Supply Chains 99 Mo Supply Chain Challenges: The majority of the reactors are between 40 and 50 years old. Several of the reactors currently used are due to be retired by the end of this decade (Seeverens (2010) and OECD Nuclear Energy Agency (2010a)). Limitations in processing capabilities make the world critically vulnerable to Molybdenum supply chain disruptions. The number of generator manufacturers is under a dozen (OECD Nuclear Energy Agency (2010b)). Long-distance transportation of the product raises safety and security risks, and also results in greater decay of the product.

76 Medical Nuclear Supply Chains In 2015, NorthStar Medical Radioisotopes LLC has received approval to begin routine production of molybdenum-99 (Mo-99) at the University of Missouri Research Reactor (MURR) facility in Columbia, Missouri. LEU rather than HEU will be used there. This transitioning of NorthStar s Mo-99 line at MURR from a development process to a routine production process is another significant step toward establishing a domestic source of Mo-99.

77 0 Radioisotope Reactors R 1 R i R nr Production Transportation Processing C 1 Facilities 1 Cj 1 C n 1 C 99 Mo Extraction and Purification C 2 1 Cj 2 C n 2 C 3 Transportation Generator G 1 anufacturing Facilities 1 G k 1 G n 1 G Generator Manufacturing G 2 1 G k 2 G n 2 G 3 Transportation Hospitals or H 1 1 H k 1 H Imaging Facilities n 1 H 99m Tc Elucitation atient Demand Points H 1 2 H k 2 H n 2 H Figure: The Medical Nuclear Supply Chain Network Topology

78 Arc Multipliers Because of the exponential decay of molybdenum, we have that the quantity of the radioisotope: N(t) = N 0 e λt so that an arc multiplier on a link a that takes t a hours of time corresponds to: α a = e ln ta.

79 Supply Chain Networks Additional Game Theory Models

80 Relationship of the Model to Others The above model is now related to several models in the literature. If the arc multipliers are all equal to 1, in which case the product is not perishable, then the model is related to the sustainable fashion supply chain network model of Nagurney and Yu in the International Journal of Production Economics 135 (2012), pp In that model, however, the other criterion, in addition to the profit maximization one, was emission minimization, rather than waste cost minimization, as in the model in this paper.

81 Relationship of the Model to Others If the product is homogeneous, and all the arc multipliers are, again, assumed to be equal to 1, and the total costs are assumed to be separable, then the above model collapses to the supply chain network oligopoly model of Nagurney (2010) in which synergies associated with mergers and acquisitions were assessed.

82 The Original Supply Chain Network Oligopoly Model M 1 1 D 1 1,1 D 1 1,2 Firm 1 1 Firm I I M 1 n M I M 1 M I n M I D 1 nd 1,1 D1,1 I D I nd I,1 D 1 n D D1,2 I D I nd I,2 R 1 R nr Figure: Supply Chain Network Structure of the Oligopoly Without Perishability; Nagurney, Computational Management Science 7(2010), pp

83 Mergers Through Coalition Formation Firm 1 Firm Firm 1 Firm n Firm n Firm I 1 n 1 I M 1 1 M 1 n 1 M n 1 M D1,1 1 D 1 M n 1 n n 1 M 1 nd 1,1 D n 1 1,1 M I n M I Dn 1 n n 3 1 D,1 D I nd I,1 D1,2 1 D 1 n 1 D,2 D n 1 1,2 Dn 1 nd 1,2 D I R 1 R nr nd I,2 Figure: Mergers of the First n 1 Firms and the Next n 2 Firms

84 A Pharmaceutical Oligopoly Model References can be found in our paper, A Supply Chain Generalized Network Oligopoly Model for Pharmaceuticals Under Brand Differentiation and Perishability, A.H. Masoumi, M. Yu, and A. Nagurney, Transportation Research E 48 (2012), pp

85 A Generalized Network Oligopoly Model for Pharmaceutical Supply Chains We consider I pharmaceutical firms, with a typical firm denoted by i. The firms compete noncooperatively, in an oligopolistic manner, and the consumers can differentiate among the products of the pharmaceutical firms through their individual product brands. The supply chain network activities include manufacturing, shipment, storage, and, ultimately, the distribution of the brand name drugs to the demand markets.

86 Pharmaceutical Firm 1 Pharmaceutical Firm I 1 I Manufacturing Options M1 1 M 1 n 1 M M 1 I M I nm I Transport Options D 1,1 1 D 1 nd 1,1 D 1,1 I D I nd I,1 Storage D 1,2 1 D 1 nd 1,2 D 1,2 I D I n D I,2 Transport R1 R nr Demand Markets Figure: The Pharmaceutical Supply Chain Network Topology

87 Our recent research has returned to food supply chains in which we are also capturing explicit quality deterioration in fresh produce using chemical formulae that capture time and temperature of various supply chain network activities. Some of our applications are to farmers markets.

88 Summary With a focus on such fresh produce items, We adopted exponential time decay for the calculation of arc multipliers, so as to handle the discarding of spoiled food products associated with the post-production supply chain activities; We considered product differentiation due to product freshness and food safety concerns; and We also allowed for the assessment of alternative technologies involved in each supply chain activity, which could affect the time durations and environmental conditions associated with that activity. We related the model to several others in the literature.

89 Further Results on Quality Deterioration in Fresh Produce In the second part of this lecture, we focus not on perishability in the form of discarding of fresh produce but on the quality deterioration over time based on temperature kinetics. We then apply the formalism to an application to Farmers markets with a case study on apples in western Massachusetts. This part of the lecture is based on the paper, Quality in Competitive Fresh Produce Supply Chains with Application to Farmers Markets, Deniz Besik and Anna Nagurney, Isenberg School of Management, UMass Amherst, 2016.

90 Further Results on Quality Deterioration in Fresh Produce The numbering of the equations, for self-containment in this part of the lecture is: (1), (2), etc.

91 Background Knowledgeable modern consumers are increasingly demanding high quality in their food products, yet, they may be unaware of the great distances the food has traveled through intricate supply chains and the length of time from the initial production or picking of the fruits and vegetables to the ultimate delivery and consumption.

92 Motivation Even though the transformation of food supply chains from local to global is remarkable, there may be some drawbacks. Consumers are facing information asymmetry. The great distances traveled create issues associated with environmental impact, sustainability, and quality since fresh produce is perishable (Nahmias (2011) and Nagurney et al. (2013)).

93 Motivation We focus on quality deterioration through kinetics in food supply chains, direct to consumer chains, and, specifically farmers markets. Consumers tend to connect the terms fresh, good quality, and tasty with locally produced foods. Farmers markets in Norway, have the potential to reduce both physical and social distances between producers and consumers, and, hence, contribute to the sustainability of local food production (Acebo et al.(2007)). There were 8,268 farmers markets in the United States in 2014, with the number having increased by 180% since 2006 (USDA(2014)).

94 Relevant Literature Various authors have emphasized quality; see Sloof, Tijskens, and Wilkinson (1996), Van der Vorst (2000), Lowe and Preckel (2004), Ahumada and Villalobos (2009, 2011), Blackburn and Scudder (2009), Akkerman, Farahani, and Grunow (2010), and Aiello, La Scalia, and Micale (2012). Yu and Nagurney (2013) propose a game theory model for oligopolistic competition in brand differentiated fresh produce supply chains with perishability. Tong, Ren, and Mack (2012) propose an optimal site selection model for farmers markets in Arizona. There is limited research on quality decay through kinetics in direct-to-consumer food supply chains.

95 What is Quality Decay? It is difficult to make a globally accepted definition of quality of fresh produce. Quality of fresh foods can be defined over the combination of their physical attributes such as: color and appearance, flavor, texture, and nutritional value. An understanding of the biochemical/physicochemical reactions can explain the quality deterioration. Taoukis and Labuza (1989) explain the rate of quality deterioration of the quality attributes as a function of microenvironment, gas composition, relative humidity, and temperature.

96 Quality as a function of time and temperature Taoukis and Labuza (1989) and Labuza (1984) show the quality decay of a food attribute Q, over time t, through the differential equation: d[q] dt = k[q] n = Ae ( E/RT ) [Q] n, (1) where k is the reaction rate defined by the Arrhenius formula: Ae ( E/RT ) [Q] n, A is the pre-exponential constant, T is temperature, E is activation energy and R is universal gas constant, n is the reaction order that belongs to the set Z = {0} Z +.

97 Types of Quality Decay Functions The deterioration function changes with respect to the reaction order of the attribute. When the initial quality is Q 0, Tijskens and Polderdijk (1996) categorize the decay functions as: Reaction Order Type Quality at Time t 0 Linear Q 0 kt 1 Exponential Q 0 e kt Table: Reaction Kinetics and Quality at Time t

98 Some Fruits, Vegetables, and Quality Decay Attribute Fresh Reaction Reference Produce Order Color Change Peaches First Toralles et al. (2005) Color Change Raspberries First Ochoa et al. (2001) Color Change Blueberries First Zhang, Guo, and Ma (2012) Nutritional (Vitamin C) Strawberries First Castro et al. (2004) Color Change Watermelons Zero Dermesonlouoglou, Giannakourou, and Taoukis (2007) Moisture Content Tomatoes First Krokida et al. (2003) Color Change Cherries First Ochoa et al. (2001) Texture Softening Apples First Tijskens (1979) Nutritional (Vitamin C) Pears First Mrad et al. (2012) Texture Softening Avocados First Maftoonazad and Ramaswamy (2008) Nutritional (Vitamin C) Pineapples First Karim and Adebowale (2009) Color Change Spinach Zero Aamir et al. (2013) Color Change Asparagus First Aamir et al. (2013) Color Change Peas First Aamir et al. (2013) Texture Softening Beans First Aamir et al. (2013) Texture Softening Brussel Sprouts First Aamir et al. (2013) Texture Softening Carrots First Aamir et al. (2013) Texture Softening Peas First Aamir et al. (2013) Color Change Coriander Leaves First Aamir et al. (2013) Table: Fresh Produce Attributes and Decay Kinetics

99 Integration of Quality Decay Into the Supply Chain Let β a denote the quality decay incurred on link a, which depends on the reaction order n, reaction rate k a and time t a on link a, as: k a t a,, if n = 0, a L β a e kata, if n 0, a L. (2) where k a = Ae ( E A/RT a). (3)

100 Integration of Quality Decay Into the Supply Chain The quality q p, over a path p, joining the origin destination farm, i, with a destination node farmers market, j, can also be shown as: q p q 0i + a p β a, if n = 0, a L, p Pj i, i, j, (4) q 0i β a, if n = 1, a L, p Pj i, i, j, a p where q 0i is the initial quality of food product at farm i, P i j represents the set of all paths that have origin i and destination j.

101 Competition The I farms compete noncooperatively in an oligopolistic manner and the products are differentiated based on quality at the farmers markets.

102 The Fresh Produce Supply Chain Topology 1 Farms i I Harvesting Processing/Packaging Storage Transportation Transportation 1 2 M Farmers Markets

103 The Fresh Produce Supply Chain Topology 1 Farms i I Harvesting Processing/Packaging Storage Transportation Transportation 1 2 M Farmers Markets 1. Fixed time horizon in a given season of the fresh fruit or vegetable, typically a week, is assumed. 2. The demand points are selected farmers markets. 3. Picking is made right before the time horizon, so that there is no storage for the first farmers market of the week. 4. Consumers can buy products that are substitutes within or across the demand points.

104 The Uncapacitated Fresh Produce Problem Nonnegativity constraint of the path flows The flow on the path, joining the farm i to the farmers markets k, is denoted by x p and it should be nonnegative: Link flows x p 0, p P i k; i = 1,..., I ; k = 1,..., n R. (5) The flow on a link a is equal to the sum of the path flows x p, on paths that include the link a, expressed as: Demand f a = p P i k x p δ ap, a L. (6) The demand at the farmers market j for the fresh produce product of farmer i is given by: x p = d ij, p Pj i ; i = 1,..., I ; j = 1,..., M. (7) p P i j

105 The Uncapacitated Fresh Produce Problem Demand Price The demand price function ρ ij for farm i s product at the farmers market j, is: ρ ij = ρ ij (d, q), i = 1,..., I ; j = 1,..., M. (8) Link cost The total operational cost of each link a, denoted by ĉ a, depends on the flows on all the links in the fresh produce supply chain network, that is, ĉ a = ĉ a (f ), a L, (9) Profit/Utility The profit/utility function of farm i, denoted by U i, is given by: U i = M ρ ij (d, q)d ij ĉ a (f ). (10) a L i j=1

106 The Uncapacitated Fresh Produce Problem Definition: Fresh Produce Supply Chain Network Cournot-Nash Equilibrium for Farmers Markets in the Uncapacitated Case A path flow pattern X K = I i=1 K i constitutes a fresh produce supply chain network Cournot-Nash equilibrium if for each farm i; i = 1,..., I : Û i (X i, ˆX i ) Ûi(X i, ˆX i ), X i K i, (11) where ˆX i (X1,..., Xi 1, X i+1,..., X I ) and K i {X i X i R n P i + }. A Cournot-Nash Equilibrium is established if no farm can unilaterally improve its profit by changing its product flows throughout its supply chain network, given the product flow decisions of the other farms.

107 The Uncapacitated Fresh Produce Problem Theorem: Variational Inequality Formulations of the Uncapacitated Model X K is a fresh produce supply chain network Cournot-Nash equilibrium for famers markets according to Definition 1 if and only if it satisfies the variational inequality: I i=1 Xi Û i (X ), X i X i 0, X K, (12) where, denotes the inner product in the corresponding Euclidean space and Xi Û i (X ) denotes the gradient of Û i (X ) with respect to X i.

108 The Uncapacitated Fresh Produce Problem The variational inequality for our uncapacitated model is equivalent to the variational inequality that determines the vector of equilibrium path flows x K 1 such that: IX 2 MX X i=1 j=1 p P j i 6 4 Ĉp(x ) ˆρ ij (x, q) x p MX l=1 ˆρ il (x, q) x p where K 1 {x x R n P + }, and for each path p; p Pi j Ĉp(x) x p X a L i X b L i 3 X xr 7 5 [x p xp ] 0, x K 1, r P i l (13) ; i = 1,..., I ; j = 1,..., M, ĉ b (f ) f a δ ap and ˆρ il (x, q) x p ρ il (d, q) d ij. (14)

109 The Uncapacitated Fresh Produce Problem The variational inequality can also be rewritten in terms of link flows as: determine the vector of equilibrium link flows and the vector of equilibrium demands (f, d ) K 2, such that: + I i=1 [ I i=1 a L i [ M ρ ij (d, q) j=1 ĉ b (f ) f a b L i M l=1 ] [f a f a ] where K 2 {(f, d) x 0, and (6) and (7) hold}. ] ρ il (d, q) dil [d ij dij ] 0, (f, d) d ik (15) Proof: (12) follows from Gabay and Moulin (1980); see, also, Masoumi, Yu, and Nagurney (2012). (13) and (15) then follow using algebraic substitutions.

110 The Uncapacitated Fresh Produce Problem Variational inequalities (13) and (15) can be put into standard form (see Nagurney (1999)): determine X K such that: F (X ), X X 0, X K, (16) where, denotes the inner product in N-dimensional Euclidean space with N = n P in our model. Let X x and [ Ĉp (x) F (X ) ˆρ ij (x, q) x p M l=1 ˆρ il (x, q) x p x r ; r P i l ] p Pj i ; i = 1,..., I ; j = 1,..., M, (17) and K K 1, then (10) can be re-expressed as (13).

111 Theorem: Existence There exists at least one solution to variational inequality (13) (equivalently, to (15)), since there exists a c > 0, such that variational inequality (17) admits a solution in K c with x c c. (18) Theorem: Uniqueness With the existence Theorem, the variational inequalities admit at least one solution. Moreover, if the function F (X ) is strictly monotone on K K 2, that is, (F (X 1 ) F (X 2 )), X 1 X 2 > 0, X 1, X 2 K, X 1 X 2, (19) then the solution to variational inequality is unique, that is, the equilibrium link flow pattern and the equilibrium demand pattern are unique.

112 The Capacitated Fresh Produce Problem Labor shortages, weather conditions, disruptions to storage or transportation can limit the supply chain activities. The objective function, the constraints, with conservation of flow equations stay the same. Link capacity constraint f a u a, a L, (20a) x p δ ap u a, a L, (20b) p P where Ki 3 {X i X i R n P i + and (20b) holds for a L i } and K 3 I i=1 K i 3.

113 The Capacitated Fresh Produce Problem The variational inequality is equivalent to the variational inequality problem: determine (x, λ ) K 4, where K 4 {x R n P +, λ R n L + }, such that: IX 2 MX X i=1 j=1 p P j i 6 4 Ĉ p (x ) ˆρ ij (x, q) x p MX l=1 ˆρ il (x, q) x p 3 X xr + X λ 7 a δ ap 5 [x p xp ] a L r P i l X a L 4u a X xp δ ap 5 [λ a λ a ] 0, (x, λ) K 4, (21) p P where Ĉ p (x) x p and ˆρ il (x,q) x p are as defined in (14).

114 The Euler Method Explicit Formulae for the Uncapacitated Model Closed form expressions for the fresh produce path flows, for each path p Pj i, i, j: x τ+1 p = max{0, x τ p +a τ (ˆρ ij (x τ, q)+ M l=1 ˆρ il (x τ, q) x p p P i j ; i = 1,..., I ; j = 1,..., M. r P i l x τ r Ĉ p (x τ ) x p )}, (23)

115 The Euler Method Explicit Formulae for the Capacitated Model For each path p Pj i, i, j, compute: x τ+1 p = max{0, x τ p + a τ (ˆρ ij (x τ, q) + MX l=1 ˆρ il (x τ, q) x p X r P i l p Pj i ; i = 1,..., I ; j = 1,..., M. x τ r Ĉp(xτ ) x p X λ τ a δ ap)}, a L (24) The Lagrange multipliers for each link a L i ; i = 1,..., I, compute: λ τ+1 a = max{0, λ τ a + a τ ( p P x τ p δ ap u a )}, a L. (25)

116 Case Study of Apple Orchards in Western Massachusetts Orchard/farms: Apex Orchards are located in Shelburne Falls. Park Hill Orchard is located in Easthampton. Sentinel Farm is located in Belchertown. Farmers markets: Northampton Farmers Market is open on Tuesdays. South Hadley Farmers Market is open on Thursdays. Amherst Farmers Market is open on Saturdays. Belchertown Farmers Market is open on Sundays.

117 Scenario 1 - Some Information Picking is made on Monday; therefore, there are no storage links for the Northampton Farmers Market. Golden Delicious apples follow first order quality decay. Harvesting is made between September and October, with average temperatures C.

118 Scenario 1 - Some Information Apex Orchards have the largest land size (170 acres), followed by Park Hill Orchard (127 acres) and Sentinel Farm (8 acres). Apex is located in a higher altitude, so that the average harvesting temperature at the orchard is lower than others. Apex uses controlled atmosphere storage which maintains the optimal temperature, 0 C. We assume that orchard/farm i; i = 1, 2, 3, in the supply chain network has initial quality, respectively, of: q 01 = 1, q 02 = 0.8, and q 03 = 0.7. Uncapacitated model is used.

119 Scenario 1- Quality Decay Operations Link a Hours Temp (C ) β a harvesting processing transportation storage (2 days) storage (4 days) storage (5 days) transportation transportation transportation harvesting processing transportation storage (2 days) storage (4 days ) storage (5 days)

120 Scenario 1 - Quality Decay Operations Link a Hours Temp (C ) β a transportation transportation transportation harvesting processing transportation storage (2 days) storage (4 days ) storage (5 days) transportation transportation transportation

121 Scenario 1- Demand Price Functions Demand Price Functions of Apex Orchards: ρ 11 (d, q) = 0.04d d d q p1 4q p5 3q p9 + 30, ρ 12 (d, q) = 0.02d d d q p2 2q p6 2q p , ρ 13 (d, q) = 0.04d d d q p3 4q p7 3q p , ρ 14 (d, q) = 0.04d d d q p4 q p8 2q p , Demand Price Functions of Park Hill Orchard: ρ 21 (d, q) = 0.04d d d q p5 2q p1 q p9 + 27, ρ 22 (d, q) = 0.04d d d q p6 2q p2 q p , ρ 23 (d, q) = 0.04d d d q p7 2q p3 q p , ρ 24 (d, q) = 0.02d d d q p8 q p4 q p , Demand Price Functions of Sentinel Farm: ρ 31 (d, q) = 0.04d d d q p9 q p1 2q p5 + 25, ρ 32 (d, q) = 0.04d d d q p10 3q p2 q p6 + 28, ρ 33 (d, q) = 0.02d d d q p11 2q p3 q p7 + 25, ρ 34 (d, q) = 0.04d d d q p12 2q p4 2q p

122 Scenario 1 - Total Link Cost Functions and Equilibrium Link Flows Operations Link a ĉ a(f ) fa harvesting f processing f transportation f storage (2 days) f storage (4 days ) f storage (5 days) f transportation f transportation f transportation f harvesting f processing f transportation f storage (2 days) f storage (4 days ) f storage (5 days) f

123 Scenario 1 - Total Link Cost Functions and Equilibrium Link Flows Operations Link a ĉ a (f ) fa transportation f transportation f transportation f harvesting f processing f transportation f storage (2 days) f storage (4 days ) f storage (5 days) f transportation f transportation f transportation f

124 Scenario 1 - Equilibrium Path Flows and Path Quality Decay Farm Path p q p xp Farmers Market Apex p Northampton Apex p South Hadley Apex p Amherst Apex p Belchertown Park Hill p Northampton Park Hill p South Hadley Park Hill p Amherst Park Hill p Belchertown Sentinel p Northampton Sentinel p South Hadley Sentinel p Amherst Sentinel p Belchertown

125 Apex Orchards price of apples per peck: ρ 11 = 27.33, ρ 12 = 24.53, ρ 13 = 30.72, ρ 14 = 25.42, Park Hill Orchard s price of apples per peck: ρ 21 = 21.25, ρ 22 = 26.13, ρ 23 = 26.34, ρ 24 = 27.40, Sentinel Farm s price of apples per peck: ρ 31 = 20.79, ρ 32 = 25.16, ρ 33 = 24.29, ρ 34 = Profits of the orchard/farms, in dollars: U 1 (X ) = , U 2 (X ) = , U 3 (X ) =

126 Scenario 2 - Some Information It is assumed that a new orchard, which was solely selling to retailers and wholesalers previously, is attracted by the demand for apples at the farmers markets.

127 Scenario 2 - Quality Decay It has similar orchard characteristics to Apex Orchards. It is located in Belchertown, which has similar seasonal temperatures to the other farm/orchards. The transportation time from the New Orchard to the farmers markets is similar to Sentinel Farm. Operations Link a Hours Temp (C ) β a harvesting processing transportation storage (2 days) storage (4 days) storage (5 days) transportation transportation transportation

128 Scenario 2 - Demand Price Functions Demand Price Functions of Apex Orchards: ρ 11 (d, q) = 0.053d d d d 41 +8q p1 2q p5 2q p9 4q p13 +30, ρ 12 (d, q) = 0.03d d d d 42 +3q p2 2q p6 2q p10 q p14 +25, ρ 13 (d, q) = 0.053d d d d 43 +8q p3 2q p7 2q p11 4q p15 +30, ρ 14 (d, q) = 0.03d d d d 44 +3q p4 q p8 2q p12 q p15 +25, Demand Price Functions of Park Hill Orchard: ρ 21 (d, q) = 0.05d d d d 41 +3q p5 q p1 q p9 q p13 +27, ρ 22 (d, q) = 0.04d d d d 42 +3q p6 2q p2 q p10 q p14 +28, ρ 23 (d, q) = 0.05d d d d 43 +4q p7 2q p3 q p11 2q p15 +27, ρ 24 (d, q) = 0.04d d d d 44 +2q p8 q p4 q p12 q p16 +28,

129 Scenario 2 - Demand Price Functions Demand Price Functions of Sentinel: ρ 21 (d, q) = 0.05d d d d 41 +3q p5 q p1 q p9 q p13 +27, ρ 22 (d, q) = 0.04d d d d 42 +3q p6 2q p2 q p10 q p14 +28, ρ 23 (d, q) = 0.05d d d d 43 +4q p7 2q p3 q p11 2q p15 +27, ρ 24 (d, q) = 0.04d d d d 44 +2q p8 q p4 q p12 q p16 +28, Demand Price Functions of New Orchard: ρ 41 (d, q) = 0.053d d d d 31 +5q p13 2q p1 q p5 q p9 +30, ρ 42 (d, q) = d d d 32 +2q p14 q p2 q p6 q p10 +25, ρ 43 (d, q) = 0.053d d d d 33 +5q p15 2q p3 q p7 q p11 +30, ρ 44 (d, q) = 0.03d d d d 34 +2q p16 q p4 q p8 q p

130 Scenario 2 - Equilibrium Path Flows and Path Quality Decay Initial quality of the apples at the orchards: q 01 = 1, q 02 = 0.8, q 03 = 0.7, and q 04 = 1. Farm Path p q p xp Farmers Market Apex p Northampton Apex p South Hadley Apex p Amherst Apex p Belchertown Park Hill p Northampton Park Hill p South Hadley Park Hill p Amherst Park Hill p Belchertown Sentinel p Northampton Sentinel p South Hadley Sentinel p Amherst Sentinel p Belchertown New Orchard p Northampton New Orchard p South Hadley New Orchard p Amherst New Orchard p Belchertown

131 Apex Orchards price of apples per peck: ρ 11 = 23.49, ρ 12 = 23.66, ρ 13 = 27.49, ρ 14 = 24.44, Park Hill Orchard s price of apples per peck: ρ 21 = 21.46, ρ 22 = 25.41, ρ 23 = 25.49, ρ 24 = 26.20, Sentinel Farm s price of apples per peck: ρ 31 = 20.38, ρ 32 = 24.38, ρ 33 = 22.91, ρ 34 = 23.08, New Orchard s price of apples per peck: ρ 41 = 23.82, ρ 42 = 23.99, ρ 43 = 27.80, ρ 44 = Profits of the orchard/farms, in dollars: U 1 (X ) = , U 2 (X ) = , U 3 (X ) = , U 4 (X ) =

132 Scenario 3 - Some Information This scenario is constructed to illustrate the apple shortage experienced in western Massachusetts in According to various news articles, the cold snap happened in May damaged the green apple buds and an apple shortage at the local markets, which includes the farmers markets, is expected. Expected shortage is assumed to be more for Apex due to being located in a higher altitude. The capacities are written according to the expected damage level of harvest at the orchard/farms. Initial quality of the apples in the orchards is q 01 = 0.4, q 02 = 0.5, and q 03 = 0.6.